Average Error: 13.9 → 0.3
Time: 1.7m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(1 \cdot \frac{1}{b + a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b - a\right)} \cdot \pi\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(1 \cdot \frac{1}{b + a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b - a\right)} \cdot \pi
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) (1.0 * ((double) (1.0 / ((double) (b + a)))))) * ((double) (((double) (1.0 / a)) - ((double) (1.0 / b)))))) / ((double) (2.0 * ((double) (b - a)))))) * ((double) M_PI)));
}

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-squares9.3

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

  4. Applied associate-/r*8.8

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

  5. Applied clear-num8.8

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

  6. Applied frac-times8.8

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

  7. Applied associate-*l/0.4

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

  9. Applied associate-*l/0.3

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

  10. Applied associate-/r/0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020113 +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))))