Average Error: 12.4 → 3.5
Time: 2.9s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.2854619819908728 \cdot 10^{-166}:\\ \;\;\;\;\frac{y - z}{y} \cdot x\\ \mathbf{elif}\;x \le 6.95759694615908471 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt[3]{1}}{\frac{y}{\sqrt[3]{1}}} \cdot \left(\sqrt[3]{1} \cdot \left(x \cdot \left(y - z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.2854619819908728 \cdot 10^{-166}:\\
\;\;\;\;\frac{y - z}{y} \cdot x\\

\mathbf{elif}\;x \le 6.95759694615908471 \cdot 10^{-203}:\\
\;\;\;\;\frac{\sqrt[3]{1}}{\frac{y}{\sqrt[3]{1}}} \cdot \left(\sqrt[3]{1} \cdot \left(x \cdot \left(y - z\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\

\end{array}
double f(double x, double y, double z) {
        double r851448 = x;
        double r851449 = y;
        double r851450 = z;
        double r851451 = r851449 - r851450;
        double r851452 = r851448 * r851451;
        double r851453 = r851452 / r851449;
        return r851453;
}
↓
double f(double x, double y, double z) {
        double r851454 = x;
        double r851455 = -6.285461981990873e-166;
        bool r851456 = r851454 <= r851455;
        double r851457 = y;
        double r851458 = z;
        double r851459 = r851457 - r851458;
        double r851460 = r851459 / r851457;
        double r851461 = r851460 * r851454;
        double r851462 = 6.957596946159085e-203;
        bool r851463 = r851454 <= r851462;
        double r851464 = 1.0;
        double r851465 = cbrt(r851464);
        double r851466 = r851457 / r851465;
        double r851467 = r851465 / r851466;
        double r851468 = r851454 * r851459;
        double r851469 = r851465 * r851468;
        double r851470 = r851467 * r851469;
        double r851471 = r851457 / r851459;
        double r851472 = r851471 / r851454;
        double r851473 = r851464 / r851472;
        double r851474 = r851463 ? r851470 : r851473;
        double r851475 = r851456 ? r851461 : r851474;
        return r851475;
}

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

Original12.4
Target3.2
Herbie3.5
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.69397660138285259 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -6.285461981990873e-166

    1. Initial program 14.2

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

    3. Applied associate-/l*1.6

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

    5. Applied clear-num1.7

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

    7. Applied div-inv1.7

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

    8. Applied add-sqr-sqrt1.7

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

    9. Applied times-frac1.8

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

    10. Simplified1.8

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

    11. Simplified1.7

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

    if -6.285461981990873e-166 < x < 6.957596946159085e-203

    1. Initial program 8.4

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

    3. Applied associate-/l*7.3

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

    5. Applied clear-num7.4

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

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

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

    8. Applied div-inv7.5

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

    9. Applied times-frac9.1

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

    10. Applied add-cube-cbrt9.1

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

    11. Applied times-frac9.1

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

    12. Simplified9.1

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

    13. Simplified8.5

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

    if 6.957596946159085e-203 < x

    1. Initial program 13.1

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

    3. Applied associate-/l*2.2

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

    5. Applied clear-num2.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{y - z}}{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.2854619819908728 \cdot 10^{-166}:\\ \;\;\;\;\frac{y - z}{y} \cdot x\\ \mathbf{elif}\;x \le 6.95759694615908471 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt[3]{1}}{\frac{y}{\sqrt[3]{1}}} \cdot \left(\sqrt[3]{1} \cdot \left(x \cdot \left(y - z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))