Average Error: 14.3 → 0.3
Time: 9.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{\frac{2 \cdot \left(b + a\right)}{\pi \cdot 1}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]
\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{1}{\frac{2 \cdot \left(b + a\right)}{\pi \cdot 1}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}
double f(double a, double b) {
        double r56230 = atan2(1.0, 0.0);
        double r56231 = 2.0;
        double r56232 = r56230 / r56231;
        double r56233 = 1.0;
        double r56234 = b;
        double r56235 = r56234 * r56234;
        double r56236 = a;
        double r56237 = r56236 * r56236;
        double r56238 = r56235 - r56237;
        double r56239 = r56233 / r56238;
        double r56240 = r56232 * r56239;
        double r56241 = r56233 / r56236;
        double r56242 = r56233 / r56234;
        double r56243 = r56241 - r56242;
        double r56244 = r56240 * r56243;
        return r56244;
}
↓
double f(double a, double b) {
        double r56245 = 1.0;
        double r56246 = 2.0;
        double r56247 = b;
        double r56248 = a;
        double r56249 = r56247 + r56248;
        double r56250 = r56246 * r56249;
        double r56251 = atan2(1.0, 0.0);
        double r56252 = 1.0;
        double r56253 = r56251 * r56252;
        double r56254 = r56250 / r56253;
        double r56255 = r56245 / r56254;
        double r56256 = r56252 / r56248;
        double r56257 = r56252 / r56247;
        double r56258 = r56256 - r56257;
        double r56259 = r56255 * r56258;
        double r56260 = r56247 - r56248;
        double r56261 = r56259 / r56260;
        return r56261;
}

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 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 associate-*r/9.0

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

  7. Applied associate-*l/0.3

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

  9. Applied frac-times0.3

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

  11. Applied clear-num0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020089 +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))))