Average Error: 14.7 → 0.3
Time: 14.0s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \left(\frac{1}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)\]
\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{\pi}{2}}{b + a} \cdot \left(\frac{1}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\right)
double f(double a, double b) {
        double r77243 = atan2(1.0, 0.0);
        double r77244 = 2.0;
        double r77245 = r77243 / r77244;
        double r77246 = 1.0;
        double r77247 = b;
        double r77248 = r77247 * r77247;
        double r77249 = a;
        double r77250 = r77249 * r77249;
        double r77251 = r77248 - r77250;
        double r77252 = r77246 / r77251;
        double r77253 = r77245 * r77252;
        double r77254 = r77246 / r77249;
        double r77255 = r77246 / r77247;
        double r77256 = r77254 - r77255;
        double r77257 = r77253 * r77256;
        return r77257;
}
↓
double f(double a, double b) {
        double r77258 = atan2(1.0, 0.0);
        double r77259 = 2.0;
        double r77260 = r77258 / r77259;
        double r77261 = b;
        double r77262 = a;
        double r77263 = r77261 + r77262;
        double r77264 = r77260 / r77263;
        double r77265 = 1.0;
        double r77266 = r77261 - r77262;
        double r77267 = r77265 / r77266;
        double r77268 = r77265 / r77262;
        double r77269 = r77265 / r77261;
        double r77270 = r77268 - r77269;
        double r77271 = r77267 * r77270;
        double r77272 = r77264 * r77271;
        return r77272;
}

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 14.7

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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-frac9.1

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

  6. Applied associate-*r*9.1

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020033 +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))))