Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 2.0 → 1.2

Time: 19.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.238363381960752745181717665585087475551 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \frac{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{\frac{y}{\sqrt[3]{x}}}\\ \mathbf{elif}\;y \le 9.283546491991937675627216509405561605054 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{\frac{y}{x}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r113653 = x;
        double r113654 = y;
        double r113655 = z;
        double r113656 = log(r113655);
        double r113657 = r113654 * r113656;
        double r113658 = t;
        double r113659 = 1.0;
        double r113660 = r113658 - r113659;
        double r113661 = a;
        double r113662 = log(r113661);
        double r113663 = r113660 * r113662;
        double r113664 = r113657 + r113663;
        double r113665 = b;
        double r113666 = r113664 - r113665;
        double r113667 = exp(r113666);
        double r113668 = r113653 * r113667;
        double r113669 = r113668 / r113654;
        return r113669;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r113670 = y;
        double r113671 = -2.2383633819607527e-257;
        bool r113672 = r113670 <= r113671;
        double r113673 = 1.0;
        double r113674 = a;
        double r113675 = sqrt(r113674);
        double r113676 = r113673 / r113675;
        double r113677 = 1.0;
        double r113678 = pow(r113676, r113677);
        double r113679 = z;
        double r113680 = r113673 / r113679;
        double r113681 = log(r113680);
        double r113682 = r113673 / r113674;
        double r113683 = log(r113682);
        double r113684 = t;
        double r113685 = b;
        double r113686 = fma(r113683, r113684, r113685);
        double r113687 = fma(r113670, r113681, r113686);
        double r113688 = exp(r113687);
        double r113689 = sqrt(r113688);
        double r113690 = r113678 / r113689;
        double r113691 = x;
        double r113692 = cbrt(r113691);
        double r113693 = r113692 * r113692;
        double r113694 = r113690 * r113693;
        double r113695 = r113670 / r113692;
        double r113696 = r113690 / r113695;
        double r113697 = r113694 * r113696;
        double r113698 = 9.283546491991938e-73;
        bool r113699 = r113670 <= r113698;
        double r113700 = pow(r113682, r113677);
        double r113701 = r113700 / r113688;
        double r113702 = r113691 * r113701;
        double r113703 = r113670 / r113702;
        double r113704 = r113673 / r113703;
        double r113705 = r113670 / r113691;
        double r113706 = r113701 / r113705;
        double r113707 = r113699 ? r113704 : r113706;
        double r113708 = r113672 ? r113697 : r113707;
        return r113708;
}

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. Split input into 3 regimes

2. if y < -2.2383633819607527e-257

   1. Initial program 1.5

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

   2. Taylor expanded around inf 1.5

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

   3. Simplified 5.0

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

   5. Applied add-cube-cbrt 5.1

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

   6. Applied *-un-lft-identity 5.1

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

   7. Applied times-frac 5.1

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

   8. Applied add-sqr-sqrt 5.1

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

   9. Applied add-sqr-sqrt 5.1

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

   10. Applied *-un-lft-identity 5.1

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

   11. Applied times-frac 5.1

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

   12. Applied unpow-prod-down 5.1

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

   13. Applied times-frac 5.1

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

   14. Applied times-frac 0.5

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

   15. Simplified 0.5

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

   if -2.2383633819607527e-257 < y < 9.283546491991938e-73

   1. Initial program 4.7

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

   2. Taylor expanded around inf 4.7

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

   3. Simplified 3.4

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

   5. Applied clear-num 3.4

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

   if 9.283546491991938e-73 < y

   1. Initial program 0.6

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

   2. Taylor expanded around inf 0.6

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

   3. Simplified 0.5

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{\frac{y}{x}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.238363381960752745181717665585087475551 \cdot 10^{-257}:\\ \;\;\;\;\left(\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \frac{\frac{{\left(\frac{1}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{\frac{y}{\sqrt[3]{x}}}\\ \mathbf{elif}\;y \le 9.283546491991937675627216509405561605054 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{\frac{y}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +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))