Average Error: 14.6 → 0.3
Time: 27.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{b - a} \cdot \frac{1}{a + b}\]
\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{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{b - a} \cdot \frac{1}{a + b}
double f(double a, double b) {
        double r1762024 = atan2(1.0, 0.0);
        double r1762025 = 2.0;
        double r1762026 = r1762024 / r1762025;
        double r1762027 = 1.0;
        double r1762028 = b;
        double r1762029 = r1762028 * r1762028;
        double r1762030 = a;
        double r1762031 = r1762030 * r1762030;
        double r1762032 = r1762029 - r1762031;
        double r1762033 = r1762027 / r1762032;
        double r1762034 = r1762026 * r1762033;
        double r1762035 = r1762027 / r1762030;
        double r1762036 = r1762027 / r1762028;
        double r1762037 = r1762035 - r1762036;
        double r1762038 = r1762034 * r1762037;
        return r1762038;
}
↓
double f(double a, double b) {
        double r1762039 = atan2(1.0, 0.0);
        double r1762040 = a;
        double r1762041 = r1762039 / r1762040;
        double r1762042 = b;
        double r1762043 = r1762039 / r1762042;
        double r1762044 = r1762041 - r1762043;
        double r1762045 = 2.0;
        double r1762046 = r1762044 / r1762045;
        double r1762047 = r1762042 - r1762040;
        double r1762048 = r1762046 / r1762047;
        double r1762049 = 1.0;
        double r1762050 = r1762040 + r1762042;
        double r1762051 = r1762049 / r1762050;
        double r1762052 = r1762048 * r1762051;
        return r1762052;
}

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. Simplified9.7

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

  4. Applied associate-/r*0.3

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

  5. Taylor expanded around 0 0.3

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

  6. Simplified0.3

    \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}}{b - a}}{a + b}\]
  7. Using strategy rm

  8. Applied div-inv0.3

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

  9. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{2}}{b - a} \cdot \frac{1}{a + b}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))