Average Error: 1.5 → 1.5
Time: 23.7s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r21508801 = x;
        double r21508802 = y;
        double r21508803 = z;
        double r21508804 = t;
        double r21508805 = r21508803 - r21508804;
        double r21508806 = a;
        double r21508807 = r21508806 - r21508804;
        double r21508808 = r21508805 / r21508807;
        double r21508809 = r21508802 * r21508808;
        double r21508810 = r21508801 + r21508809;
        return r21508810;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r21508811 = z;
        double r21508812 = t;
        double r21508813 = r21508811 - r21508812;
        double r21508814 = a;
        double r21508815 = r21508814 - r21508812;
        double r21508816 = r21508813 / r21508815;
        double r21508817 = y;
        double r21508818 = x;
        double r21508819 = fma(r21508816, r21508817, r21508818);
        return r21508819;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.5
Target0.5
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.5

    \[x + y \cdot \frac{z - t}{a - t}\]

  2. Simplified1.5

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

  3. Final simplification1.5

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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