Average Error: 15.0 → 0.3
Time: 10.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot 1\right)}{b + a} \cdot \frac{1}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\frac{\pi}{2} \cdot 1\right)}{b + a} \cdot \frac{1}{b - a}
double f(double a, double b) {
        double r50184 = atan2(1.0, 0.0);
        double r50185 = 2.0;
        double r50186 = r50184 / r50185;
        double r50187 = 1.0;
        double r50188 = b;
        double r50189 = r50188 * r50188;
        double r50190 = a;
        double r50191 = r50190 * r50190;
        double r50192 = r50189 - r50191;
        double r50193 = r50187 / r50192;
        double r50194 = r50186 * r50193;
        double r50195 = r50187 / r50190;
        double r50196 = r50187 / r50188;
        double r50197 = r50195 - r50196;
        double r50198 = r50194 * r50197;
        return r50198;
}
↓
double f(double a, double b) {
        double r50199 = 1.0;
        double r50200 = a;
        double r50201 = r50199 / r50200;
        double r50202 = b;
        double r50203 = r50199 / r50202;
        double r50204 = r50201 - r50203;
        double r50205 = atan2(1.0, 0.0);
        double r50206 = 2.0;
        double r50207 = r50205 / r50206;
        double r50208 = r50207 * r50199;
        double r50209 = r50204 * r50208;
        double r50210 = r50202 + r50200;
        double r50211 = r50209 / r50210;
        double r50212 = 1.0;
        double r50213 = r50202 - r50200;
        double r50214 = r50212 / r50213;
        double r50215 = r50211 * r50214;
        return r50215;
}

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-squares10.3

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

  4. Applied add-sqr-sqrt10.3

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

  5. Applied times-frac9.8

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

  6. Applied associate-*r*9.8

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

  8. Applied associate-*r/9.8

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

  9. Applied associate-*l/0.3

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

  10. Simplified0.3

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

  12. Applied div-inv0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))