Average Error: 13.0 → 1.4
Time: 4.2s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.446414607682033344844893055916604547379 \cdot 10^{-250}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}\\ \mathbf{elif}\;y \le 3572580006165377287364755677987733504:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.446414607682033344844893055916604547379 \cdot 10^{-250}:\\
\;\;\;\;\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}\\

\mathbf{elif}\;y \le 3572580006165377287364755677987733504:\\
\;\;\;\;x - \frac{x \cdot z}{y}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{y - z}{y}\\

\end{array}
double f(double x, double y, double z) {
        double r848545 = x;
        double r848546 = y;
        double r848547 = z;
        double r848548 = r848546 - r848547;
        double r848549 = r848545 * r848548;
        double r848550 = r848549 / r848546;
        return r848550;
}
↓
double f(double x, double y, double z) {
        double r848551 = y;
        double r848552 = -1.4464146076820333e-250;
        bool r848553 = r848551 <= r848552;
        double r848554 = x;
        double r848555 = z;
        double r848556 = r848551 - r848555;
        double r848557 = cbrt(r848556);
        double r848558 = r848557 * r848557;
        double r848559 = cbrt(r848551);
        double r848560 = r848559 * r848559;
        double r848561 = r848558 / r848560;
        double r848562 = r848554 * r848561;
        double r848563 = r848557 / r848559;
        double r848564 = r848562 * r848563;
        double r848565 = 3.572580006165377e+36;
        bool r848566 = r848551 <= r848565;
        double r848567 = r848554 * r848555;
        double r848568 = r848567 / r848551;
        double r848569 = r848554 - r848568;
        double r848570 = r848556 / r848551;
        double r848571 = r848554 * r848570;
        double r848572 = r848566 ? r848569 : r848571;
        double r848573 = r848553 ? r848564 : r848572;
        return r848573;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target3.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.4464146076820333e-250

    1. Initial program 13.1

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

    3. Applied *-un-lft-identity13.1

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

    4. Applied times-frac2.5

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

    5. Simplified2.5

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

    7. Applied add-cube-cbrt3.7

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

    8. Applied add-cube-cbrt3.0

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

    9. Applied times-frac3.0

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

    10. Applied associate-*r*0.8

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

    if -1.4464146076820333e-250 < y < 3.572580006165377e+36

    1. Initial program 6.3

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

    3. Applied *-un-lft-identity6.3

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

    4. Applied times-frac8.1

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

    5. Simplified8.1

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

    7. Applied add-cube-cbrt9.2

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

    8. Applied associate-*l*9.2

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

    9. Taylor expanded around 0 3.7

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

    if 3.572580006165377e+36 < y

    1. Initial program 19.6

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

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

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

      \[\leadsto \color{blue}{x} \cdot \frac{y - z}{y}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.446414607682033344844893055916604547379 \cdot 10^{-250}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}\\ \mathbf{elif}\;y \le 3572580006165377287364755677987733504:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

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