Average Error: 14.6 → 0.3
Time: 4.2m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{\frac{\frac{2}{\frac{1}{a} - \frac{1}{b}}}{\frac{{\left(\sqrt[3]{1}\right)}^{3}}{b + a}}}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi}{\frac{\frac{2}{\frac{1}{a} - \frac{1}{b}}}{\frac{{\left(\sqrt[3]{1}\right)}^{3}}{b + a}}}}{b - a}
double f(double a, double b) {
        double r1070295 = atan2(1.0, 0.0);
        double r1070296 = 2.0;
        double r1070297 = r1070295 / r1070296;
        double r1070298 = 1.0;
        double r1070299 = b;
        double r1070300 = r1070299 * r1070299;
        double r1070301 = a;
        double r1070302 = r1070301 * r1070301;
        double r1070303 = r1070300 - r1070302;
        double r1070304 = r1070298 / r1070303;
        double r1070305 = r1070297 * r1070304;
        double r1070306 = r1070298 / r1070301;
        double r1070307 = r1070298 / r1070299;
        double r1070308 = r1070306 - r1070307;
        double r1070309 = r1070305 * r1070308;
        return r1070309;
}
↓
double f(double a, double b) {
        double r1070310 = atan2(1.0, 0.0);
        double r1070311 = 2.0;
        double r1070312 = 1.0;
        double r1070313 = a;
        double r1070314 = r1070312 / r1070313;
        double r1070315 = b;
        double r1070316 = r1070312 / r1070315;
        double r1070317 = r1070314 - r1070316;
        double r1070318 = r1070311 / r1070317;
        double r1070319 = cbrt(r1070312);
        double r1070320 = 3.0;
        double r1070321 = pow(r1070319, r1070320);
        double r1070322 = r1070315 + r1070313;
        double r1070323 = r1070321 / r1070322;
        double r1070324 = r1070318 / r1070323;
        double r1070325 = r1070310 / r1070324;
        double r1070326 = r1070315 - r1070313;
        double r1070327 = r1070325 / r1070326;
        return r1070327;
}

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 14.6

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 add-cube-cbrt9.6

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(b + a\right) \cdot \left(b - a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  5. Applied times-frac9.1

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a} \cdot \frac{\sqrt[3]{1}}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  6. Applied associate-*r*9.1

    \[\leadsto \color{blue}{\left(\left(\frac{\pi}{2} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \frac{\sqrt[3]{1}}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  7. Using strategy rm

  8. Applied associate-*l/9.1

    \[\leadsto \left(\color{blue}{\frac{\pi \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}}{2}} \cdot \frac{\sqrt[3]{1}}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  9. Applied frac-times9.1

    \[\leadsto \color{blue}{\frac{\left(\pi \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \sqrt[3]{1}}{2 \cdot \left(b - a\right)}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  10. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\left(\pi \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \sqrt[3]{1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b - a\right)}}\]
  11. Using strategy rm

  12. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{\left(\left(\pi \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a}\right) \cdot \sqrt[3]{1}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2}}{b - a}}\]

  13. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{\pi}{\frac{\frac{2}{\frac{1}{a} - \frac{1}{b}}}{\frac{{\left(\sqrt[3]{1}\right)}^{3}}{b + a}}}}}{b - a}\]

  14. Final simplification0.3

    \[\leadsto \frac{\frac{\pi}{\frac{\frac{2}{\frac{1}{a} - \frac{1}{b}}}{\frac{{\left(\sqrt[3]{1}\right)}^{3}}{b + a}}}}{b - a}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))