Average Error: 14.4 → 0.3
Time: 4.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(1 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{2 \cdot \left(b + a\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(1 \cdot \frac{\pi}{a \cdot b}\right) \cdot \frac{1}{2 \cdot \left(b + a\right)}
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) (1.0 * ((double) (((double) M_PI) / ((double) (a * b)))))) * ((double) (1.0 / ((double) (2.0 * ((double) (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 14.4

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

  3. Applied difference-of-squares9.7

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

  4. Applied add-sqr-sqrt9.7

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

  5. Applied times-frac9.2

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

  6. Applied associate-*r*9.2

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

  8. Applied frac-times9.2

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

  9. Applied associate-*l/9.2

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

  10. Applied associate-*l/0.3

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

  11. Simplified0.3

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

  12. Taylor expanded around 0 0.2

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

  14. Applied div-inv0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2020130 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))