Average Error: 4.3 → 3.6
Time: 19.1s
Precision: binary64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -60120.550694716185 \lor \neg \left(t \leq 8.362446977804171 \cdot 10^{-141}\right):\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) + t \cdot \left(3 \cdot \left(\frac{5}{6} \cdot \frac{5}{6} - a \cdot a\right)\right)\right)\right)}{t \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)}\right)}}\\ \end{array}\]
\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 \leq -60120.550694716185 \lor \neg \left(t \leq 8.362446977804171 \cdot 10^{-141}\right):\\
\;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}}\\

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

\end{array}
double code(double x, double y, double z, double t, double a, double b, double c) {
	return (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) ((((double) (z * ((double) sqrt(((double) (t + a)))))) / t) - ((double) (((double) (b - c)) * ((double) (((double) (a + (5.0 / 6.0))) - (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 <= -60120.550694716185) || !(t <= 8.362446977804171e-141))) {
		VAR = (x / ((double) (x + ((double) (y * ((double) pow(((double) exp(2.0)), ((double) (((double) (z * (((double) sqrt(((double) (t + a)))) / t))) + ((double) (((double) (b - c)) * ((double) ((2.0 / ((double) (t * 3.0))) - ((double) (a + (5.0 / 6.0))))))))))))))));
	} else {
		VAR = (x / ((double) (x + ((double) (y * ((double) pow(((double) exp(2.0)), (((double) (((double) (z * ((double) (((double) sqrt(((double) (t + a)))) * ((double) (t * ((double) (3.0 * ((double) (a - (5.0 / 6.0))))))))))) + ((double) (t * ((double) (((double) (b - c)) * ((double) (((double) (2.0 * ((double) (a - (5.0 / 6.0))))) + ((double) (t * ((double) (3.0 * ((double) (((double) ((5.0 / 6.0) * (5.0 / 6.0))) - ((double) (a * a)))))))))))))))) / ((double) (t * ((double) (t * ((double) (3.0 * ((double) (a - (5.0 / 6.0)))))))))))))))));
	}
	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

Target

Original4.3
Target3.1
Herbie3.6
\[\begin{array}{l} \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -60120.5506947161848 or 8.36244697780417073e-141 < t

    1. Initial program 2.9

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

    2. Simplified0.7

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

    if -60120.5506947161848 < t < 8.36244697780417073e-141

    1. Initial program 7.0

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

    2. Simplified8.8

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

    4. Applied flip-+11.6

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

    5. Applied frac-sub11.6

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

    6. Applied associate-*r/11.6

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

    7. Applied associate-*r/10.1

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

    8. Applied frac-add7.5

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

    9. Simplified9.4

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

    10. Simplified9.4

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

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -60120.550694716185 \lor \neg \left(t \leq 8.362446977804171 \cdot 10^{-141}\right):\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) + t \cdot \left(3 \cdot \left(\frac{5}{6} \cdot \frac{5}{6} - a \cdot a\right)\right)\right)\right)}{t \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))