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

↓

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

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 r512126 = x;
        double r512127 = y;
        double r512128 = r512127 - r512126;
        double r512129 = z;
        double r512130 = t;
        double r512131 = r512129 / r512130;
        double r512132 = r512128 * r512131;
        double r512133 = r512126 + r512132;
        return r512133;
}

↓

double f(double x, double y, double z, double t) {
        double r512134 = y;
        double r512135 = x;
        double r512136 = r512134 - r512135;
        double r512137 = z;
        double r512138 = t;
        double r512139 = r512137 / r512138;
        double r512140 = fma(r512136, r512139, r512135);
        return r512140;
}

Error

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

Target

Original	1.9
Target	1.9
Herbie	1.9

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

Derivation

Initial program 1.9
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
Simplified1.9
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification1.9
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

Reproduce

herbie shell --seed 2020046 +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))))