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

↓

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

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 r595043 = x;
        double r595044 = y;
        double r595045 = z;
        double r595046 = r595044 - r595045;
        double r595047 = t;
        double r595048 = r595046 * r595047;
        double r595049 = a;
        double r595050 = r595049 - r595045;
        double r595051 = r595048 / r595050;
        double r595052 = r595043 + r595051;
        return r595052;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r595053 = y;
        double r595054 = z;
        double r595055 = r595053 - r595054;
        double r595056 = a;
        double r595057 = r595056 - r595054;
        double r595058 = r595055 / r595057;
        double r595059 = t;
        double r595060 = x;
        double r595061 = fma(r595058, r595059, r595060);
        return r595061;
}

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))))