Average Error: 7.7 → 3.9
Time: 4.1s
Precision: binary64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.7805821512846415 \cdot 10^{+125} \lor \neg \left(z \leq 2.2177898291412525 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\left(z \cdot t - x\right) \cdot \frac{1}{z \cdot y - x}}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -3.7805821512846415 \cdot 10^{+125} \lor \neg \left(z \leq 2.2177898291412525 \cdot 10^{+185}\right):\\
\;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{1}{\left(z \cdot t - x\right) \cdot \frac{1}{z \cdot y - x}}}{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 (or (<= z -3.7805821512846415e+125) (not (<= z 2.2177898291412525e+185)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (+ x (/ 1.0 (* (- (* z t) x) (/ 1.0 (- (* z y) x))))) (+ 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 ((z <= -3.7805821512846415e+125) || !(z <= 2.2177898291412525e+185)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x + (1.0 / (((z * t) - x) * (1.0 / ((z * y) - x))))) / (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

Original7.7
Target0.4
Herbie3.9
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.7805821512846415e125 or 2.21778982914125251e185 < z

    1. Initial program 22.3

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

    2. Taylor expanded around inf 6.5

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]

    if -3.7805821512846415e125 < z < 2.21778982914125251e185

    1. Initial program 2.9

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

    3. Applied clear-num_binary643.0

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

    4. Simplified3.0

      \[\leadsto \frac{x + \frac{1}{\color{blue}{\frac{z \cdot t - x}{y \cdot z - x}}}}{x + 1}\]
    5. Using strategy rm

    6. Applied div-inv_binary643.0

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

    7. Simplified3.0

      \[\leadsto \frac{x + \frac{1}{\left(z \cdot t - x\right) \cdot \color{blue}{\frac{1}{z \cdot y - x}}}}{x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7805821512846415 \cdot 10^{+125} \lor \neg \left(z \leq 2.2177898291412525 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\left(z \cdot t - x\right) \cdot \frac{1}{z \cdot y - x}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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)))