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

↓

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

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

↓

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

double f(double x, double y, double z, double t) {
        double r61560136 = x;
        double r61560137 = y;
        double r61560138 = r61560137 - r61560136;
        double r61560139 = z;
        double r61560140 = t;
        double r61560141 = r61560139 / r61560140;
        double r61560142 = r61560138 * r61560141;
        double r61560143 = r61560136 + r61560142;
        return r61560143;
}

↓

double f(double x, double y, double z, double t) {
        double r61560144 = z;
        double r61560145 = t;
        double r61560146 = r61560144 / r61560145;
        double r61560147 = y;
        double r61560148 = x;
        double r61560149 = r61560147 - r61560148;
        double r61560150 = fma(r61560146, r61560149, r61560148);
        return r61560150;
}

Error

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

Target

Original	2.1
Target	2.3
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.88671875:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

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

Reproduce

herbie shell --seed 2019173 +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))))