Average Error: 7.6 → 3.7
Time: 4.2s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 1.2889474225732965 \cdot 10^{-305} \lor \neg \left(z \le 3.938006923925878 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le 1.2889474225732965 \cdot 10^{-305} \lor \neg \left(z \le 3.938006923925878 \cdot 10^{-109}\right):\\
\;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\

\end{array}
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 VAR;
	if (((z <= 1.2889474225732965e-305) || !(z <= 3.938006923925878e-109))) {
		VAR = (1.0 / ((x + 1.0) / (fma((y / ((t * z) - x)), z, x) - (x / ((t * z) - x)))));
	} else {
		VAR = ((x + (((y * z) - x) * (1.0 / ((t * z) - x)))) / (x + 1.0));
	}
	return VAR;
}

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

Original7.6
Target0.3
Herbie3.7
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 1.2889474225732965e-305 or 3.938006923925878e-109 < z

    1. Initial program 9.0

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

    3. Applied div-sub9.0

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

    4. Applied associate-+r-9.0

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

    5. Simplified4.4

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

    7. Applied clear-num4.4

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

    if 1.2889474225732965e-305 < z < 3.938006923925878e-109

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 1.2889474225732965 \cdot 10^{-305} \lor \neg \left(z \le 3.938006923925878 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{1}{\frac{x + 1}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(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))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))