Average Error: 2.0 → 1.0
Time: 33.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{{\left(\frac{\sqrt{1}}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y} \cdot \left(\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right)\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\frac{{\left(\frac{\sqrt{1}}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y} \cdot \left(\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r144121 = x;
        double r144122 = y;
        double r144123 = z;
        double r144124 = log(r144123);
        double r144125 = r144122 * r144124;
        double r144126 = t;
        double r144127 = 1.0;
        double r144128 = r144126 - r144127;
        double r144129 = a;
        double r144130 = log(r144129);
        double r144131 = r144128 * r144130;
        double r144132 = r144125 + r144131;
        double r144133 = b;
        double r144134 = r144132 - r144133;
        double r144135 = exp(r144134);
        double r144136 = r144121 * r144135;
        double r144137 = r144136 / r144122;
        return r144137;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r144138 = 1.0;
        double r144139 = sqrt(r144138);
        double r144140 = a;
        double r144141 = sqrt(r144140);
        double r144142 = r144139 / r144141;
        double r144143 = 1.0;
        double r144144 = pow(r144142, r144143);
        double r144145 = y;
        double r144146 = z;
        double r144147 = r144138 / r144146;
        double r144148 = log(r144147);
        double r144149 = r144138 / r144140;
        double r144150 = log(r144149);
        double r144151 = t;
        double r144152 = b;
        double r144153 = fma(r144150, r144151, r144152);
        double r144154 = fma(r144145, r144148, r144153);
        double r144155 = exp(r144154);
        double r144156 = sqrt(r144155);
        double r144157 = r144144 / r144156;
        double r144158 = r144157 / r144145;
        double r144159 = sqrt(r144149);
        double r144160 = pow(r144159, r144143);
        double r144161 = r144160 / r144156;
        double r144162 = x;
        double r144163 = r144161 * r144162;
        double r144164 = r144158 * r144163;
        return r144164;
}

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

Derivation

  1. Initial program 2.0

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

  2. Taylor expanded around inf 2.0

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

  3. Simplified6.2

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

  5. Applied div-inv6.2

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

  6. Applied add-sqr-sqrt6.2

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

  7. Applied add-sqr-sqrt6.2

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

  8. Applied add-sqr-sqrt6.2

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

  9. Applied times-frac6.2

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

  10. Applied unpow-prod-down6.2

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

  11. Applied times-frac6.2

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

  12. Applied times-frac1.0

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

  13. Simplified1.0

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

  15. Applied sqrt-undiv1.0

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

  16. Final simplification1.0

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))