Average Error: 4.0 → 2.8
Time: 56.8s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.235874327363358100249014052036578048109 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{x + e^{\log \left(e^{\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\frac{5}{6} + \left(a - \frac{2}{t \cdot 3}\right)\right)}\right) \cdot 2} \cdot y}\\ \mathbf{elif}\;t \le 3.340725403067544451185955686851105101177 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{x + e^{2 \cdot \frac{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot \left(\sqrt{t + a} \cdot z\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot t}} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\log \left(e^{\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\frac{5}{6} + \left(a - \frac{2}{t \cdot 3}\right)\right)}\right) \cdot 2} \cdot y}\\ \end{array}\]
\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 -6.235874327363358100249014052036578048109 \cdot 10^{-226}:\\
\;\;\;\;\frac{x}{x + e^{\log \left(e^{\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\frac{5}{6} + \left(a - \frac{2}{t \cdot 3}\right)\right)}\right) \cdot 2} \cdot y}\\

\mathbf{elif}\;t \le 3.340725403067544451185955686851105101177 \cdot 10^{-267}:\\
\;\;\;\;\frac{x}{x + e^{2 \cdot \frac{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot \left(\sqrt{t + a} \cdot z\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)}{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot t}} \cdot y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3443068 = x;
        double r3443069 = y;
        double r3443070 = 2.0;
        double r3443071 = z;
        double r3443072 = t;
        double r3443073 = a;
        double r3443074 = r3443072 + r3443073;
        double r3443075 = sqrt(r3443074);
        double r3443076 = r3443071 * r3443075;
        double r3443077 = r3443076 / r3443072;
        double r3443078 = b;
        double r3443079 = c;
        double r3443080 = r3443078 - r3443079;
        double r3443081 = 5.0;
        double r3443082 = 6.0;
        double r3443083 = r3443081 / r3443082;
        double r3443084 = r3443073 + r3443083;
        double r3443085 = 3.0;
        double r3443086 = r3443072 * r3443085;
        double r3443087 = r3443070 / r3443086;
        double r3443088 = r3443084 - r3443087;
        double r3443089 = r3443080 * r3443088;
        double r3443090 = r3443077 - r3443089;
        double r3443091 = r3443070 * r3443090;
        double r3443092 = exp(r3443091);
        double r3443093 = r3443069 * r3443092;
        double r3443094 = r3443068 + r3443093;
        double r3443095 = r3443068 / r3443094;
        return r3443095;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3443096 = t;
        double r3443097 = -6.235874327363358e-226;
        bool r3443098 = r3443096 <= r3443097;
        double r3443099 = x;
        double r3443100 = z;
        double r3443101 = a;
        double r3443102 = r3443096 + r3443101;
        double r3443103 = sqrt(r3443102);
        double r3443104 = r3443096 / r3443103;
        double r3443105 = r3443100 / r3443104;
        double r3443106 = b;
        double r3443107 = c;
        double r3443108 = r3443106 - r3443107;
        double r3443109 = 5.0;
        double r3443110 = 6.0;
        double r3443111 = r3443109 / r3443110;
        double r3443112 = 2.0;
        double r3443113 = 3.0;
        double r3443114 = r3443096 * r3443113;
        double r3443115 = r3443112 / r3443114;
        double r3443116 = r3443101 - r3443115;
        double r3443117 = r3443111 + r3443116;
        double r3443118 = r3443108 * r3443117;
        double r3443119 = r3443105 - r3443118;
        double r3443120 = exp(r3443119);
        double r3443121 = log(r3443120);
        double r3443122 = r3443121 * r3443112;
        double r3443123 = exp(r3443122);
        double r3443124 = y;
        double r3443125 = r3443123 * r3443124;
        double r3443126 = r3443099 + r3443125;
        double r3443127 = r3443099 / r3443126;
        double r3443128 = 3.3407254030675445e-267;
        bool r3443129 = r3443096 <= r3443128;
        double r3443130 = r3443101 - r3443111;
        double r3443131 = r3443130 * r3443114;
        double r3443132 = r3443103 * r3443100;
        double r3443133 = r3443131 * r3443132;
        double r3443134 = r3443101 * r3443101;
        double r3443135 = r3443111 * r3443111;
        double r3443136 = r3443134 - r3443135;
        double r3443137 = r3443136 * r3443114;
        double r3443138 = r3443130 * r3443112;
        double r3443139 = r3443137 - r3443138;
        double r3443140 = r3443108 * r3443139;
        double r3443141 = r3443096 * r3443140;
        double r3443142 = r3443133 - r3443141;
        double r3443143 = r3443131 * r3443096;
        double r3443144 = r3443142 / r3443143;
        double r3443145 = r3443112 * r3443144;
        double r3443146 = exp(r3443145);
        double r3443147 = r3443146 * r3443124;
        double r3443148 = r3443099 + r3443147;
        double r3443149 = r3443099 / r3443148;
        double r3443150 = r3443129 ? r3443149 : r3443127;
        double r3443151 = r3443098 ? r3443127 : r3443150;
        return r3443151;
}

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 < -6.235874327363358e-226 or 3.3407254030675445e-267 < t

    1. Initial program 3.3

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

    3. Applied add-cbrt-cube3.3

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

    4. Applied add-cbrt-cube6.3

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

    5. Applied cbrt-unprod6.3

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

    6. Applied add-cbrt-cube6.3

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

    7. Applied cbrt-undiv6.4

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

    8. Simplified6.4

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

    10. Applied add-log-exp8.9

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

    11. Applied add-log-exp16.9

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

    12. Applied diff-log16.9

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

    13. Simplified2.0

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

    if -6.235874327363358e-226 < t < 3.3407254030675445e-267

    1. Initial program 10.6

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

    3. Applied flip-+14.0

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

    4. Applied frac-sub14.0

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

    5. Applied associate-*r/14.0

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

    6. Applied frac-sub10.2

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.235874327363358100249014052036578048109 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{x + e^{\log \left(e^{\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\frac{5}{6} + \left(a - \frac{2}{t \cdot 3}\right)\right)}\right) \cdot 2} \cdot y}\\ \mathbf{elif}\;t \le 3.340725403067544451185955686851105101177 \cdot 10^{-267}:\\ \;\;\;\;\frac{x}{x + e^{2 \cdot \frac{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot \left(\sqrt{t + a} \cdot z\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot t}} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\log \left(e^{\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\frac{5}{6} + \left(a - \frac{2}{t \cdot 3}\right)\right)}\right) \cdot 2} \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))