Average Error: 7.5 → 2.5
Time: 10.8s
Precision: binary64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;1 + \frac{z}{x + 1} \cdot \left(\frac{t}{x} - \frac{y}{x}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 8.7209616902523 \cdot 10^{+264}:\\ \;\;\;\;\frac{x + \left(\frac{y \cdot z}{z \cdot t - x} - \frac{x}{z \cdot t - x}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + \frac{y}{t}\right) - \frac{x}{z \cdot t}}{x + 1}\\ \end{array}\]
\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:\\
\;\;\;\;1 + \frac{z}{x + 1} \cdot \left(\frac{t}{x} - \frac{y}{x}\right)\\

\mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 8.7209616902523 \cdot 10^{+264}:\\
\;\;\;\;\frac{x + \left(\frac{y \cdot z}{z \cdot t - x} - \frac{x}{z \cdot t - x}\right)}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x + \frac{y}{t}\right) - \frac{x}{z \cdot 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))
   (+ 1.0 (* (/ z (+ x 1.0)) (- (/ t x) (/ y x))))
   (if (<=
        (/ (+ x (/ (- (* y z) x) (- (* z t) x))) (+ x 1.0))
        8.7209616902523e+264)
     (/ (+ x (- (/ (* y z) (- (* z t) x)) (/ x (- (* z t) x)))) (+ x 1.0))
     (/ (- (+ x (/ y t)) (/ x (* z t))) (+ x 1.0)))))
double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= -((double) INFINITY)) {
		tmp = 1.0 + ((z / (x + 1.0)) * ((t / x) - (y / x)));
	} else if (((x + (((y * z) - x) / ((z * t) - x))) / (x + 1.0)) <= 8.7209616902523e+264) {
		tmp = (x + (((y * z) / ((z * t) - x)) - (x / ((z * t) - x)))) / (x + 1.0);
	} else {
		tmp = ((x + (y / t)) - (x / (z * t))) / (x + 1.0);
	}
	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

Original7.5
Target0.3
Herbie2.5
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -inf.0

    1. Initial program 64.0

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

    2. Taylor expanded around 0 64.0

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

    3. Simplified38.0

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

    if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 8.72096169025229998e264

    1. Initial program 0.6

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

    3. Applied div-sub_binary64_188380.6

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

    if 8.72096169025229998e264 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

    1. Initial program 60.2

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

      \[\leadsto \frac{\color{blue}{\left(x + \frac{y}{t}\right) - \frac{x}{t \cdot z}}}{x + 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty:\\ \;\;\;\;1 + \frac{z}{x + 1} \cdot \left(\frac{t}{x} - \frac{y}{x}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 8.7209616902523 \cdot 10^{+264}:\\ \;\;\;\;\frac{x + \left(\frac{y \cdot z}{z \cdot t - x} - \frac{x}{z \cdot t - x}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + \frac{y}{t}\right) - \frac{x}{z \cdot t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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)))