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

↓

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

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 r506733 = x;
        double r506734 = y;
        double r506735 = r506734 - r506733;
        double r506736 = z;
        double r506737 = t;
        double r506738 = r506736 / r506737;
        double r506739 = r506735 * r506738;
        double r506740 = r506733 + r506739;
        return r506740;
}

↓

double f(double x, double y, double z, double t) {
        double r506741 = y;
        double r506742 = x;
        double r506743 = r506741 - r506742;
        double r506744 = z;
        double r506745 = t;
        double r506746 = r506744 / r506745;
        double r506747 = fma(r506743, r506746, r506742);
        return r506747;
}

Error

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

Target

Original	1.8
Target	2.0
Herbie	1.8

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

Derivation

Initial program 1.8
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
Simplified1.8
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification1.8
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

Reproduce

herbie shell --seed 2020033 +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))))