Average Error: 10.9 → 1.4
Time: 23.7s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\]
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 r399576 = x;
        double r399577 = y;
        double r399578 = z;
        double r399579 = r399577 - r399578;
        double r399580 = t;
        double r399581 = r399579 * r399580;
        double r399582 = a;
        double r399583 = r399582 - r399578;
        double r399584 = r399581 / r399583;
        double r399585 = r399576 + r399584;
        return r399585;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r399586 = y;
        double r399587 = z;
        double r399588 = r399586 - r399587;
        double r399589 = a;
        double r399590 = r399589 - r399587;
        double r399591 = r399588 / r399590;
        double r399592 = t;
        double r399593 = x;
        double r399594 = fma(r399591, r399592, r399593);
        return r399594;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target0.6
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 10.9

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

  2. Simplified1.4

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

  3. Final simplification1.4

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

Reproduce

herbie shell --seed 2019198 +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))))