Average Error: 12.4 → 1.8
Time: 16.0s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.290655561206783908563838071823474030955 \cdot 10^{97}:\\ \;\;\;\;\left(1 - \frac{z}{y}\right) \cdot x\\ \mathbf{elif}\;x \le 2.731727571395544206998641716133386673057 \cdot 10^{-15}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.290655561206783908563838071823474030955 \cdot 10^{97}:\\
\;\;\;\;\left(1 - \frac{z}{y}\right) \cdot x\\

\mathbf{elif}\;x \le 2.731727571395544206998641716133386673057 \cdot 10^{-15}:\\
\;\;\;\;x - \frac{x \cdot z}{y}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r489335 = x;
        double r489336 = y;
        double r489337 = z;
        double r489338 = r489336 - r489337;
        double r489339 = r489335 * r489338;
        double r489340 = r489339 / r489336;
        return r489340;
}
↓
double f(double x, double y, double z) {
        double r489341 = x;
        double r489342 = -1.290655561206784e+97;
        bool r489343 = r489341 <= r489342;
        double r489344 = 1.0;
        double r489345 = z;
        double r489346 = y;
        double r489347 = r489345 / r489346;
        double r489348 = r489344 - r489347;
        double r489349 = r489348 * r489341;
        double r489350 = 2.7317275713955442e-15;
        bool r489351 = r489341 <= r489350;
        double r489352 = r489341 * r489345;
        double r489353 = r489352 / r489346;
        double r489354 = r489341 - r489353;
        double r489355 = r489346 - r489345;
        double r489356 = r489346 / r489355;
        double r489357 = expm1(r489356);
        double r489358 = log1p(r489357);
        double r489359 = r489341 / r489358;
        double r489360 = r489351 ? r489354 : r489359;
        double r489361 = r489343 ? r489349 : r489360;
        return r489361;
}

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
Target3.3
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.290655561206784e+97

    1. Initial program 31.2

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

    3. Applied associate-/l*0.1

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

    5. Applied clear-num0.2

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

    7. Applied div-inv0.3

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

    8. Applied add-cube-cbrt0.3

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

    9. Applied times-frac0.3

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

    10. Simplified0.2

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

    11. Simplified0.1

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

    if -1.290655561206784e+97 < x < 2.7317275713955442e-15

    1. Initial program 5.6

      \[\frac{x \cdot \left(y - z\right)}{y}\]

    2. Taylor expanded around 0 2.7

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

    if 2.7317275713955442e-15 < x

    1. Initial program 20.8

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

    3. Applied associate-/l*0.1

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

    5. Applied log1p-expm1-u0.1

      \[\leadsto \frac{x}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.290655561206783908563838071823474030955 \cdot 10^{97}:\\ \;\;\;\;\left(1 - \frac{z}{y}\right) \cdot x\\ \mathbf{elif}\;x \le 2.731727571395544206998641716133386673057 \cdot 10^{-15}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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 -2.060202331921739e104) (- x (/ (* z x) y)) (if (< z 1.69397660138285259e213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

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