Average Error: 7.4 → 1.1
Time: 21.4s
Precision: 64
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{\sqrt[3]{t - z}} \cdot \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{\sqrt[3]{t - z}} \cdot \frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{\sqrt[3]{t - z}}\right)\]
\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}
↓
\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{\sqrt[3]{t - z}} \cdot \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y - z}}}{\sqrt[3]{t - 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 r30155366 = x;
        double r30155367 = y;
        double r30155368 = z;
        double r30155369 = r30155367 - r30155368;
        double r30155370 = t;
        double r30155371 = r30155370 - r30155368;
        double r30155372 = r30155369 * r30155371;
        double r30155373 = r30155366 / r30155372;
        return r30155373;
}
↓
double f(double x, double y, double z, double t) {
        double r30155374 = x;
        double r30155375 = cbrt(r30155374);
        double r30155376 = y;
        double r30155377 = z;
        double r30155378 = r30155376 - r30155377;
        double r30155379 = cbrt(r30155378);
        double r30155380 = r30155375 / r30155379;
        double r30155381 = t;
        double r30155382 = r30155381 - r30155377;
        double r30155383 = cbrt(r30155382);
        double r30155384 = r30155380 / r30155383;
        double r30155385 = r30155384 * r30155384;
        double r30155386 = r30155384 * r30155385;
        return r30155386;
}

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

Original7.4
Target8.4
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \lt 0.0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array}\]

Derivation

  1. Initial program 7.4

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

  3. Applied *-un-lft-identity7.4

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

  4. Applied times-frac2.2

    \[\leadsto \color{blue}{\frac{1}{y - z} \cdot \frac{x}{t - z}}\]
  5. Using strategy rm

  6. Applied associate-*l/2.1

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

  7. Simplified2.1

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

  9. Applied add-cube-cbrt2.7

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

  10. Applied add-cube-cbrt2.9

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

  11. Applied add-cube-cbrt3.0

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

  12. Applied times-frac3.0

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

  13. Applied times-frac1.2

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

  14. Simplified1.1

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

  15. Simplified1.1

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

  16. Final simplification1.1

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

Reproduce

herbie shell --seed 2019169 +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))))