Average Error: 15.9 → 9.1
Time: 7.4s
Precision: binary64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -8.703056610822989 \cdot 10^{+228}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq 6.965175920449405 \cdot 10^{+180}:\\ \;\;\;\;\left(x + y\right) - y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \leq -8.703056610822989 \cdot 10^{+228}:\\
\;\;\;\;x + \frac{z \cdot y}{t}\\

\mathbf{elif}\;t \leq 6.965175920449405 \cdot 10^{+180}:\\
\;\;\;\;\left(x + y\right) - y \cdot \frac{z - t}{a - t}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{z}{\frac{t}{y}}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (- (+ x y) (/ (* (- z t) y) (- a t))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8.703056610822989e+228)
   (+ x (/ (* z y) t))
   (if (<= t 6.965175920449405e+180)
     (- (+ x y) (* y (/ (- z t) (- a t))))
     (+ x (/ z (/ t y))))))
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 tmp;
	if (t <= -8.703056610822989e+228) {
		tmp = x + ((z * y) / t);
	} else if (t <= 6.965175920449405e+180) {
		tmp = (x + y) - (y * ((z - t) / (a - t)));
	} else {
		tmp = x + (z / (t / y));
	}
	return tmp;
}

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

Original15.9
Target8.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -8.7030566108229891e228

    1. Initial program 33.4

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

    2. Taylor expanded around inf 14.6

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

    3. Simplified14.6

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

    if -8.7030566108229891e228 < t < 6.9651759204494053e180

    1. Initial program 12.0

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

    3. Applied associate-/l*_binary64_165368.5

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

    5. Applied associate-/r/_binary64_165378.4

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

    if 6.9651759204494053e180 < t

    1. Initial program 33.2

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

    3. Applied associate-/l*_binary64_1653624.4

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

    4. Taylor expanded around inf 15.8

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

    5. Simplified10.7

      \[\leadsto \color{blue}{x + \frac{z}{\frac{t}{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.703056610822989 \cdot 10^{+228}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq 6.965175920449405 \cdot 10^{+180}:\\ \;\;\;\;\left(x + y\right) - y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

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