Average Error: 14.3 → 0.3
Time: 6.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{-\pi}{b + a}}{\left(b - a\right) \cdot \frac{-2}{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{-\pi}{b + a}}{\left(b - a\right) \cdot \frac{-2}{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}
double code(double a, double b) {
	return (((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b)));
}
↓
double code(double a, double b) {
	return ((-((double) M_PI) / (b + a)) / ((b - a) * (-2.0 / (1.0 * ((1.0 / a) - (1.0 / b))))));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.3

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  2. Using strategy rm

  3. Applied difference-of-squares9.5

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  4. Applied associate-/r*8.9

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\frac{1}{b + a}}{b - a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  5. Applied frac-2neg8.9

    \[\leadsto \left(\color{blue}{\frac{-\pi}{-2}} \cdot \frac{\frac{1}{b + a}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  6. Applied frac-times8.9

    \[\leadsto \color{blue}{\frac{\left(-\pi\right) \cdot \frac{1}{b + a}}{\left(-2\right) \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  7. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\left(-\pi\right) \cdot \frac{1}{b + a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\left(-2\right) \cdot \left(b - a\right)}}\]
  8. Using strategy rm

  9. Applied *-commutative0.3

    \[\leadsto \frac{\left(\left(-\pi\right) \cdot \frac{1}{b + a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\color{blue}{\left(b - a\right) \cdot \left(-2\right)}}\]

  10. Applied associate-*l*0.3

    \[\leadsto \frac{\color{blue}{\left(-\pi\right) \cdot \left(\frac{1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)}}{\left(b - a\right) \cdot \left(-2\right)}\]

  11. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{-\pi}{b - a} \cdot \frac{\frac{1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{-2}}\]
  12. Using strategy rm

  13. Applied div-inv0.3

    \[\leadsto \frac{-\pi}{b - a} \cdot \frac{\color{blue}{\left(1 \cdot \frac{1}{b + a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{-2}\]
  14. Using strategy rm

  15. Applied *-commutative0.3

    \[\leadsto \frac{-\pi}{b - a} \cdot \frac{\color{blue}{\left(\frac{1}{b + a} \cdot 1\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{-2}\]

  16. Applied associate-*l*0.3

    \[\leadsto \frac{-\pi}{b - a} \cdot \frac{\color{blue}{\frac{1}{b + a} \cdot \left(1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)}}{-2}\]

  17. Applied associate-/l*0.3

    \[\leadsto \frac{-\pi}{b - a} \cdot \color{blue}{\frac{\frac{1}{b + a}}{\frac{-2}{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}}\]

  18. Applied frac-times0.3

    \[\leadsto \color{blue}{\frac{\left(-\pi\right) \cdot \frac{1}{b + a}}{\left(b - a\right) \cdot \frac{-2}{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}}\]

  19. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{-\pi}{b + a}}}{\left(b - a\right) \cdot \frac{-2}{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}\]

  20. Final simplification0.3

    \[\leadsto \frac{\frac{-\pi}{b + a}}{\left(b - a\right) \cdot \frac{-2}{1 \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))