Average Error: 1.4 → 1.4
Time: 3.9s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r599826 = x;
        double r599827 = y;
        double r599828 = z;
        double r599829 = t;
        double r599830 = r599828 - r599829;
        double r599831 = a;
        double r599832 = r599831 - r599829;
        double r599833 = r599830 / r599832;
        double r599834 = r599827 * r599833;
        double r599835 = r599826 + r599834;
        return r599835;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r599836 = y;
        double r599837 = z;
        double r599838 = t;
        double r599839 = r599837 - r599838;
        double r599840 = a;
        double r599841 = r599840 - r599838;
        double r599842 = r599839 / r599841;
        double r599843 = x;
        double r599844 = fma(r599836, r599842, r599843);
        return r599844;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target0.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.4

    \[x + y \cdot \frac{z - t}{a - t}\]

  2. Simplified1.4

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

  3. Final simplification1.4

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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