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

↓

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

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

↓

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

double f(double x, double y, double z, double t, double a) {
        double r3178 = x;
        double r3179 = y;
        double r3180 = z;
        double r3181 = r3179 - r3180;
        double r3182 = t;
        double r3183 = r3181 * r3182;
        double r3184 = a;
        double r3185 = r3184 - r3180;
        double r3186 = r3183 / r3185;
        double r3187 = r3178 + r3186;
        return r3187;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r3188 = y;
        double r3189 = a;
        double r3190 = z;
        double r3191 = r3189 - r3190;
        double r3192 = r3188 / r3191;
        double r3193 = r3190 / r3191;
        double r3194 = r3192 - r3193;
        double r3195 = t;
        double r3196 = x;
        double r3197 = fma(r3194, r3195, r3196);
        return r3197;
}

Error

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

Target

Original	11.3
Target	0.5
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

Initial program 11.3
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
Simplified1.2
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)}\]
Using strategy rm
Applied div-sub1.2
\[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a - z} - \frac{z}{a - z}}, t, x\right)\]
Final simplification1.2
\[\leadsto \mathsf{fma}\left(\frac{y}{a - z} - \frac{z}{a - z}, t, x\right)\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

  (+ x (/ (* (- y z) t) (- a z))))