Average Error: 14.1 → 0.2
Time: 47.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{\left(a + b\right) \cdot \left(2 \cdot a\right)} + \frac{\frac{\pi}{a + b} \cdot \frac{-1}{2}}{b}}{b - a}\]
\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{\pi}{\left(a + b\right) \cdot \left(2 \cdot a\right)} + \frac{\frac{\pi}{a + b} \cdot \frac{-1}{2}}{b}}{b - a}
double f(double a, double b) {
        double r1525050 = atan2(1.0, 0.0);
        double r1525051 = 2.0;
        double r1525052 = r1525050 / r1525051;
        double r1525053 = 1.0;
        double r1525054 = b;
        double r1525055 = r1525054 * r1525054;
        double r1525056 = a;
        double r1525057 = r1525056 * r1525056;
        double r1525058 = r1525055 - r1525057;
        double r1525059 = r1525053 / r1525058;
        double r1525060 = r1525052 * r1525059;
        double r1525061 = r1525053 / r1525056;
        double r1525062 = r1525053 / r1525054;
        double r1525063 = r1525061 - r1525062;
        double r1525064 = r1525060 * r1525063;
        return r1525064;
}
↓
double f(double a, double b) {
        double r1525065 = atan2(1.0, 0.0);
        double r1525066 = a;
        double r1525067 = b;
        double r1525068 = r1525066 + r1525067;
        double r1525069 = 2.0;
        double r1525070 = r1525069 * r1525066;
        double r1525071 = r1525068 * r1525070;
        double r1525072 = r1525065 / r1525071;
        double r1525073 = r1525065 / r1525068;
        double r1525074 = -0.5;
        double r1525075 = r1525073 * r1525074;
        double r1525076 = r1525075 / r1525067;
        double r1525077 = r1525072 + r1525076;
        double r1525078 = r1525067 - r1525066;
        double r1525079 = r1525077 / r1525078;
        return r1525079;
}

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 14.1

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

  2. Simplified0.3

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

  4. Applied associate-/l/0.3

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

  6. Applied *-un-lft-identity0.3

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

  7. Applied *-un-lft-identity0.3

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

  8. Applied distribute-lft-out--0.3

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

  9. Applied associate-/r*0.3

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))