Average Error: 1.2 → 1.2
Time: 9.8s
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 r2078 = x;
        double r2079 = y;
        double r2080 = z;
        double r2081 = t;
        double r2082 = r2080 - r2081;
        double r2083 = a;
        double r2084 = r2083 - r2081;
        double r2085 = r2082 / r2084;
        double r2086 = r2079 * r2085;
        double r2087 = r2078 + r2086;
        return r2087;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2088 = y;
        double r2089 = z;
        double r2090 = t;
        double r2091 = r2089 - r2090;
        double r2092 = a;
        double r2093 = r2092 - r2090;
        double r2094 = r2091 / r2093;
        double r2095 = x;
        double r2096 = fma(r2088, r2094, r2095);
        return r2096;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.2
Target0.4
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.2

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

  2. Simplified1.2

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

  3. Final simplification1.2

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

Reproduce

herbie shell --seed 2020025 +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)))))