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

↓

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

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 r9457265 = x;
        double r9457266 = y;
        double r9457267 = r9457266 - r9457265;
        double r9457268 = z;
        double r9457269 = t;
        double r9457270 = r9457268 / r9457269;
        double r9457271 = r9457267 * r9457270;
        double r9457272 = r9457265 + r9457271;
        return r9457272;
}

↓

double f(double x, double y, double z, double t) {
        double r9457273 = y;
        double r9457274 = x;
        double r9457275 = r9457273 - r9457274;
        double r9457276 = z;
        double r9457277 = t;
        double r9457278 = r9457276 / r9457277;
        double r9457279 = fma(r9457275, r9457278, r9457274);
        return r9457279;
}

Error

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

Target

Original	2.0
Target	2.1
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

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

Reproduce

herbie shell --seed 2019156 +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))))