Average Error: 3.9 → 1.7
Time: 8.7s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 3.92873909809889365 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z \cdot \sqrt{t + a}}{t}\right) + \left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt{t}}, \frac{\sqrt{t + a}}{\sqrt{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]
\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
↓
\begin{array}{l}
\mathbf{if}\;t \le 3.92873909809889365 \cdot 10^{-219}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z \cdot \sqrt{t + a}}{t}\right) + \left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt{t}}, \frac{\sqrt{t + a}}{\sqrt{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\

\end{array}
double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (((double) (z * ((double) sqrt(((double) (t + a)))))) / t)) - ((double) (((double) (b - c)) * ((double) (((double) (a + ((double) (5.0 / 6.0)))) - ((double) (2.0 / ((double) (t * 3.0))))))))))))))))))));
}
↓
double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((t <= 3.9287390980988937e-219)) {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) fma(((double) (a + ((double) (((double) (5.0 / 6.0)) - ((double) (2.0 / ((double) (t * 3.0)))))))), ((double) -(((double) (b - c)))), ((double) (((double) (z * ((double) sqrt(((double) (t + a)))))) / t)))) + ((double) (((double) (a + ((double) (((double) (5.0 / 6.0)) - ((double) (2.0 / ((double) (t * 3.0)))))))) * ((double) (((double) -(((double) (b - c)))) + ((double) (b - c))))))))))))))))));
	} else {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) fma(((double) (z / ((double) sqrt(t)))), ((double) (((double) sqrt(((double) (t + a)))) / ((double) sqrt(t)))), ((double) -(((double) (((double) (b - c)) * ((double) (((double) (a + ((double) (5.0 / 6.0)))) - ((double) (2.0 / ((double) (t * 3.0))))))))))))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if t < 3.9287390980988937e-219

    1. Initial program 6.8

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
    2. Using strategy rm

    3. Applied log1p-expm1-u12.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)}\right)\right)}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt36.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\sqrt{\frac{z \cdot \sqrt{t + a}}{t}} \cdot \sqrt{\frac{z \cdot \sqrt{t + a}}{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right)\right)}}\]

    6. Applied prod-diff61.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{z \cdot \sqrt{t + a}}{t}}, \sqrt{\frac{z \cdot \sqrt{t + a}}{t}}, -\left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right) \cdot \left(b - c\right)\right) + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right), b - c, \left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right) \cdot \left(b - c\right)\right)\right)}}}\]

    7. Simplified58.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z \cdot \sqrt{t + a}}{t}\right)} + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right), b - c, \left(\left(a + \frac{5}{6}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{t \cdot 3}\right)\right)\right) \cdot \left(b - c\right)\right)\right)}}\]

    8. Simplified4.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z \cdot \sqrt{t + a}}{t}\right) + \color{blue}{\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)}\right)}}\]

    if 3.9287390980988937e-219 < t

    1. Initial program 2.4

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt2.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\sqrt{t} \cdot \sqrt{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    4. Applied times-frac1.0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt{t}} \cdot \frac{\sqrt{t + a}}{\sqrt{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    5. Applied fma-neg0.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{\sqrt{t}}, \frac{\sqrt{t + a}}{\sqrt{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.92873909809889365 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right), -\left(b - c\right), \frac{z \cdot \sqrt{t + a}}{t}\right) + \left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt{t}}, \frac{\sqrt{t + a}}{\sqrt{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))