Average Error: 4.0 → 5.1
Time: 9.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.0159840576688351 \cdot 10^{48}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 4.062741574579635 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \frac{t}{\sqrt{t + a}} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]
\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 -2.0159840576688351 \cdot 10^{48}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\

\mathbf{elif}\;t \le 4.062741574579635 \cdot 10^{-103}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \frac{t}{\sqrt{t + a}} \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)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r375975 = x;
        double r375976 = y;
        double r375977 = 2.0;
        double r375978 = z;
        double r375979 = t;
        double r375980 = a;
        double r375981 = r375979 + r375980;
        double r375982 = sqrt(r375981);
        double r375983 = r375978 * r375982;
        double r375984 = r375983 / r375979;
        double r375985 = b;
        double r375986 = c;
        double r375987 = r375985 - r375986;
        double r375988 = 5.0;
        double r375989 = 6.0;
        double r375990 = r375988 / r375989;
        double r375991 = r375980 + r375990;
        double r375992 = 3.0;
        double r375993 = r375979 * r375992;
        double r375994 = r375977 / r375993;
        double r375995 = r375991 - r375994;
        double r375996 = r375987 * r375995;
        double r375997 = r375984 - r375996;
        double r375998 = r375977 * r375997;
        double r375999 = exp(r375998);
        double r376000 = r375976 * r375999;
        double r376001 = r375975 + r376000;
        double r376002 = r375975 / r376001;
        return r376002;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r376003 = t;
        double r376004 = -2.015984057668835e+48;
        bool r376005 = r376003 <= r376004;
        double r376006 = x;
        double r376007 = y;
        double r376008 = 2.0;
        double r376009 = a;
        double r376010 = c;
        double r376011 = r376009 * r376010;
        double r376012 = 0.8333333333333334;
        double r376013 = r376012 * r376010;
        double r376014 = r376011 + r376013;
        double r376015 = b;
        double r376016 = r376009 * r376015;
        double r376017 = r376014 - r376016;
        double r376018 = r376008 * r376017;
        double r376019 = exp(r376018);
        double r376020 = r376007 * r376019;
        double r376021 = r376006 + r376020;
        double r376022 = r376006 / r376021;
        double r376023 = 4.062741574579635e-103;
        bool r376024 = r376003 <= r376023;
        double r376025 = z;
        double r376026 = 5.0;
        double r376027 = 6.0;
        double r376028 = r376026 / r376027;
        double r376029 = r376009 - r376028;
        double r376030 = 3.0;
        double r376031 = r376003 * r376030;
        double r376032 = r376029 * r376031;
        double r376033 = r376025 * r376032;
        double r376034 = r376003 + r376009;
        double r376035 = sqrt(r376034);
        double r376036 = r376003 / r376035;
        double r376037 = r376015 - r376010;
        double r376038 = r376009 * r376009;
        double r376039 = r376028 * r376028;
        double r376040 = r376038 - r376039;
        double r376041 = r376040 * r376031;
        double r376042 = r376029 * r376008;
        double r376043 = r376041 - r376042;
        double r376044 = r376037 * r376043;
        double r376045 = r376036 * r376044;
        double r376046 = r376033 - r376045;
        double r376047 = r376036 * r376032;
        double r376048 = r376046 / r376047;
        double r376049 = r376008 * r376048;
        double r376050 = exp(r376049);
        double r376051 = r376007 * r376050;
        double r376052 = r376006 + r376051;
        double r376053 = r376006 / r376052;
        double r376054 = r376025 / r376036;
        double r376055 = r376009 + r376028;
        double r376056 = r376008 / r376031;
        double r376057 = 3.0;
        double r376058 = pow(r376056, r376057);
        double r376059 = 0.3333333333333333;
        double r376060 = pow(r376058, r376059);
        double r376061 = r376055 - r376060;
        double r376062 = r376037 * r376061;
        double r376063 = r376054 - r376062;
        double r376064 = r376008 * r376063;
        double r376065 = exp(r376064);
        double r376066 = r376007 * r376065;
        double r376067 = r376006 + r376066;
        double r376068 = r376006 / r376067;
        double r376069 = r376024 ? r376053 : r376068;
        double r376070 = r376005 ? r376022 : r376069;
        return r376070;
}

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

Original4.0
Target3.1
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -2.015984057668835e+48

    1. Initial program 5.3

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

    2. Taylor expanded around inf 7.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}}\]

    if -2.015984057668835e+48 < t < 4.062741574579635e-103

    1. Initial program 6.4

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

    3. Applied associate-/l*7.6

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

    5. Applied flip-+10.2

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

    6. Applied frac-sub10.6

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

    7. Applied associate-*r/10.6

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

    8. Applied frac-sub11.4

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

    if 4.062741574579635e-103 < t

    1. Initial program 2.1

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

    3. Applied associate-/l*0.3

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

    5. Applied add-cbrt-cube0.3

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

    6. Applied add-cbrt-cube0.3

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

    7. Applied cbrt-unprod0.3

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

    8. Applied add-cbrt-cube0.3

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

    9. Applied cbrt-undiv0.3

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

    10. Simplified0.3

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

    12. Applied pow1/30.3

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.0159840576688351 \cdot 10^{48}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 4.062741574579635 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \frac{t}{\sqrt{t + a}} \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\frac{t}{\sqrt{t + a}} \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-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)))))))))))