Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Average Error: 3.8 → 3.3

Time: 28.3s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r269457 = x;
        double r269458 = y;
        double r269459 = 2.0;
        double r269460 = z;
        double r269461 = t;
        double r269462 = a;
        double r269463 = r269461 + r269462;
        double r269464 = sqrt(r269463);
        double r269465 = r269460 * r269464;
        double r269466 = r269465 / r269461;
        double r269467 = b;
        double r269468 = c;
        double r269469 = r269467 - r269468;
        double r269470 = 5.0;
        double r269471 = 6.0;
        double r269472 = r269470 / r269471;
        double r269473 = r269462 + r269472;
        double r269474 = 3.0;
        double r269475 = r269461 * r269474;
        double r269476 = r269459 / r269475;
        double r269477 = r269473 - r269476;
        double r269478 = r269469 * r269477;
        double r269479 = r269466 - r269478;
        double r269480 = r269459 * r269479;
        double r269481 = exp(r269480);
        double r269482 = r269458 * r269481;
        double r269483 = r269457 + r269482;
        double r269484 = r269457 / r269483;
        return r269484;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r269485 = x;
        double r269486 = y;
        double r269487 = 2.0;
        double r269488 = z;
        double r269489 = t;
        double r269490 = a;
        double r269491 = r269489 + r269490;
        double r269492 = sqrt(r269491);
        double r269493 = r269489 / r269492;
        double r269494 = r269488 / r269493;
        double r269495 = b;
        double r269496 = c;
        double r269497 = r269495 - r269496;
        double r269498 = 5.0;
        double r269499 = 6.0;
        double r269500 = r269498 / r269499;
        double r269501 = r269490 + r269500;
        double r269502 = 3.0;
        double r269503 = r269489 * r269502;
        double r269504 = r269487 / r269503;
        double r269505 = r269501 - r269504;
        double r269506 = r269497 * r269505;
        double r269507 = r269494 - r269506;
        double r269508 = r269487 * r269507;
        double r269509 = exp(r269508);
        double r269510 = r269486 * r269509;
        double r269511 = r269485 + r269510;
        double r269512 = r269485 / r269511;
        return r269512;
}

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

Bits error versus c

Target

Original	3.8
Target	2.8
Herbie	3.3

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581057561884576920117070548 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333333703407674875052180141211 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547088010424937268931048836 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Initial program 3.8

   \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
2. Using strategy rm

3. Applied associate-/l* 3.3

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\frac{t}{\sqrt{t + a}}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

4. Final simplification 3.3

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z}{\frac{t}{\sqrt{t + a}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.1183266448915811e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.83333333333333337 c)) (* a b))))))) (if (< t 5.19658877065154709e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))