Average Error: 14.4 → 0.3
Time: 5.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi \cdot 1}{2 \cdot \left(b + a\right)} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi \cdot 1}{2 \cdot \left(b + a\right)} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}
double f(double a, double b) {
        double r40164 = atan2(1.0, 0.0);
        double r40165 = 2.0;
        double r40166 = r40164 / r40165;
        double r40167 = 1.0;
        double r40168 = b;
        double r40169 = r40168 * r40168;
        double r40170 = a;
        double r40171 = r40170 * r40170;
        double r40172 = r40169 - r40171;
        double r40173 = r40167 / r40172;
        double r40174 = r40166 * r40173;
        double r40175 = r40167 / r40170;
        double r40176 = r40167 / r40168;
        double r40177 = r40175 - r40176;
        double r40178 = r40174 * r40177;
        return r40178;
}
↓
double f(double a, double b) {
        double r40179 = atan2(1.0, 0.0);
        double r40180 = 1.0;
        double r40181 = r40179 * r40180;
        double r40182 = 2.0;
        double r40183 = b;
        double r40184 = a;
        double r40185 = r40183 + r40184;
        double r40186 = r40182 * r40185;
        double r40187 = r40181 / r40186;
        double r40188 = r40180 / r40184;
        double r40189 = r40180 / r40183;
        double r40190 = r40188 - r40189;
        double r40191 = r40183 - r40184;
        double r40192 = r40190 / r40191;
        double r40193 = r40187 * r40192;
        return r40193;
}

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 14.4

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

  3. Applied difference-of-squares9.8

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

  4. Applied associate-/r*9.3

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

  6. Applied associate-*r/9.2

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

  7. Applied associate-*l/0.3

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

  9. Applied frac-times0.3

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

  11. Applied *-un-lft-identity0.3

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

  12. Applied times-frac0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019362 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))