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

Average Error: 2.1 → 1.4

Time: 38.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r318299 = x;
        double r318300 = y;
        double r318301 = z;
        double r318302 = log(r318301);
        double r318303 = r318300 * r318302;
        double r318304 = t;
        double r318305 = 1.0;
        double r318306 = r318304 - r318305;
        double r318307 = a;
        double r318308 = log(r318307);
        double r318309 = r318306 * r318308;
        double r318310 = r318303 + r318309;
        double r318311 = b;
        double r318312 = r318310 - r318311;
        double r318313 = exp(r318312);
        double r318314 = r318299 * r318313;
        double r318315 = r318314 / r318300;
        return r318315;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r318316 = 1.0;
        double r318317 = cbrt(r318316);
        double r318318 = r318317 * r318317;
        double r318319 = a;
        double r318320 = cbrt(r318319);
        double r318321 = r318320 * r318320;
        double r318322 = r318318 / r318321;
        double r318323 = 1.0;
        double r318324 = pow(r318322, r318323);
        double r318325 = y;
        double r318326 = z;
        double r318327 = r318316 / r318326;
        double r318328 = log(r318327);
        double r318329 = r318325 * r318328;
        double r318330 = r318316 / r318319;
        double r318331 = log(r318330);
        double r318332 = t;
        double r318333 = r318331 * r318332;
        double r318334 = b;
        double r318335 = r318333 + r318334;
        double r318336 = r318329 + r318335;
        double r318337 = exp(r318336);
        double r318338 = sqrt(r318337);
        double r318339 = r318324 / r318338;
        double r318340 = r318339 / r318325;
        double r318341 = r318317 / r318320;
        double r318342 = pow(r318341, r318323);
        double r318343 = r318342 / r318338;
        double r318344 = r318340 * r318343;
        double r318345 = x;
        double r318346 = r318344 * r318345;
        return r318346;
}

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	11.4
Herbie	1.4

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

Derivation

1. Initial program 2.1

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

2. Taylor expanded around inf 2.1

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

3. Simplified 6.2

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

5. Applied div-inv 6.2

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

6. Applied add-sqr-sqrt 6.2

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

7. Applied add-cube-cbrt 6.4

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

8. Applied add-cube-cbrt 6.4

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

9. Applied times-frac 6.4

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

10. Applied unpow-prod-down 6.4

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

11. Applied times-frac 6.4

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

12. Applied times-frac 1.2

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

13. Final simplification 1.4

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

Reproduce

herbie shell --seed 2019298 
(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))