Average Error: 15.0 → 0.3
Time: 37.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{a + b}}{2 \cdot \left(b - a\right)}\]
\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 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{a + b}}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r2875099 = atan2(1.0, 0.0);
        double r2875100 = 2.0;
        double r2875101 = r2875099 / r2875100;
        double r2875102 = 1.0;
        double r2875103 = b;
        double r2875104 = r2875103 * r2875103;
        double r2875105 = a;
        double r2875106 = r2875105 * r2875105;
        double r2875107 = r2875104 - r2875106;
        double r2875108 = r2875102 / r2875107;
        double r2875109 = r2875101 * r2875108;
        double r2875110 = r2875102 / r2875105;
        double r2875111 = r2875102 / r2875103;
        double r2875112 = r2875110 - r2875111;
        double r2875113 = r2875109 * r2875112;
        return r2875113;
}
↓
double f(double a, double b) {
        double r2875114 = 1.0;
        double r2875115 = atan2(1.0, 0.0);
        double r2875116 = a;
        double r2875117 = r2875115 / r2875116;
        double r2875118 = b;
        double r2875119 = r2875115 / r2875118;
        double r2875120 = r2875117 - r2875119;
        double r2875121 = r2875114 * r2875120;
        double r2875122 = r2875116 + r2875118;
        double r2875123 = r2875121 / r2875122;
        double r2875124 = 2.0;
        double r2875125 = r2875118 - r2875116;
        double r2875126 = r2875124 * r2875125;
        double r2875127 = r2875123 / r2875126;
        return r2875127;
}

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 15.0

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

  3. Applied difference-of-squares10.0

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

  4. Applied associate-/r*9.4

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

  6. Applied frac-times9.4

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

  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. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))