Assuming the robots were 4 times as productive as humans. There would also be 3 billion robotaxis and robotrucks making transportation 10 times cheaper and 2-3 times faster for ground transportation. There would be global 6G space communication via satellites, a 3-fold increase in energy, and superintelligent AI with ample training data from robots and vehicles.

The speed of reaching 10 billion humanoid robots would be to make 1 billion bots per year for 10 years. A humanoid bot is about 3% of the mass of a car. The world makes almost 100 million cars per year. The same mass of humanoid bot production would be 3 billion bots per year. Besides production capacity for the actuators there needs be the production of chips at the billions per year.

Current Baseline
As of 2024, the global GDP is approximately $100 trillion. Without the transformative technologies mentioned, GDP might grow at a historical rate of 3% per year. Over 16 years (2024 to 2040), this would yield:
GDP2040=100×(1+0.03)16 which is about 100×1.605
160.5 trillion dollars

Impact of Humanoid Robots
The scenario includes 10 billion humanoid robots, each 4 times as productive as a human. The current global workforce is about 3.5 billion people. If each robot's productivity is equivalent to 4 humans, 10 billion robots would provide:

10 billion×4=40 billion human-equivalent worker

This is over 10 times the current workforce. However, effective productivity gains may be lower due to coordination overheads and diminishing returns. Let's conservatively assume each robot effectively contributes 2-3 times a human's productivity in practice, still resulting in a massive boost—equivalent to 20-30 billion human workers. If an average human worker contributes roughly $50,000 to GDP annually,

this could add:
10 billion × 2 × 50,000 = 1 quadrillion dollars

to annual GDP, though integration into the economy would moderate this figure.

Impact of Robotaxis and Robotrucks
Three billion robotaxis and robotrucks make transportation 10 times cheaper and 2-3 times faster. Transportation costs currently account for about 10% of global GDP ($10 trillion).
Reducing this by a factor of 10 saves:
10 trillion×(1−110) = 9 trillion dollars

These savings could be reinvested, boosting economic activity. Faster transportation also increases the efficiency of goods and people movement, potentially amplifying commerce velocity by 2-3 times, further enhancing GDP.