Average Error: 1.9 → 1.4
Time: 21.5s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\left(x \cdot \frac{{\left(\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{1}{\sqrt[3]{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\left(x \cdot \frac{{\left(\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{1}{\sqrt[3]{a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}
double f(double x, double y, double z, double t, double a, double b) {
        double r414133 = x;
        double r414134 = y;
        double r414135 = z;
        double r414136 = log(r414135);
        double r414137 = r414134 * r414136;
        double r414138 = t;
        double r414139 = 1.0;
        double r414140 = r414138 - r414139;
        double r414141 = a;
        double r414142 = log(r414141);
        double r414143 = r414140 * r414142;
        double r414144 = r414137 + r414143;
        double r414145 = b;
        double r414146 = r414144 - r414145;
        double r414147 = exp(r414146);
        double r414148 = r414133 * r414147;
        double r414149 = r414148 / r414134;
        return r414149;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r414150 = x;
        double r414151 = 1.0;
        double r414152 = a;
        double r414153 = cbrt(r414152);
        double r414154 = r414153 * r414153;
        double r414155 = r414151 / r414154;
        double r414156 = 1.0;
        double r414157 = pow(r414155, r414156);
        double r414158 = y;
        double r414159 = z;
        double r414160 = r414151 / r414159;
        double r414161 = log(r414160);
        double r414162 = r414158 * r414161;
        double r414163 = r414151 / r414152;
        double r414164 = log(r414163);
        double r414165 = t;
        double r414166 = r414164 * r414165;
        double r414167 = b;
        double r414168 = r414166 + r414167;
        double r414169 = r414162 + r414168;
        double r414170 = exp(r414169);
        double r414171 = sqrt(r414170);
        double r414172 = r414157 / r414171;
        double r414173 = r414150 * r414172;
        double r414174 = r414151 / r414153;
        double r414175 = pow(r414174, r414156);
        double r414176 = r414175 / r414171;
        double r414177 = r414173 * r414176;
        double r414178 = r414177 / r414158;
        return r414178;
}

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

Target

Original1.9
Target10.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

  1. Initial program 1.9

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

  2. Taylor expanded around inf 1.9

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

  3. Simplified1.2

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

  5. Applied add-sqr-sqrt1.3

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

  6. Applied add-cube-cbrt1.4

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

  7. Applied *-un-lft-identity1.4

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

  8. Applied times-frac1.4

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

  9. Applied unpow-prod-down1.4

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

  10. Applied times-frac1.4

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

  11. Applied associate-*r*1.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.88458485041274715) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.22883740731) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))