Average Error: 14.5 → 0.3
Time: 9.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot 1}{b + a}}{\frac{2 \cdot \left(b - a\right)}{\frac{1}{a} - \frac{1}{b}}}\]
\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 \cdot 1}{b + a}}{\frac{2 \cdot \left(b - a\right)}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r54349 = atan2(1.0, 0.0);
        double r54350 = 2.0;
        double r54351 = r54349 / r54350;
        double r54352 = 1.0;
        double r54353 = b;
        double r54354 = r54353 * r54353;
        double r54355 = a;
        double r54356 = r54355 * r54355;
        double r54357 = r54354 - r54356;
        double r54358 = r54352 / r54357;
        double r54359 = r54351 * r54358;
        double r54360 = r54352 / r54355;
        double r54361 = r54352 / r54353;
        double r54362 = r54360 - r54361;
        double r54363 = r54359 * r54362;
        return r54363;
}
↓
double f(double a, double b) {
        double r54364 = atan2(1.0, 0.0);
        double r54365 = 1.0;
        double r54366 = r54364 * r54365;
        double r54367 = b;
        double r54368 = a;
        double r54369 = r54367 + r54368;
        double r54370 = r54366 / r54369;
        double r54371 = 2.0;
        double r54372 = r54367 - r54368;
        double r54373 = r54371 * r54372;
        double r54374 = r54365 / r54368;
        double r54375 = r54365 / r54367;
        double r54376 = r54374 - r54375;
        double r54377 = r54373 / r54376;
        double r54378 = r54370 / r54377;
        return r54378;
}

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 14.5

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

  3. Applied difference-of-squares9.6

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

  4. Applied associate-/r*9.0

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

  6. Applied frac-times9.0

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  11. Applied associate-/l*0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020020 +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))))