Average Error: 14.3 → 0.3
Time: 6.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1 \cdot \left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right)}{a} + \frac{\left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right) \cdot \left(-1\right)}{b}}{2 \cdot \left(b + a\right)}}{b - a}\]
\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{\frac{1 \cdot \left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right)}{a} + \frac{\left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right) \cdot \left(-1\right)}{b}}{2 \cdot \left(b + a\right)}}{b - a}
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 (((((1.0 * (pow(cbrt(1.0), 3.0) * ((double) M_PI))) / a) + (((pow(cbrt(1.0), 3.0) * ((double) M_PI)) * -1.0) / b)) / (2.0 * (b + a))) / (b - a));
}

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 add-cube-cbrt9.5

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(b + a\right) \cdot \left(b - a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  5. Applied times-frac9.0

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a} \cdot \frac{\sqrt[3]{1}}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  6. Applied associate-*r*8.9

    \[\leadsto \color{blue}{\left(\left(\frac{\pi}{2} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \frac{\sqrt[3]{1}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  7. Using strategy rm

  8. Applied associate-*r/8.9

    \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \sqrt[3]{1}}{b - a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  9. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\left(\frac{\pi}{2} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \sqrt[3]{1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}}\]
  10. Using strategy rm

  11. Applied frac-times0.3

    \[\leadsto \frac{\left(\color{blue}{\frac{\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{2 \cdot \left(b + a\right)}} \cdot \sqrt[3]{1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]

  12. Applied associate-*l/0.3

    \[\leadsto \frac{\color{blue}{\frac{\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \sqrt[3]{1}}{2 \cdot \left(b + a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]

  13. Applied associate-*l/0.3

    \[\leadsto \frac{\color{blue}{\frac{\left(\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \sqrt[3]{1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b + a\right)}}}{b - a}\]
  14. Using strategy rm

  15. Applied sub-neg0.3

    \[\leadsto \frac{\frac{\left(\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \sqrt[3]{1}\right) \cdot \color{blue}{\left(\frac{1}{a} + \left(-\frac{1}{b}\right)\right)}}{2 \cdot \left(b + a\right)}}{b - a}\]

  16. Applied distribute-lft-in0.3

    \[\leadsto \frac{\frac{\color{blue}{\left(\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \sqrt[3]{1}\right) \cdot \frac{1}{a} + \left(\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \sqrt[3]{1}\right) \cdot \left(-\frac{1}{b}\right)}}{2 \cdot \left(b + a\right)}}{b - a}\]

  17. Simplified0.3

    \[\leadsto \frac{\frac{\color{blue}{\frac{1 \cdot \left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right)}{a}} + \left(\left(\pi \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) \cdot \sqrt[3]{1}\right) \cdot \left(-\frac{1}{b}\right)}{2 \cdot \left(b + a\right)}}{b - a}\]

  18. Simplified0.3

    \[\leadsto \frac{\frac{\frac{1 \cdot \left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right)}{a} + \color{blue}{\frac{\left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right) \cdot \left(-1\right)}{b}}}{2 \cdot \left(b + a\right)}}{b - a}\]

  19. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{1 \cdot \left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right)}{a} + \frac{\left({\left(\sqrt[3]{1}\right)}^{3} \cdot \pi\right) \cdot \left(-1\right)}{b}}{2 \cdot \left(b + a\right)}}{b - a}\]

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))))