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

Average Error: 4.1 → 5.6

Time: 14.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.3617424863871238 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 9.2554044737280137 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \left(e^{\frac{\frac{2}{t}}{3}}\right)\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r485407 = x;
        double r485408 = y;
        double r485409 = 2.0;
        double r485410 = z;
        double r485411 = t;
        double r485412 = a;
        double r485413 = r485411 + r485412;
        double r485414 = sqrt(r485413);
        double r485415 = r485410 * r485414;
        double r485416 = r485415 / r485411;
        double r485417 = b;
        double r485418 = c;
        double r485419 = r485417 - r485418;
        double r485420 = 5.0;
        double r485421 = 6.0;
        double r485422 = r485420 / r485421;
        double r485423 = r485412 + r485422;
        double r485424 = 3.0;
        double r485425 = r485411 * r485424;
        double r485426 = r485409 / r485425;
        double r485427 = r485423 - r485426;
        double r485428 = r485419 * r485427;
        double r485429 = r485416 - r485428;
        double r485430 = r485409 * r485429;
        double r485431 = exp(r485430);
        double r485432 = r485408 * r485431;
        double r485433 = r485407 + r485432;
        double r485434 = r485407 / r485433;
        return r485434;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r485435 = t;
        double r485436 = -2.3617424863871238e-92;
        bool r485437 = r485435 <= r485436;
        double r485438 = x;
        double r485439 = y;
        double r485440 = 2.0;
        double r485441 = c;
        double r485442 = a;
        double r485443 = 0.8333333333333334;
        double r485444 = r485442 + r485443;
        double r485445 = r485441 * r485444;
        double r485446 = b;
        double r485447 = r485442 * r485446;
        double r485448 = r485445 - r485447;
        double r485449 = r485440 * r485448;
        double r485450 = exp(r485449);
        double r485451 = r485439 * r485450;
        double r485452 = r485438 + r485451;
        double r485453 = r485438 / r485452;
        double r485454 = 9.255404473728014e-17;
        bool r485455 = r485435 <= r485454;
        double r485456 = z;
        double r485457 = r485435 + r485442;
        double r485458 = sqrt(r485457);
        double r485459 = r485456 * r485458;
        double r485460 = 5.0;
        double r485461 = 6.0;
        double r485462 = r485460 / r485461;
        double r485463 = r485442 - r485462;
        double r485464 = 3.0;
        double r485465 = r485435 * r485464;
        double r485466 = r485463 * r485465;
        double r485467 = r485459 * r485466;
        double r485468 = r485446 - r485441;
        double r485469 = r485442 * r485442;
        double r485470 = r485462 * r485462;
        double r485471 = r485469 - r485470;
        double r485472 = r485471 * r485465;
        double r485473 = r485463 * r485440;
        double r485474 = r485472 - r485473;
        double r485475 = r485468 * r485474;
        double r485476 = r485435 * r485475;
        double r485477 = r485467 - r485476;
        double r485478 = r485435 * r485466;
        double r485479 = r485477 / r485478;
        double r485480 = r485440 * r485479;
        double r485481 = exp(r485480);
        double r485482 = r485439 * r485481;
        double r485483 = r485438 + r485482;
        double r485484 = r485438 / r485483;
        double r485485 = r485459 / r485435;
        double r485486 = r485442 + r485462;
        double r485487 = r485440 / r485435;
        double r485488 = r485487 / r485464;
        double r485489 = exp(r485488);
        double r485490 = log(r485489);
        double r485491 = 3.0;
        double r485492 = pow(r485490, r485491);
        double r485493 = 0.3333333333333333;
        double r485494 = pow(r485492, r485493);
        double r485495 = r485486 - r485494;
        double r485496 = r485468 * r485495;
        double r485497 = r485485 - r485496;
        double r485498 = r485440 * r485497;
        double r485499 = exp(r485498);
        double r485500 = r485439 * r485499;
        double r485501 = r485438 + r485500;
        double r485502 = r485438 / r485501;
        double r485503 = r485455 ? r485484 : r485502;
        double r485504 = r485437 ? r485453 : r485503;
        return r485504;
}

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	4.1
Target	3.3
Herbie	5.6

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -2.3617424863871238e-92

   1. Initial program 3.6

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

   2. Taylor expanded around inf 8.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}}\]

   3. Simplified 8.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\]

   if -2.3617424863871238e-92 < t < 9.255404473728014e-17

   1. Initial program 5.4

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

   3. Applied flip-+ 8.1

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

   4. Applied frac-sub 8.1

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

   5. Applied associate-*r/ 8.2

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

   6. Applied frac-sub 6.7

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

   if 9.255404473728014e-17 < t

   1. Initial program 2.9

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

   3. Applied add-cbrt-cube 2.9

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

   4. Applied add-cbrt-cube 2.9

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

   5. Applied cbrt-unprod 2.9

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

   6. Applied add-cbrt-cube 2.9

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

   7. Applied cbrt-undiv 2.9

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

   8. Simplified 2.9

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

   10. Applied pow1/3 2.9

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{{\left({\left(\frac{2}{t \cdot 3}\right)}^{3}\right)}^{\frac{1}{3}}}\right)\right)}}\]
   11. Using strategy rm

   12. Applied add-log-exp 3.7

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\color{blue}{\left(\log \left(e^{\frac{2}{t \cdot 3}}\right)\right)}}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\]

   13. Simplified 3.7

       \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \color{blue}{\left(e^{\frac{\frac{2}{t}}{3}}\right)}\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.3617424863871238 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \le 9.2554044737280137 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - {\left({\left(\log \left(e^{\frac{\frac{2}{t}}{3}}\right)\right)}^{3}\right)}^{\frac{1}{3}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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 -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-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)))))))))))