Average Error: 16.2 → 11.9
Time: 5.1s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1999581429001337180000:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot \left(\sqrt[3]{\frac{1}{a - t}} \cdot \sqrt[3]{\frac{1}{a - t}}\right)\right) \cdot \sqrt[3]{\frac{1}{a - t}}, t - z, x + y\right)\\ \mathbf{elif}\;t \le 979.44708308574286:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;t \le 1.3799551143033571 \cdot 10^{201}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1999581429001337180000:\\
\;\;\;\;\mathsf{fma}\left(\left(y \cdot \left(\sqrt[3]{\frac{1}{a - t}} \cdot \sqrt[3]{\frac{1}{a - t}}\right)\right) \cdot \sqrt[3]{\frac{1}{a - t}}, t - z, x + y\right)\\

\mathbf{elif}\;t \le 979.44708308574286:\\
\;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\

\mathbf{elif}\;t \le 1.3799551143033571 \cdot 10^{201}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((t <= -1.9995814290013372e+21)) {
		VAR = ((double) fma(((double) (((double) (y * ((double) (((double) cbrt(((double) (1.0 / ((double) (a - t)))))) * ((double) cbrt(((double) (1.0 / ((double) (a - t)))))))))) * ((double) cbrt(((double) (1.0 / ((double) (a - t)))))))), ((double) (t - z)), ((double) (x + y))));
	} else {
		double VAR_1;
		if ((t <= 979.4470830857429)) {
			VAR_1 = ((double) (((double) (x + y)) - ((double) (((double) (((double) (z - t)) * y)) / ((double) (a - t))))));
		} else {
			double VAR_2;
			if ((t <= 1.379955114303357e+201)) {
				VAR_2 = ((double) fma(((double) (y * ((double) (1.0 / ((double) (a - t)))))), ((double) (t - z)), ((double) (x + y))));
			} else {
				VAR_2 = x;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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

Original16.2
Target8.2
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if t < -1.9995814290013372e+21

    1. Initial program 25.9

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

    2. Simplified18.5

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

    4. Applied div-inv18.5

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

    6. Applied add-cube-cbrt18.7

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

    7. Applied associate-*r*18.7

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

    if -1.9995814290013372e+21 < t < 979.4470830857429

    1. Initial program 6.6

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

    if 979.4470830857429 < t < 1.379955114303357e+201

    1. Initial program 20.5

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

    2. Simplified13.7

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

    4. Applied div-inv13.8

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

    if 1.379955114303357e+201 < t

    1. Initial program 34.1

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

    2. Simplified26.0

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

    3. Taylor expanded around 0 18.5

      \[\leadsto \color{blue}{x}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1999581429001337180000:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot \left(\sqrt[3]{\frac{1}{a - t}} \cdot \sqrt[3]{\frac{1}{a - t}}\right)\right) \cdot \sqrt[3]{\frac{1}{a - t}}, t - z, x + y\right)\\ \mathbf{elif}\;t \le 979.44708308574286:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;t \le 1.3799551143033571 \cdot 10^{201}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +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))))