Average Error: 9.9 → 1.3
Time: 21.8s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\mathsf{fma}\left(t, \frac{y - z}{a - z}, x\right)\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\mathsf{fma}\left(t, \frac{y - z}{a - z}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r26538074 = x;
        double r26538075 = y;
        double r26538076 = z;
        double r26538077 = r26538075 - r26538076;
        double r26538078 = t;
        double r26538079 = r26538077 * r26538078;
        double r26538080 = a;
        double r26538081 = r26538080 - r26538076;
        double r26538082 = r26538079 / r26538081;
        double r26538083 = r26538074 + r26538082;
        return r26538083;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r26538084 = t;
        double r26538085 = y;
        double r26538086 = z;
        double r26538087 = r26538085 - r26538086;
        double r26538088 = a;
        double r26538089 = r26538088 - r26538086;
        double r26538090 = r26538087 / r26538089;
        double r26538091 = x;
        double r26538092 = fma(r26538084, r26538090, r26538091);
        return r26538092;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original9.9
Target0.6
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 9.9

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

  2. Simplified1.3

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

  3. Final simplification1.3

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"

  :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))))