Average Error: 3.8 → 3.0
Time: 30.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.333804143538084395085776415371440122739 \cdot 10^{-42} \lor \neg \left(t \le 1.029580973042603122977889054947443375466 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \frac{\sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]
\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 -1.333804143538084395085776415371440122739 \cdot 10^{-42} \lor \neg \left(t \le 1.029580973042603122977889054947443375466 \cdot 10^{-146}\right):\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \frac{\sqrt{t + a}}{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(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)}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r325776 = x;
        double r325777 = y;
        double r325778 = 2.0;
        double r325779 = z;
        double r325780 = t;
        double r325781 = a;
        double r325782 = r325780 + r325781;
        double r325783 = sqrt(r325782);
        double r325784 = r325779 * r325783;
        double r325785 = r325784 / r325780;
        double r325786 = b;
        double r325787 = c;
        double r325788 = r325786 - r325787;
        double r325789 = 5.0;
        double r325790 = 6.0;
        double r325791 = r325789 / r325790;
        double r325792 = r325781 + r325791;
        double r325793 = 3.0;
        double r325794 = r325780 * r325793;
        double r325795 = r325778 / r325794;
        double r325796 = r325792 - r325795;
        double r325797 = r325788 * r325796;
        double r325798 = r325785 - r325797;
        double r325799 = r325778 * r325798;
        double r325800 = exp(r325799);
        double r325801 = r325777 * r325800;
        double r325802 = r325776 + r325801;
        double r325803 = r325776 / r325802;
        return r325803;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r325804 = t;
        double r325805 = -1.3338041435380844e-42;
        bool r325806 = r325804 <= r325805;
        double r325807 = 1.0295809730426031e-146;
        bool r325808 = r325804 <= r325807;
        double r325809 = !r325808;
        bool r325810 = r325806 || r325809;
        double r325811 = x;
        double r325812 = y;
        double r325813 = 2.0;
        double r325814 = z;
        double r325815 = a;
        double r325816 = r325804 + r325815;
        double r325817 = sqrt(r325816);
        double r325818 = r325817 / r325804;
        double r325819 = r325814 * r325818;
        double r325820 = b;
        double r325821 = c;
        double r325822 = r325820 - r325821;
        double r325823 = 5.0;
        double r325824 = 6.0;
        double r325825 = r325823 / r325824;
        double r325826 = r325815 + r325825;
        double r325827 = 3.0;
        double r325828 = r325804 * r325827;
        double r325829 = r325813 / r325828;
        double r325830 = r325826 - r325829;
        double r325831 = r325822 * r325830;
        double r325832 = r325819 - r325831;
        double r325833 = r325813 * r325832;
        double r325834 = exp(r325833);
        double r325835 = r325812 * r325834;
        double r325836 = r325811 + r325835;
        double r325837 = r325811 / r325836;
        double r325838 = r325814 * r325817;
        double r325839 = r325815 - r325825;
        double r325840 = r325839 * r325828;
        double r325841 = r325838 * r325840;
        double r325842 = r325815 * r325815;
        double r325843 = r325825 * r325825;
        double r325844 = r325842 - r325843;
        double r325845 = r325844 * r325828;
        double r325846 = r325839 * r325813;
        double r325847 = r325845 - r325846;
        double r325848 = r325822 * r325847;
        double r325849 = r325804 * r325848;
        double r325850 = r325841 - r325849;
        double r325851 = r325804 * r325840;
        double r325852 = r325850 / r325851;
        double r325853 = r325813 * r325852;
        double r325854 = exp(r325853);
        double r325855 = r325812 * r325854;
        double r325856 = r325811 + r325855;
        double r325857 = r325811 / r325856;
        double r325858 = r325810 ? r325837 : r325857;
        return r325858;
}

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
Target3.0
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.3338041435380844e-42 or 1.0295809730426031e-146 < 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 *-un-lft-identity2.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    4. Applied times-frac0.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    5. Simplified0.6

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

    if -1.3338041435380844e-42 < t < 1.0295809730426031e-146

    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. Using strategy rm

    3. Applied flip-+10.9

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

    4. Applied frac-sub10.9

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

    5. Applied associate-*r/10.9

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

    6. Applied frac-sub8.4

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.333804143538084395085776415371440122739 \cdot 10^{-42} \lor \neg \left(t \le 1.029580973042603122977889054947443375466 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(z \cdot \frac{\sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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 -2.1183266448915811e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.83333333333333337 c)) (* a b))))))) (if (< t 5.19658877065154709e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))