Average Error: 2.0 → 2.0
Time: 22.1s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7160.278861583339676144532859325408935547:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;t \le -5.485219336659275411636551846957666382716 \cdot 10^{-236}:\\ \;\;\;\;x + \left(\frac{y \cdot z}{t} - \frac{z \cdot x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x}{\frac{t}{z}}\right)\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7160.278861583339676144532859325408935547:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

\mathbf{elif}\;t \le -5.485219336659275411636551846957666382716 \cdot 10^{-236}:\\
\;\;\;\;x + \left(\frac{y \cdot z}{t} - \frac{z \cdot x}{t}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x}{\frac{t}{z}}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r22352966 = x;
        double r22352967 = y;
        double r22352968 = r22352967 - r22352966;
        double r22352969 = z;
        double r22352970 = t;
        double r22352971 = r22352969 / r22352970;
        double r22352972 = r22352968 * r22352971;
        double r22352973 = r22352966 + r22352972;
        return r22352973;
}
↓
double f(double x, double y, double z, double t) {
        double r22352974 = t;
        double r22352975 = -7160.27886158334;
        bool r22352976 = r22352974 <= r22352975;
        double r22352977 = y;
        double r22352978 = x;
        double r22352979 = r22352977 - r22352978;
        double r22352980 = r22352979 / r22352974;
        double r22352981 = z;
        double r22352982 = fma(r22352980, r22352981, r22352978);
        double r22352983 = -5.485219336659275e-236;
        bool r22352984 = r22352974 <= r22352983;
        double r22352985 = r22352977 * r22352981;
        double r22352986 = r22352985 / r22352974;
        double r22352987 = r22352981 * r22352978;
        double r22352988 = r22352987 / r22352974;
        double r22352989 = r22352986 - r22352988;
        double r22352990 = r22352978 + r22352989;
        double r22352991 = r22352981 / r22352974;
        double r22352992 = r22352974 / r22352981;
        double r22352993 = r22352978 / r22352992;
        double r22352994 = r22352978 - r22352993;
        double r22352995 = fma(r22352991, r22352977, r22352994);
        double r22352996 = r22352984 ? r22352990 : r22352995;
        double r22352997 = r22352976 ? r22352982 : r22352996;
        return r22352997;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.2
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.88671875:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -7160.27886158334

    1. Initial program 1.1

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

    2. Simplified1.2

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

    if -7160.27886158334 < t < -5.485219336659275e-236

    1. Initial program 2.2

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

    2. Taylor expanded around 0 1.5

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

    if -5.485219336659275e-236 < t

    1. Initial program 2.5

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

    2. Taylor expanded around 0 6.7

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

    3. Simplified2.5

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7160.278861583339676144532859325408935547:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;t \le -5.485219336659275411636551846957666382716 \cdot 10^{-236}:\\ \;\;\;\;x + \left(\frac{y \cdot z}{t} - \frac{z \cdot x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x}{\frac{t}{z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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