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

↓

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

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

↓

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

double f(double x, double y, double z, double t, double a) {
        double r618697 = x;
        double r618698 = y;
        double r618699 = z;
        double r618700 = t;
        double r618701 = r618699 - r618700;
        double r618702 = a;
        double r618703 = r618699 - r618702;
        double r618704 = r618701 / r618703;
        double r618705 = r618698 * r618704;
        double r618706 = r618697 + r618705;
        return r618706;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r618707 = y;
        double r618708 = z;
        double r618709 = t;
        double r618710 = r618708 - r618709;
        double r618711 = 1.0;
        double r618712 = a;
        double r618713 = r618708 - r618712;
        double r618714 = r618711 / r618713;
        double r618715 = r618710 * r618714;
        double r618716 = x;
        double r618717 = fma(r618707, r618715, r618716);
        return r618717;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Original	1.4
Target	1.3
Herbie	1.5

Derivation

Initial program 1.4
\[x + y \cdot \frac{z - t}{z - a}\]
Simplified1.4
\[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right)}\]
Using strategy rm
Applied div-inv1.5
\[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(z - t\right) \cdot \frac{1}{z - a}}, x\right)\]
Final simplification1.5
\[\leadsto \mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{z - a}, x\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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

Error

Target

Derivation

Reproduce