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

↓

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

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 r516029 = x;
        double r516030 = y;
        double r516031 = r516030 - r516029;
        double r516032 = z;
        double r516033 = t;
        double r516034 = r516032 / r516033;
        double r516035 = r516031 * r516034;
        double r516036 = r516029 + r516035;
        return r516036;
}

↓

double f(double x, double y, double z, double t) {
        double r516037 = y;
        double r516038 = x;
        double r516039 = r516037 - r516038;
        double r516040 = z;
        double r516041 = t;
        double r516042 = r516040 / r516041;
        double r516043 = fma(r516039, r516042, r516038);
        return r516043;
}

Error

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

Target

Original	1.9
Target	2.0
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 2020081 +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))))