Average Error: 16.4 → 8.8
Time: 4.5s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.31751108397336504 \cdot 10^{111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le -1.68819575296857856 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{elif}\;t \le 22.968365102893742:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;t \le 1.50644189407964551 \cdot 10^{164}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.31751108397336504 \cdot 10^{111}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;t \le -1.68819575296857856 \cdot 10^{-99}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\

\mathbf{elif}\;t \le 22.968365102893742:\\
\;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\

\mathbf{elif}\;t \le 1.50644189407964551 \cdot 10^{164}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\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 <= -2.317511083973365e+111)) {
		VAR = fma((z / t), y, x);
	} else {
		double VAR_1;
		if ((t <= -1.6881957529685786e-99)) {
			VAR_1 = fma((y * (1.0 / (a - t))), (t - z), (x + y));
		} else {
			double VAR_2;
			if ((t <= 22.968365102893742)) {
				VAR_2 = ((x + y) - (((z - t) * y) / (a - t)));
			} else {
				double VAR_3;
				if ((t <= 1.5064418940796455e+164)) {
					VAR_3 = fma((y * (1.0 / (a - t))), (t - z), (x + y));
				} else {
					VAR_3 = fma((z / t), y, x);
				}
				VAR_2 = VAR_3;
			}
			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

Original16.4
Target8.3
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -2.317511083973365e+111 or 1.5064418940796455e+164 < t

    1. Initial program 31.0

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

    2. Simplified23.2

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

    4. Applied fma-udef23.3

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

    6. Applied div-inv23.3

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

    7. Applied associate-*l*21.7

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

    8. Simplified21.7

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

    9. Taylor expanded around inf 16.3

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

    10. Simplified11.2

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

    if -2.317511083973365e+111 < t < -1.6881957529685786e-99 or 22.968365102893742 < t < 1.5064418940796455e+164

    1. Initial program 16.3

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

    2. Simplified11.1

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

    4. Applied div-inv11.2

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

    if -1.6881957529685786e-99 < t < 22.968365102893742

    1. Initial program 5.1

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.31751108397336504 \cdot 10^{111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le -1.68819575296857856 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{elif}\;t \le 22.968365102893742:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;t \le 1.50644189407964551 \cdot 10^{164}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +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))))