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

↓

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

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

↓

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

double f(double x, double y, double z, double t) {
        double r18505821 = x;
        double r18505822 = y;
        double r18505823 = r18505822 - r18505821;
        double r18505824 = z;
        double r18505825 = t;
        double r18505826 = r18505824 / r18505825;
        double r18505827 = r18505823 * r18505826;
        double r18505828 = r18505821 + r18505827;
        return r18505828;
}

↓

double f(double x, double y, double z, double t) {
        double r18505829 = z;
        double r18505830 = t;
        double r18505831 = r18505829 / r18505830;
        double r18505832 = y;
        double r18505833 = x;
        double r18505834 = r18505832 - r18505833;
        double r18505835 = fma(r18505831, r18505834, r18505833);
        return r18505835;
}

Error

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

Target

Original	2.0
Target	2.2
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.88671875:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ 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))))