Average Error: 14.2 → 0.3
Time: 40.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{2} \cdot \frac{\frac{1}{b \cdot a}}{a + b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{2} \cdot \frac{\frac{1}{b \cdot a}}{a + b}
double f(double a, double b) {
        double r2839139 = atan2(1.0, 0.0);
        double r2839140 = 2.0;
        double r2839141 = r2839139 / r2839140;
        double r2839142 = 1.0;
        double r2839143 = b;
        double r2839144 = r2839143 * r2839143;
        double r2839145 = a;
        double r2839146 = r2839145 * r2839145;
        double r2839147 = r2839144 - r2839146;
        double r2839148 = r2839142 / r2839147;
        double r2839149 = r2839141 * r2839148;
        double r2839150 = r2839142 / r2839145;
        double r2839151 = r2839142 / r2839143;
        double r2839152 = r2839150 - r2839151;
        double r2839153 = r2839149 * r2839152;
        return r2839153;
}
↓
double f(double a, double b) {
        double r2839154 = atan2(1.0, 0.0);
        double r2839155 = 2.0;
        double r2839156 = r2839154 / r2839155;
        double r2839157 = 1.0;
        double r2839158 = b;
        double r2839159 = a;
        double r2839160 = r2839158 * r2839159;
        double r2839161 = r2839157 / r2839160;
        double r2839162 = r2839159 + r2839158;
        double r2839163 = r2839161 / r2839162;
        double r2839164 = r2839156 * r2839163;
        return r2839164;
}

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 14.2

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

  3. Applied difference-of-squares9.4

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

  4. Applied *-un-lft-identity9.4

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified8.9

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

  9. Applied associate-*l*0.3

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

  10. Taylor expanded around 0 0.3

    \[\leadsto \frac{\frac{\pi}{2}}{a + b} \cdot \color{blue}{\frac{1}{a \cdot b}}\]
  11. Using strategy rm

  12. Applied div-inv0.3

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

  13. Applied associate-*l*0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +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))))