Average Error: 2.0 → 2.0
Time: 22.3s
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 r23200934 = x;
        double r23200935 = y;
        double r23200936 = r23200935 - r23200934;
        double r23200937 = z;
        double r23200938 = t;
        double r23200939 = r23200937 / r23200938;
        double r23200940 = r23200936 * r23200939;
        double r23200941 = r23200934 + r23200940;
        return r23200941;
}
↓
double f(double x, double y, double z, double t) {
        double r23200942 = t;
        double r23200943 = -7160.27886158334;
        bool r23200944 = r23200942 <= r23200943;
        double r23200945 = y;
        double r23200946 = x;
        double r23200947 = r23200945 - r23200946;
        double r23200948 = r23200947 / r23200942;
        double r23200949 = z;
        double r23200950 = fma(r23200948, r23200949, r23200946);
        double r23200951 = -5.485219336659275e-236;
        bool r23200952 = r23200942 <= r23200951;
        double r23200953 = r23200945 * r23200949;
        double r23200954 = r23200953 / r23200942;
        double r23200955 = r23200949 * r23200946;
        double r23200956 = r23200955 / r23200942;
        double r23200957 = r23200954 - r23200956;
        double r23200958 = r23200946 + r23200957;
        double r23200959 = r23200949 / r23200942;
        double r23200960 = r23200942 / r23200949;
        double r23200961 = r23200946 / r23200960;
        double r23200962 = r23200946 - r23200961;
        double r23200963 = fma(r23200959, r23200945, r23200962);
        double r23200964 = r23200952 ? r23200958 : r23200963;
        double r23200965 = r23200944 ? r23200950 : r23200964;
        return r23200965;
}

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))))