Average Error: 2.0 → 1.5
Time: 23.7s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -6.746333275120044 \cdot 10^{-196}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{z}{t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{z}{t} \le -6.746333275120044 \cdot 10^{-196}:\\
\;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\

\mathbf{elif}\;\frac{z}{t} \le 0.0:\\
\;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r21438987 = x;
        double r21438988 = y;
        double r21438989 = r21438988 - r21438987;
        double r21438990 = z;
        double r21438991 = t;
        double r21438992 = r21438990 / r21438991;
        double r21438993 = r21438989 * r21438992;
        double r21438994 = r21438987 + r21438993;
        return r21438994;
}
↓
double f(double x, double y, double z, double t) {
        double r21438995 = z;
        double r21438996 = t;
        double r21438997 = r21438995 / r21438996;
        double r21438998 = -6.746333275120044e-196;
        bool r21438999 = r21438997 <= r21438998;
        double r21439000 = x;
        double r21439001 = y;
        double r21439002 = r21439001 - r21439000;
        double r21439003 = r21439002 * r21438997;
        double r21439004 = r21439000 + r21439003;
        double r21439005 = 0.0;
        bool r21439006 = r21438997 <= r21439005;
        double r21439007 = r21439002 / r21438996;
        double r21439008 = fma(r21439007, r21438995, r21439000);
        double r21439009 = r21439006 ? r21439008 : r21439004;
        double r21439010 = r21438999 ? r21439004 : r21439009;
        return r21439010;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.2
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ z t) < -6.746333275120044e-196 or 0.0 < (/ z t)

    1. Initial program 1.9

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

    if -6.746333275120044e-196 < (/ z t) < 0.0

    1. Initial program 2.3

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

    2. Simplified0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \le -6.746333275120044 \cdot 10^{-196}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;\frac{z}{t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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))))