Average Error: 24.0 → 14.5
Time: 17.7s
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \frac{y}{\sqrt[3]{y + \left(b - y\right) \cdot z}}\right) - \left(z \cdot \frac{1}{y + \left(b - y\right) \cdot z}\right) \cdot \left(a - t\right)\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\frac{\sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \frac{y}{\sqrt[3]{y + \left(b - y\right) \cdot z}}\right) - \left(z \cdot \frac{1}{y + \left(b - y\right) \cdot z}\right) \cdot \left(a - t\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r406768 = x;
        double r406769 = y;
        double r406770 = r406768 * r406769;
        double r406771 = z;
        double r406772 = t;
        double r406773 = a;
        double r406774 = r406772 - r406773;
        double r406775 = r406771 * r406774;
        double r406776 = r406770 + r406775;
        double r406777 = b;
        double r406778 = r406777 - r406769;
        double r406779 = r406771 * r406778;
        double r406780 = r406769 + r406779;
        double r406781 = r406776 / r406780;
        return r406781;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r406782 = x;
        double r406783 = cbrt(r406782);
        double r406784 = y;
        double r406785 = b;
        double r406786 = r406785 - r406784;
        double r406787 = z;
        double r406788 = r406786 * r406787;
        double r406789 = r406784 + r406788;
        double r406790 = cbrt(r406789);
        double r406791 = r406783 / r406790;
        double r406792 = r406783 * r406783;
        double r406793 = r406792 / r406790;
        double r406794 = r406784 / r406790;
        double r406795 = r406793 * r406794;
        double r406796 = r406791 * r406795;
        double r406797 = 1.0;
        double r406798 = r406797 / r406789;
        double r406799 = r406787 * r406798;
        double r406800 = a;
        double r406801 = t;
        double r406802 = r406800 - r406801;
        double r406803 = r406799 * r406802;
        double r406804 = r406796 - r406803;
        return r406804;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.0
Target18.5
Herbie14.5
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Initial program 24.0

    \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]

  2. Simplified24.0

    \[\leadsto \color{blue}{\frac{y \cdot x - z \cdot \left(a - t\right)}{y + z \cdot \left(b - y\right)}}\]
  3. Using strategy rm

  4. Applied div-sub24.0

    \[\leadsto \color{blue}{\frac{y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{z \cdot \left(a - t\right)}{y + z \cdot \left(b - y\right)}}\]

  5. Simplified24.0

    \[\leadsto \color{blue}{\frac{y \cdot x}{z \cdot \left(b - y\right) + y}} - \frac{z \cdot \left(a - t\right)}{y + z \cdot \left(b - y\right)}\]

  6. Simplified19.6

    \[\leadsto \frac{y \cdot x}{z \cdot \left(b - y\right) + y} - \color{blue}{\frac{z}{z \cdot \left(b - y\right) + y} \cdot \left(a - t\right)}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity19.6

    \[\leadsto \frac{y \cdot x}{\color{blue}{1 \cdot \left(z \cdot \left(b - y\right) + y\right)}} - \frac{z}{z \cdot \left(b - y\right) + y} \cdot \left(a - t\right)\]

  9. Applied times-frac18.4

    \[\leadsto \color{blue}{\frac{y}{1} \cdot \frac{x}{z \cdot \left(b - y\right) + y}} - \frac{z}{z \cdot \left(b - y\right) + y} \cdot \left(a - t\right)\]

  10. Simplified18.4

    \[\leadsto \color{blue}{y} \cdot \frac{x}{z \cdot \left(b - y\right) + y} - \frac{z}{z \cdot \left(b - y\right) + y} \cdot \left(a - t\right)\]

  11. Simplified18.4

    \[\leadsto y \cdot \color{blue}{\frac{x}{y + \left(b - y\right) \cdot z}} - \frac{z}{z \cdot \left(b - y\right) + y} \cdot \left(a - t\right)\]
  12. Using strategy rm

  13. Applied div-inv18.4

    \[\leadsto y \cdot \frac{x}{y + \left(b - y\right) \cdot z} - \color{blue}{\left(z \cdot \frac{1}{z \cdot \left(b - y\right) + y}\right)} \cdot \left(a - t\right)\]

  14. Simplified18.4

    \[\leadsto y \cdot \frac{x}{y + \left(b - y\right) \cdot z} - \left(z \cdot \color{blue}{\frac{1}{y + z \cdot \left(b - y\right)}}\right) \cdot \left(a - t\right)\]
  15. Using strategy rm

  16. Applied add-cube-cbrt18.7

    \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\sqrt[3]{y + \left(b - y\right) \cdot z} \cdot \sqrt[3]{y + \left(b - y\right) \cdot z}\right) \cdot \sqrt[3]{y + \left(b - y\right) \cdot z}}} - \left(z \cdot \frac{1}{y + z \cdot \left(b - y\right)}\right) \cdot \left(a - t\right)\]

  17. Applied add-cube-cbrt18.8

    \[\leadsto y \cdot \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{y + \left(b - y\right) \cdot z} \cdot \sqrt[3]{y + \left(b - y\right) \cdot z}\right) \cdot \sqrt[3]{y + \left(b - y\right) \cdot z}} - \left(z \cdot \frac{1}{y + z \cdot \left(b - y\right)}\right) \cdot \left(a - t\right)\]

  18. Applied times-frac18.8

    \[\leadsto y \cdot \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z} \cdot \sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}}\right)} - \left(z \cdot \frac{1}{y + z \cdot \left(b - y\right)}\right) \cdot \left(a - t\right)\]

  19. Applied associate-*r*15.2

    \[\leadsto \color{blue}{\left(y \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z} \cdot \sqrt[3]{y + \left(b - y\right) \cdot z}}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}}} - \left(z \cdot \frac{1}{y + z \cdot \left(b - y\right)}\right) \cdot \left(a - t\right)\]

  20. Simplified14.5

    \[\leadsto \color{blue}{\left(\frac{y}{\sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}}\right)} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}} - \left(z \cdot \frac{1}{y + z \cdot \left(b - y\right)}\right) \cdot \left(a - t\right)\]

  21. Final simplification14.5

    \[\leadsto \frac{\sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y + \left(b - y\right) \cdot z}} \cdot \frac{y}{\sqrt[3]{y + \left(b - y\right) \cdot z}}\right) - \left(z \cdot \frac{1}{y + \left(b - y\right) \cdot z}\right) \cdot \left(a - t\right)\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))