Average Error: 2.0 → 2.0
Time: 3.8s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.64219457899372803 \cdot 10^{-224} \lor \neg \left(x \le 1.6257691218099716 \cdot 10^{-308}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.64219457899372803 \cdot 10^{-224} \lor \neg \left(x \le 1.6257691218099716 \cdot 10^{-308}\right):\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r544724 = x;
        double r544725 = y;
        double r544726 = r544725 - r544724;
        double r544727 = z;
        double r544728 = t;
        double r544729 = r544727 / r544728;
        double r544730 = r544726 * r544729;
        double r544731 = r544724 + r544730;
        return r544731;
}
↓
double f(double x, double y, double z, double t) {
        double r544732 = x;
        double r544733 = -8.642194578993728e-224;
        bool r544734 = r544732 <= r544733;
        double r544735 = 1.6257691218099716e-308;
        bool r544736 = r544732 <= r544735;
        double r544737 = !r544736;
        bool r544738 = r544734 || r544737;
        double r544739 = y;
        double r544740 = r544739 - r544732;
        double r544741 = z;
        double r544742 = t;
        double r544743 = r544741 / r544742;
        double r544744 = fma(r544740, r544743, r544732);
        double r544745 = r544740 * r544741;
        double r544746 = r544745 / r544742;
        double r544747 = r544746 + r544732;
        double r544748 = r544738 ? r544744 : r544747;
        return r544748;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.2
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -8.642194578993728e-224 or 1.6257691218099716e-308 < x

    1. Initial program 1.7

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

    2. Simplified1.7

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

    if -8.642194578993728e-224 < x < 1.6257691218099716e-308

    1. Initial program 6.0

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

    2. Simplified6.0

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

    4. Applied fma-udef6.0

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

    6. Applied associate-*r/5.5

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.64219457899372803 \cdot 10^{-224} \lor \neg \left(x \le 1.6257691218099716 \cdot 10^{-308}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ 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))))