Average Error: 2.1 → 0.6
Time: 45.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot {\left(e^{\sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)} \cdot \sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)}}\right)}^{\left(\sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)}\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot {\left(e^{\sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)} \cdot \sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)}}\right)}^{\left(\sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)}\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r7726108 = x;
        double r7726109 = y;
        double r7726110 = z;
        double r7726111 = log(r7726110);
        double r7726112 = t;
        double r7726113 = r7726111 - r7726112;
        double r7726114 = r7726109 * r7726113;
        double r7726115 = a;
        double r7726116 = 1.0;
        double r7726117 = r7726116 - r7726110;
        double r7726118 = log(r7726117);
        double r7726119 = b;
        double r7726120 = r7726118 - r7726119;
        double r7726121 = r7726115 * r7726120;
        double r7726122 = r7726114 + r7726121;
        double r7726123 = exp(r7726122);
        double r7726124 = r7726108 * r7726123;
        return r7726124;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r7726125 = x;
        double r7726126 = z;
        double r7726127 = log(r7726126);
        double r7726128 = t;
        double r7726129 = r7726127 - r7726128;
        double r7726130 = y;
        double r7726131 = r7726129 * r7726130;
        double r7726132 = 1.0;
        double r7726133 = log(r7726132);
        double r7726134 = r7726126 * r7726132;
        double r7726135 = r7726133 - r7726134;
        double r7726136 = a;
        double r7726137 = r7726135 * r7726136;
        double r7726138 = b;
        double r7726139 = r7726136 * r7726138;
        double r7726140 = r7726137 - r7726139;
        double r7726141 = r7726131 + r7726140;
        double r7726142 = cbrt(r7726141);
        double r7726143 = r7726142 * r7726142;
        double r7726144 = exp(r7726143);
        double r7726145 = pow(r7726144, r7726142);
        double r7726146 = r7726125 * r7726145;
        return r7726146;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.1

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(a \cdot \log 1 - \left(1 \cdot \left(a \cdot z\right) + a \cdot b\right)\right)}}\]

  3. Simplified0.6

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + \color{blue}{\left(a \cdot \left(\log 1 - 1 \cdot z\right) - b \cdot a\right)}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.6

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

  6. Applied exp-prod0.6

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

  7. Final simplification0.6

    \[\leadsto x \cdot {\left(e^{\sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)} \cdot \sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)}}\right)}^{\left(\sqrt[3]{\left(\log z - t\right) \cdot y + \left(\left(\log 1 - z \cdot 1\right) \cdot a - a \cdot b\right)}\right)}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))