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

↓

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

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

↓

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

double f(double x, double y, double z, double t, double a) {
        double r475173 = x;
        double r475174 = y;
        double r475175 = z;
        double r475176 = r475174 - r475175;
        double r475177 = t;
        double r475178 = r475176 * r475177;
        double r475179 = a;
        double r475180 = r475179 - r475175;
        double r475181 = r475178 / r475180;
        double r475182 = r475173 + r475181;
        return r475182;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r475183 = y;
        double r475184 = z;
        double r475185 = r475183 - r475184;
        double r475186 = a;
        double r475187 = r475186 - r475184;
        double r475188 = r475185 / r475187;
        double r475189 = t;
        double r475190 = x;
        double r475191 = fma(r475188, r475189, r475190);
        return r475191;
}

Error

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

Target

Original	10.7
Target	0.6
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

Initial program 10.7
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
Simplified1.3
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)}\]
Final simplification1.3
\[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\]

Reproduce

herbie shell --seed 2020001 +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))))