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

↓

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

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 r477769 = x;
        double r477770 = y;
        double r477771 = r477770 - r477769;
        double r477772 = z;
        double r477773 = t;
        double r477774 = r477772 / r477773;
        double r477775 = r477771 * r477774;
        double r477776 = r477769 + r477775;
        return r477776;
}

↓

double f(double x, double y, double z, double t) {
        double r477777 = y;
        double r477778 = x;
        double r477779 = r477777 - r477778;
        double r477780 = z;
        double r477781 = t;
        double r477782 = r477780 / r477781;
        double r477783 = fma(r477779, r477782, r477778);
        return r477783;
}

Error

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

Target

Original	2.2
Target	2.3
Herbie	2.2

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

Derivation

Initial program 2.2
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
Simplified2.2
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification2.2
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

Reproduce

herbie shell --seed 2020039 +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))))