Average Error: 12.7 → 2.4
Time: 12.2s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.580679840002735045330812359603634219556 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \left(-\frac{z}{y}\right) \cdot x\right)\\ \mathbf{elif}\;x \le 1.524563898325377264226038471226668164466 \cdot 10^{-157}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.580679840002735045330812359603634219556 \cdot 10^{-172}:\\
\;\;\;\;\mathsf{fma}\left(1, x, \left(-\frac{z}{y}\right) \cdot x\right)\\

\mathbf{elif}\;x \le 1.524563898325377264226038471226668164466 \cdot 10^{-157}:\\
\;\;\;\;x - \frac{x \cdot z}{y}\\

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

\end{array}
double f(double x, double y, double z) {
        double r492524 = x;
        double r492525 = y;
        double r492526 = z;
        double r492527 = r492525 - r492526;
        double r492528 = r492524 * r492527;
        double r492529 = r492528 / r492525;
        return r492529;
}
↓
double f(double x, double y, double z) {
        double r492530 = x;
        double r492531 = -2.580679840002735e-172;
        bool r492532 = r492530 <= r492531;
        double r492533 = 1.0;
        double r492534 = z;
        double r492535 = y;
        double r492536 = r492534 / r492535;
        double r492537 = -r492536;
        double r492538 = r492537 * r492530;
        double r492539 = fma(r492533, r492530, r492538);
        double r492540 = 1.5245638983253773e-157;
        bool r492541 = r492530 <= r492540;
        double r492542 = r492530 * r492534;
        double r492543 = r492542 / r492535;
        double r492544 = r492530 - r492543;
        double r492545 = r492535 - r492534;
        double r492546 = r492535 / r492545;
        double r492547 = r492530 / r492546;
        double r492548 = r492541 ? r492544 : r492547;
        double r492549 = r492532 ? r492539 : r492548;
        return r492549;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original12.7
Target3.2
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.580679840002735e-172

    1. Initial program 13.9

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

    3. Applied associate-/l*1.2

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

    4. Taylor expanded around 0 4.8

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

    6. Applied *-un-lft-identity4.8

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

    7. Applied fma-neg4.8

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

    8. Simplified1.4

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

    if -2.580679840002735e-172 < x < 1.5245638983253773e-157

    1. Initial program 8.9

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

    3. Applied associate-/l*7.2

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

    4. Taylor expanded around 0 4.6

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

    if 1.5245638983253773e-157 < x

    1. Initial program 14.3

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

    3. Applied associate-/l*1.7

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.580679840002735045330812359603634219556 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \left(-\frac{z}{y}\right) \cdot x\right)\\ \mathbf{elif}\;x \le 1.524563898325377264226038471226668164466 \cdot 10^{-157}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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))