Average Error: 16.3 → 10.8
Time: 7.4s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 3.4126882859688763 \cdot 10^{232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{a - t} \cdot \left(t - z\right), x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le 3.4126882859688763 \cdot 10^{232}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{a - t} \cdot \left(t - z\right), x + y\right)\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((x + y) - (((z - t) * y) / (a - t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((t <= 3.4126882859688763e+232)) {
		VAR = fma(((cbrt(y) * cbrt(y)) / 1.0), (((cbrt((cbrt(y) * cbrt(y))) * cbrt(cbrt(y))) / (a - t)) * (t - z)), (x + y));
	} else {
		VAR = x;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original16.3
Target8.6
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < 3.4126882859688763e+232

    1. Initial program 14.7

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

    2. Simplified10.7

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

    4. Applied fma-udef10.7

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

    6. Applied *-un-lft-identity10.7

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

    7. Applied add-cube-cbrt10.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot \left(a - t\right)} \cdot \left(t - z\right) + \left(x + y\right)\]

    8. Applied times-frac10.9

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{a - t}\right)} \cdot \left(t - z\right) + \left(x + y\right)\]

    9. Applied associate-*l*10.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \left(\frac{\sqrt[3]{y}}{a - t} \cdot \left(t - z\right)\right)} + \left(x + y\right)\]
    10. Using strategy rm

    11. Applied fma-def10.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{y}}{a - t} \cdot \left(t - z\right), x + y\right)}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt10.2

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}{a - t} \cdot \left(t - z\right), x + y\right)\]

    14. Applied cbrt-prod10.2

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\color{blue}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}{a - t} \cdot \left(t - z\right), x + y\right)\]

    if 3.4126882859688763e+232 < t

    1. Initial program 37.5

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

    2. Simplified26.6

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

    3. Taylor expanded around 0 18.3

      \[\leadsto \color{blue}{x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.4126882859688763 \cdot 10^{232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}, \frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}{a - t} \cdot \left(t - z\right), x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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