Average Error: 7.6 → 1.2
Time: 21.5s
Precision: 64
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
↓
\[\left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t - z}}\right) \cdot \left(\frac{\frac{1}{\sqrt[3]{y - z}}}{\sqrt[3]{y - z}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{\sqrt[3]{t - z}}\right)\]
\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}
↓
\left(\frac{\sqrt[3]{x}}{\sqrt[3]{t - z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{t - z}}\right) \cdot \left(\frac{\frac{1}{\sqrt[3]{y - z}}}{\sqrt[3]{y - z}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{\sqrt[3]{t - z}}\right)
double f(double x, double y, double z, double t) {
        double r31088115 = x;
        double r31088116 = y;
        double r31088117 = z;
        double r31088118 = r31088116 - r31088117;
        double r31088119 = t;
        double r31088120 = r31088119 - r31088117;
        double r31088121 = r31088118 * r31088120;
        double r31088122 = r31088115 / r31088121;
        return r31088122;
}
↓
double f(double x, double y, double z, double t) {
        double r31088123 = x;
        double r31088124 = cbrt(r31088123);
        double r31088125 = t;
        double r31088126 = z;
        double r31088127 = r31088125 - r31088126;
        double r31088128 = cbrt(r31088127);
        double r31088129 = r31088124 / r31088128;
        double r31088130 = r31088129 * r31088129;
        double r31088131 = 1.0;
        double r31088132 = y;
        double r31088133 = r31088132 - r31088126;
        double r31088134 = cbrt(r31088133);
        double r31088135 = r31088131 / r31088134;
        double r31088136 = r31088135 / r31088134;
        double r31088137 = r31088124 / r31088134;
        double r31088138 = r31088137 / r31088128;
        double r31088139 = r31088136 * r31088138;
        double r31088140 = r31088130 * r31088139;
        return r31088140;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.6
Target8.3
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \lt 0.0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array}\]

Derivation

  1. Initial program 7.6

    \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
  2. Using strategy rm

  3. Applied associate-/r*2.1

    \[\leadsto \color{blue}{\frac{\frac{x}{y - z}}{t - z}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt2.7

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

  6. Applied *-un-lft-identity2.7

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

  7. Applied add-cube-cbrt2.8

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

  8. Applied times-frac2.8

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

  9. Applied times-frac1.2

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}}{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}} \cdot \frac{\frac{\sqrt[3]{x}}{y - z}}{\sqrt[3]{t - z}}}\]

  10. Simplified1.3

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

  12. Applied *-un-lft-identity1.3

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

  13. Applied cbrt-prod1.3

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

  14. Applied add-cube-cbrt1.4

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

  15. Applied *-un-lft-identity1.4

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

  16. Applied cbrt-prod1.4

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

  17. Applied times-frac1.4

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

  18. Applied times-frac1.2

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

  19. Simplified1.2

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

  20. Final simplification1.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"

  :herbie-target
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 (* (- y z) (- t z)))))

  (/ x (* (- y z) (- t z))))