Average Error: 14.6 → 0.2
Time: 12.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{0.5 \cdot \pi}{b + a}}{a \cdot b}\]
\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{0.5 \cdot \pi}{b + a}}{a \cdot b}
double f(double a, double b) {
        double r56482 = atan2(1.0, 0.0);
        double r56483 = 2.0;
        double r56484 = r56482 / r56483;
        double r56485 = 1.0;
        double r56486 = b;
        double r56487 = r56486 * r56486;
        double r56488 = a;
        double r56489 = r56488 * r56488;
        double r56490 = r56487 - r56489;
        double r56491 = r56485 / r56490;
        double r56492 = r56484 * r56491;
        double r56493 = r56485 / r56488;
        double r56494 = r56485 / r56486;
        double r56495 = r56493 - r56494;
        double r56496 = r56492 * r56495;
        return r56496;
}
↓
double f(double a, double b) {
        double r56497 = 0.5;
        double r56498 = atan2(1.0, 0.0);
        double r56499 = r56497 * r56498;
        double r56500 = b;
        double r56501 = a;
        double r56502 = r56500 + r56501;
        double r56503 = r56499 / r56502;
        double r56504 = r56501 * r56500;
        double r56505 = r56503 / r56504;
        return r56505;
}

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 14.6

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 add-sqr-sqrt9.6

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.1

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

  8. Applied frac-sub9.1

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

  9. Applied associate-*r/9.1

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

  10. Simplified0.2

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \pi}}{b + a}}{a \cdot b}\]

  12. Final simplification0.2

    \[\leadsto \frac{\frac{0.5 \cdot \pi}{b + a}}{a \cdot b}\]

Reproduce

herbie shell --seed 2020042 +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))))