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

↓

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

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 r567048 = x;
        double r567049 = y;
        double r567050 = r567049 - r567048;
        double r567051 = z;
        double r567052 = t;
        double r567053 = r567051 / r567052;
        double r567054 = r567050 * r567053;
        double r567055 = r567048 + r567054;
        return r567055;
}

↓

double f(double x, double y, double z, double t) {
        double r567056 = y;
        double r567057 = x;
        double r567058 = r567056 - r567057;
        double r567059 = z;
        double r567060 = t;
        double r567061 = r567059 / r567060;
        double r567062 = fma(r567058, r567061, r567057);
        return r567062;
}

Error

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

Target

Original	2.1
Target	2.3
Herbie	2.1

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

Derivation

Initial program 2.1
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
Simplified2.1
\[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)}\]
Final simplification2.1
\[\leadsto \mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\]

Reproduce

herbie shell --seed 2020035 +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))))