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

↓

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

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

↓

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

double f(double x, double y, double z, double t) {
        double r1269755 = x;
        double r1269756 = y;
        double r1269757 = r1269756 - r1269755;
        double r1269758 = z;
        double r1269759 = t;
        double r1269760 = r1269758 / r1269759;
        double r1269761 = r1269757 * r1269760;
        double r1269762 = r1269755 + r1269761;
        return r1269762;
}

↓

double f(double x, double y, double z, double t) {
        double r1269763 = y;
        double r1269764 = x;
        double r1269765 = r1269763 - r1269764;
        double r1269766 = z;
        double r1269767 = t;
        double r1269768 = r1269766 / r1269767;
        double r1269769 = fma(r1269765, r1269768, r1269764);
        return r1269769;
}

Error

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

Target

Original	2.0
Target	2.2
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

Initial program 2.0
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
Simplified2.0
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification2.0
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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