Average Error: 11.0 → 1.5
Time: 24.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - t}}, x\right)\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - t}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r421141 = x;
        double r421142 = y;
        double r421143 = z;
        double r421144 = t;
        double r421145 = r421143 - r421144;
        double r421146 = r421142 * r421145;
        double r421147 = a;
        double r421148 = r421147 - r421144;
        double r421149 = r421146 / r421148;
        double r421150 = r421141 + r421149;
        return r421150;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r421151 = y;
        double r421152 = 1.0;
        double r421153 = a;
        double r421154 = t;
        double r421155 = r421153 - r421154;
        double r421156 = z;
        double r421157 = r421156 - r421154;
        double r421158 = r421155 / r421157;
        double r421159 = r421152 / r421158;
        double r421160 = x;
        double r421161 = fma(r421151, r421159, r421160);
        return r421161;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target1.3
Herbie1.5
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Initial program 11.0

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

  2. Simplified1.5

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

  4. Applied clear-num1.5

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{a - t}{z - t}}}, x\right)\]

  5. Final simplification1.5

    \[\leadsto \mathsf{fma}\left(y, \frac{1}{\frac{a - t}{z - t}}, x\right)\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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