Average Error: 7.7 → 1.1
Time: 20.5s
Precision: 64
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
↓
\[\frac{\sqrt[3]{x}}{\sqrt[3]{y - z} \cdot \sqrt[3]{t - z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y - z} \cdot \sqrt[3]{t - z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y - z} \cdot \sqrt[3]{t - z}}\right)\]
\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}
↓
\frac{\sqrt[3]{x}}{\sqrt[3]{y - z} \cdot \sqrt[3]{t - z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y - z} \cdot \sqrt[3]{t - z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y - z} \cdot \sqrt[3]{t - z}}\right)
double f(double x, double y, double z, double t) {
        double r36728383 = x;
        double r36728384 = y;
        double r36728385 = z;
        double r36728386 = r36728384 - r36728385;
        double r36728387 = t;
        double r36728388 = r36728387 - r36728385;
        double r36728389 = r36728386 * r36728388;
        double r36728390 = r36728383 / r36728389;
        return r36728390;
}
↓
double f(double x, double y, double z, double t) {
        double r36728391 = x;
        double r36728392 = cbrt(r36728391);
        double r36728393 = y;
        double r36728394 = z;
        double r36728395 = r36728393 - r36728394;
        double r36728396 = cbrt(r36728395);
        double r36728397 = t;
        double r36728398 = r36728397 - r36728394;
        double r36728399 = cbrt(r36728398);
        double r36728400 = r36728396 * r36728399;
        double r36728401 = r36728392 / r36728400;
        double r36728402 = r36728401 * r36728401;
        double r36728403 = r36728401 * r36728402;
        return r36728403;
}

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

Original7.7
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 7.7

    \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
  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 add-cube-cbrt2.9

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

  7. Applied add-cube-cbrt3.0

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

  8. Applied times-frac3.0

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

  9. Applied times-frac1.2

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

  10. Simplified1.1

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

  11. Simplified1.1

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

  12. Final simplification1.1

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

Reproduce

herbie shell --seed 2019163 +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 (* (- y z) (- t z)))))

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