Average Error: 13.9 → 0.3
Time: 3.9m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{2} \cdot \frac{\sqrt[3]{1} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{b - a}\]
\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{\pi}{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)}{2} \cdot \frac{\sqrt[3]{1} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{b - a}
double code(double a, double b) {
	return ((double) (((double) (((double) (((double) M_PI) / 2.0)) * ((double) (1.0 / ((double) (((double) (b * b)) - ((double) (a * a)))))))) * ((double) (((double) (1.0 / a)) - ((double) (1.0 / b))))));
}
↓
double code(double a, double b) {
	return ((double) (((double) (((double) (((double) (((double) (((double) M_PI) / 1.0)) * ((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))))) / 2.0)) * ((double) (((double) (((double) cbrt(1.0)) * ((double) (((double) (1.0 / a)) - ((double) (1.0 / b)))))) / ((double) (b + a)))))) / ((double) (b - a))));
}

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.9

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

  3. Applied difference-of-squares9.4

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

  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 associate-*r/8.9

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

  6. Applied associate-*l/0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied associate-/r*0.3

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

  10. Applied frac-times0.3

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

  11. Applied associate-*l/0.3

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

  13. Applied add-cube-cbrt0.3

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

  14. Applied associate-*r*0.3

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

  15. Applied associate-*l*0.3

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

  16. Applied times-frac0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))