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

Average Error: 2.0 → 1.3

Time: 31.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r366378 = x;
        double r366379 = y;
        double r366380 = z;
        double r366381 = log(r366380);
        double r366382 = r366379 * r366381;
        double r366383 = t;
        double r366384 = 1.0;
        double r366385 = r366383 - r366384;
        double r366386 = a;
        double r366387 = log(r366386);
        double r366388 = r366385 * r366387;
        double r366389 = r366382 + r366388;
        double r366390 = b;
        double r366391 = r366389 - r366390;
        double r366392 = exp(r366391);
        double r366393 = r366378 * r366392;
        double r366394 = r366393 / r366379;
        return r366394;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r366395 = a;
        double r366396 = 1.0;
        double r366397 = -r366396;
        double r366398 = pow(r366395, r366397);
        double r366399 = b;
        double r366400 = log(r366395);
        double r366401 = t;
        double r366402 = r366400 * r366401;
        double r366403 = z;
        double r366404 = log(r366403);
        double r366405 = y;
        double r366406 = r366404 * r366405;
        double r366407 = r366402 + r366406;
        double r366408 = -r366407;
        double r366409 = r366399 + r366408;
        double r366410 = exp(r366409);
        double r366411 = sqrt(r366410);
        double r366412 = r366398 / r366411;
        double r366413 = x;
        double r366414 = r366399 - r366402;
        double r366415 = r366414 - r366406;
        double r366416 = exp(r366415);
        double r366417 = sqrt(r366416);
        double r366418 = r366413 / r366417;
        double r366419 = r366412 * r366418;
        double r366420 = r366419 / r366405;
        return r366420;
}

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	2.0
Target	10.8
Herbie	1.3

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

Derivation

1. Initial program 2.0

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

2. Taylor expanded around inf 2.0

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

3. Simplified 1.3

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

5. Applied add-sqr-sqrt 1.3

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

6. Applied *-un-lft-identity 1.3

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

7. Applied unpow-prod-down 1.3

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

8. Applied times-frac 1.3

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

9. Applied associate-*r* 1.3

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

10. Simplified 1.3

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

11. Final simplification 1.3

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

Reproduce

herbie shell --seed 2019305 
(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))