Average Error: 14.5 → 0.3
Time: 22.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{a \cdot b}\]
\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{\pi}{2}}{b + a} \cdot \frac{1}{a \cdot b}
double f(double a, double b) {
        double r74186 = atan2(1.0, 0.0);
        double r74187 = 2.0;
        double r74188 = r74186 / r74187;
        double r74189 = 1.0;
        double r74190 = b;
        double r74191 = r74190 * r74190;
        double r74192 = a;
        double r74193 = r74192 * r74192;
        double r74194 = r74191 - r74193;
        double r74195 = r74189 / r74194;
        double r74196 = r74188 * r74195;
        double r74197 = r74189 / r74192;
        double r74198 = r74189 / r74190;
        double r74199 = r74197 - r74198;
        double r74200 = r74196 * r74199;
        return r74200;
}
↓
double f(double a, double b) {
        double r74201 = atan2(1.0, 0.0);
        double r74202 = 2.0;
        double r74203 = r74201 / r74202;
        double r74204 = b;
        double r74205 = a;
        double r74206 = r74204 + r74205;
        double r74207 = r74203 / r74206;
        double r74208 = 1.0;
        double r74209 = r74205 * r74204;
        double r74210 = r74208 / r74209;
        double r74211 = r74207 * r74210;
        return r74211;
}

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.5

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 *-un-lft-identity9.6

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

  5. Applied times-frac9.2

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

  6. Applied associate-*r*9.2

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

  7. Simplified9.1

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

  9. Applied associate-*l*0.3

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

  10. Simplified0.3

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

  11. Taylor expanded around 0 0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019304 +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))))