Average Error: 7.4 → 1.5
Time: 4.6s
Precision: binary64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + \left(z \cdot \frac{y}{z \cdot t - x} - \frac{x}{z \cdot t - x}\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2.953727335544004 \cdot 10^{+286}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\
\;\;\;\;\frac{x + \left(z \cdot \frac{y}{z \cdot t - x} - \frac{x}{z \cdot t - x}\right)}{x + 1}\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2.953727335544004 \cdot 10^{+286}:\\
\;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\end{array}
(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0)) (- INFINITY))
   (/ (+ x (- (* z (/ y (- (* z t) x))) (/ x (- (* z t) x)))) (+ x 1.0))
   (if (<=
        (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))
        2.953727335544004e+286)
     (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))
     (/ (+ x (/ y t)) (+ x 1.0)))))
double code(double x, double y, double z, double t) {
	return (((double) (x + (((double) (((double) (y * z)) - x)) / ((double) (((double) (t * z)) - x))))) / ((double) (x + 1.0)));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (((((double) (x + (((double) (((double) (y * z)) - x)) / ((double) (((double) (z * t)) - x))))) / ((double) (x + 1.0))) <= ((double) -(((double) INFINITY))))) {
		tmp = (((double) (x + ((double) (((double) (z * (y / ((double) (((double) (z * t)) - x))))) - (x / ((double) (((double) (z * t)) - x))))))) / ((double) (x + 1.0)));
	} else {
		double tmp_1;
		if (((((double) (x + (((double) (((double) (y * z)) - x)) / ((double) (((double) (z * t)) - x))))) / ((double) (x + 1.0))) <= 2.953727335544004e+286)) {
			tmp_1 = (((double) (x + (((double) (((double) (y * z)) - x)) / ((double) (((double) (z * t)) - x))))) / ((double) (x + 1.0)));
		} else {
			tmp_1 = (((double) (x + (y / t))) / ((double) (x + 1.0)));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.4
Target0.3
Herbie1.5
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0

    1. Initial program Error: 64.0 bits

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
    2. Using strategy rm

    3. Applied div-subError: 64.0 bits

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

    4. SimplifiedError: 4.7 bits

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

    5. SimplifiedError: 4.7 bits

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

    if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 2.95372733554400404e286

    1. Initial program Error: 0.8 bits

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

    if 2.95372733554400404e286 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

    1. Initial program Error: 62.1 bits

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

    2. Taylor expanded around inf Error: 8.1 bits

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplificationError: 1.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;\frac{x + \left(z \cdot \frac{y}{z \cdot t - x} - \frac{x}{z \cdot t - x}\right)}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2.953727335544004 \cdot 10^{+286}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))