Average Error: 1.2 → 1.2
Time: 7.2s
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 r799068 = x;
        double r799069 = y;
        double r799070 = z;
        double r799071 = t;
        double r799072 = r799070 - r799071;
        double r799073 = a;
        double r799074 = r799073 - r799071;
        double r799075 = r799072 / r799074;
        double r799076 = r799069 * r799075;
        double r799077 = r799068 + r799076;
        return r799077;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r799078 = y;
        double r799079 = z;
        double r799080 = t;
        double r799081 = r799079 - r799080;
        double r799082 = a;
        double r799083 = r799082 - r799080;
        double r799084 = r799081 / r799083;
        double r799085 = x;
        double r799086 = fma(r799078, r799084, r799085);
        return r799086;
}

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)))))