Average Error: 3.8 → 1.9
Time: 7.2s
Precision: binary64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -3.433275005074214 \cdot 10^{-144} \lor \neg \left(t \leq 2.5292981124674526 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right) - 2 \cdot \left(a - \frac{5}{6}\right)\right)\right)}{t \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right)}}}\\ \end{array}\]
\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 -3.433275005074214 \cdot 10^{-144} \lor \neg \left(t \leq 2.5292981124674526 \cdot 10^{-127}\right):\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\

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

\end{array}
(FPCore (x y z t a b c)
 :precision binary64
 (/
  x
  (+
   x
   (*
    y
    (exp
     (*
      2.0
      (-
       (/ (* z (sqrt (+ t a))) t)
       (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))
↓
(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= t -3.433275005074214e-144) (not (<= t 2.5292981124674526e-127)))
   (/
    x
    (+
     x
     (*
      y
      (exp
       (*
        2.0
        (-
         (* (/ z (* (cbrt t) (cbrt t))) (/ (sqrt (+ t a)) (cbrt t)))
         (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))
   (/
    x
    (+
     x
     (*
      y
      (exp
       (*
        2.0
        (/
         (-
          (* (* z (sqrt (+ t a))) (* (* t 3.0) (- a (/ 5.0 6.0))))
          (*
           t
           (*
            (- b c)
            (-
             (* (+ a (/ 5.0 6.0)) (* (* t 3.0) (- a (/ 5.0 6.0))))
             (* 2.0 (- a (/ 5.0 6.0)))))))
         (* t (* (* t 3.0) (- a (/ 5.0 6.0))))))))))))
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 tmp;
	if (((t <= -3.433275005074214e-144) || !(t <= 2.5292981124674526e-127))) {
		tmp = (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) ((z / ((double) (((double) cbrt(t)) * ((double) cbrt(t))))) * (((double) sqrt(((double) (t + a)))) / ((double) cbrt(t))))) - ((double) (((double) (b - c)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))))))))))));
	} else {
		tmp = (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * (((double) (((double) (((double) (z * ((double) sqrt(((double) (t + a)))))) * ((double) (((double) (t * 3.0)) * ((double) (a - (5.0 / 6.0))))))) - ((double) (t * ((double) (((double) (b - c)) * ((double) (((double) (((double) (a + (5.0 / 6.0))) * ((double) (((double) (t * 3.0)) * ((double) (a - (5.0 / 6.0))))))) - ((double) (2.0 * ((double) (a - (5.0 / 6.0))))))))))))) / ((double) (t * ((double) (((double) (t * 3.0)) * ((double) (a - (5.0 / 6.0)))))))))))))))));
	}
	return tmp;
}

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

Original3.8
Target3.1
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.43327500507421395e-144 or 2.5292981124674526e-127 < t

    1. Initial program 2.3

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

    3. Applied add-cube-cbrt_binary642.3

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

    4. Applied times-frac_binary640.6

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

    if -3.43327500507421395e-144 < t < 2.5292981124674526e-127

    1. Initial program 7.3

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

    3. Applied flip-+_binary6411.3

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

    4. Applied frac-sub_binary6411.3

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

    5. Applied associate-*r/_binary6411.3

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

    6. Applied frac-sub_binary648.4

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

    7. Simplified4.8

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

    8. Simplified4.8

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.433275005074214 \cdot 10^{-144} \lor \neg \left(t \leq 2.5292981124674526 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right) - 2 \cdot \left(a - \frac{5}{6}\right)\right)\right)}{t \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))))))))))