Average Error: 7.3 → 2.2
Time: 3.8s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 6.8077680963010561 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(\frac{z}{t \cdot z - x} \cdot y + x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
\mathbf{if}\;z \le 6.8077680963010561 \cdot 10^{135}:\\
\;\;\;\;\frac{\left(\frac{z}{t \cdot z - x} \cdot y + x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \frac{y}{t}}{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 <= 6.807768096301056e+135)) {
		VAR = (((((z / ((t * z) - x)) * y) + x) - (x / ((t * z) - x))) / (x + 1.0));
	} else {
		VAR = ((x + (y / t)) / (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

Original7.3
Target0.4
Herbie2.2
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 6.807768096301056e+135

    1. Initial program 4.9

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

    3. Applied div-sub4.9

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

    4. Applied associate-+r-4.9

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

    5. Simplified4.0

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

    7. Applied clear-num4.1

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

    9. Applied fma-udef4.1

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

    10. Simplified1.5

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

    if 6.807768096301056e+135 < z

    1. Initial program 23.2

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

    2. Taylor expanded around inf 6.9

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 6.8077680963010561 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(\frac{z}{t \cdot z - x} \cdot y + x\right) - \frac{x}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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)))