Average Error: 1.4 → 1.4
Time: 4.6s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r507804 = x;
        double r507805 = y;
        double r507806 = z;
        double r507807 = t;
        double r507808 = r507806 - r507807;
        double r507809 = a;
        double r507810 = r507809 - r507807;
        double r507811 = r507808 / r507810;
        double r507812 = r507805 * r507811;
        double r507813 = r507804 + r507812;
        return r507813;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r507814 = y;
        double r507815 = z;
        double r507816 = a;
        double r507817 = t;
        double r507818 = r507816 - r507817;
        double r507819 = r507815 / r507818;
        double r507820 = 1.0;
        double r507821 = r507818 / r507817;
        double r507822 = r507820 / r507821;
        double r507823 = r507819 - r507822;
        double r507824 = x;
        double r507825 = fma(r507814, r507823, r507824);
        return r507825;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target0.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  4. Applied div-sub1.4

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

  6. Applied clear-num1.4

    \[\leadsto \mathsf{fma}\left(y, \frac{z}{a - t} - \color{blue}{\frac{1}{\frac{a - t}{t}}}, x\right)\]

  7. Final simplification1.4

    \[\leadsto \mathsf{fma}\left(y, \frac{z}{a - t} - \frac{1}{\frac{a - t}{t}}, x\right)\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :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)))))