Average Error: 1.3 → 1.3
Time: 19.9s
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 r31543674 = x;
        double r31543675 = y;
        double r31543676 = z;
        double r31543677 = t;
        double r31543678 = r31543676 - r31543677;
        double r31543679 = a;
        double r31543680 = r31543679 - r31543677;
        double r31543681 = r31543678 / r31543680;
        double r31543682 = r31543675 * r31543681;
        double r31543683 = r31543674 + r31543682;
        return r31543683;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r31543684 = z;
        double r31543685 = t;
        double r31543686 = r31543684 - r31543685;
        double r31543687 = a;
        double r31543688 = r31543687 - r31543685;
        double r31543689 = r31543686 / r31543688;
        double r31543690 = y;
        double r31543691 = x;
        double r31543692 = fma(r31543689, r31543690, r31543691);
        return r31543692;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.5
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

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

  3. Final simplification1.3

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

Reproduce

herbie shell --seed 2019168 +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 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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