Average Error: 14.7 → 0.7
Time: 1.6m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}} \cdot \frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}}}}{\frac{a}{\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{a + b}}}}{b - a}}} - \frac{\frac{1}{b - a}}{b} \cdot \frac{\frac{\pi}{2}}{a + b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}} \cdot \frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}}}}{\frac{a}{\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{a + b}}}}{b - a}}} - \frac{\frac{1}{b - a}}{b} \cdot \frac{\frac{\pi}{2}}{a + b}
double f(double a, double b) {
        double r3346559 = atan2(1.0, 0.0);
        double r3346560 = 2.0;
        double r3346561 = r3346559 / r3346560;
        double r3346562 = 1.0;
        double r3346563 = b;
        double r3346564 = r3346563 * r3346563;
        double r3346565 = a;
        double r3346566 = r3346565 * r3346565;
        double r3346567 = r3346564 - r3346566;
        double r3346568 = r3346562 / r3346567;
        double r3346569 = r3346561 * r3346568;
        double r3346570 = r3346562 / r3346565;
        double r3346571 = r3346562 / r3346563;
        double r3346572 = r3346570 - r3346571;
        double r3346573 = r3346569 * r3346572;
        return r3346573;
}
↓
double f(double a, double b) {
        double r3346574 = atan2(1.0, 0.0);
        double r3346575 = a;
        double r3346576 = b;
        double r3346577 = r3346575 + r3346576;
        double r3346578 = cbrt(r3346577);
        double r3346579 = r3346578 * r3346578;
        double r3346580 = r3346574 / r3346579;
        double r3346581 = 0.5;
        double r3346582 = sqrt(r3346581);
        double r3346583 = cbrt(r3346579);
        double r3346584 = r3346582 / r3346583;
        double r3346585 = cbrt(r3346578);
        double r3346586 = r3346582 / r3346585;
        double r3346587 = r3346576 - r3346575;
        double r3346588 = r3346586 / r3346587;
        double r3346589 = r3346575 / r3346588;
        double r3346590 = r3346584 / r3346589;
        double r3346591 = r3346580 * r3346590;
        double r3346592 = 1.0;
        double r3346593 = r3346592 / r3346587;
        double r3346594 = r3346593 / r3346576;
        double r3346595 = 2.0;
        double r3346596 = r3346574 / r3346595;
        double r3346597 = r3346596 / r3346577;
        double r3346598 = r3346594 * r3346597;
        double r3346599 = r3346591 - r3346598;
        return r3346599;
}

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. Simplified9.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{a} - \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{b}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity9.1

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

  5. Applied div-inv9.1

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

  6. Applied times-frac4.8

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

  8. Applied *-un-lft-identity4.8

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

  9. Applied *-un-lft-identity4.8

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

  10. Applied add-cube-cbrt5.0

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

  11. Applied div-inv5.0

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

  12. Applied times-frac5.0

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

  13. Applied times-frac5.0

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

  14. Applied times-frac1.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{\pi}{\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}}}{1}}{1} \cdot \frac{\frac{\frac{\frac{1}{2}}{\sqrt[3]{a + b}}}{b - a}}{a}} - \frac{\frac{\frac{\pi}{2}}{a + b}}{1} \cdot \frac{\frac{1}{b - a}}{b}\]
  15. Using strategy rm

  16. Applied *-un-lft-identity1.5

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

  17. Applied add-cube-cbrt1.5

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

  18. Applied cbrt-prod1.6

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

  19. Applied add-sqr-sqrt1.5

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

  20. Applied times-frac1.5

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

  21. Applied times-frac1.5

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

  22. Applied associate-/l*0.7

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

  23. Final simplification0.7

    \[\leadsto \frac{\pi}{\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}} \cdot \frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{a + b} \cdot \sqrt[3]{a + b}}}}{\frac{a}{\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{a + b}}}}{b - a}}} - \frac{\frac{1}{b - a}}{b} \cdot \frac{\frac{\pi}{2}}{a + b}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))