Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 2.1 → 26.9

Time: 37.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r259297 = x;
        double r259298 = y;
        double r259299 = z;
        double r259300 = log(r259299);
        double r259301 = r259298 * r259300;
        double r259302 = t;
        double r259303 = 1.0;
        double r259304 = r259302 - r259303;
        double r259305 = a;
        double r259306 = log(r259305);
        double r259307 = r259304 * r259306;
        double r259308 = r259301 + r259307;
        double r259309 = b;
        double r259310 = r259308 - r259309;
        double r259311 = exp(r259310);
        double r259312 = r259297 * r259311;
        double r259313 = r259312 / r259298;
        return r259313;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r259314 = 1.0;
        double r259315 = a;
        double r259316 = r259314 / r259315;
        double r259317 = 1.0;
        double r259318 = pow(r259316, r259317);
        double r259319 = y;
        double r259320 = r259318 / r259319;
        double r259321 = x;
        double r259322 = r259320 * r259321;
        double r259323 = z;
        double r259324 = r259314 / r259323;
        double r259325 = pow(r259324, r259319);
        double r259326 = b;
        double r259327 = exp(r259326);
        double r259328 = r259325 * r259327;
        double r259329 = t;
        double r259330 = pow(r259315, r259329);
        double r259331 = r259328 / r259330;
        double r259332 = r259322 / r259331;
        return r259332;
}

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

Target

Original	2.1
Target	11.2
Herbie	26.9

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

Derivation

1. Split input into 2 regimes

2. if a < 2.018699136318795e+64

   1. Initial program 0.8

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

   2. Taylor expanded around inf 0.8

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

   3. Simplified 0.2

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

   5. Applied div-inv 0.2

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

   7. Applied add-sqr-sqrt 0.2

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

   8. Applied associate-*r* 0.2

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

   if 2.018699136318795e+64 < a

   1. Initial program 3.8

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

   2. Taylor expanded around inf 3.8

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

   3. Simplified 3.0

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

   5. Applied div-inv 3.0

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

   7. Applied add-sqr-sqrt 3.1

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

   8. Applied associate-*r* 3.1

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

   10. Applied sqrt-div 3.1

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

   11. Applied sqrt-div 3.1

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

   12. Applied associate-*r/ 3.1

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

   13. Applied frac-times 3.1

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

   14. Applied associate-*l/ 4.3

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

   15. Simplified 1.3

       \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y}{{\left(\frac{1}{a}\right)}^{1}}}}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9

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

Reproduce

herbie shell --seed 2019294 
(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))