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

Average Error: 3.6 → 1.8

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.382061142785191097588738309999822145457 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x}\\ \mathbf{elif}\;t \le 4.339752338022243962882841013808555445154 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\sqrt{t + a} \cdot z\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(\left(\frac{5}{6} + a\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right) \cdot \left(b - c\right)\right)}{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot t} \cdot 2} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r23446562 = x;
        double r23446563 = y;
        double r23446564 = 2.0;
        double r23446565 = z;
        double r23446566 = t;
        double r23446567 = a;
        double r23446568 = r23446566 + r23446567;
        double r23446569 = sqrt(r23446568);
        double r23446570 = r23446565 * r23446569;
        double r23446571 = r23446570 / r23446566;
        double r23446572 = b;
        double r23446573 = c;
        double r23446574 = r23446572 - r23446573;
        double r23446575 = 5.0;
        double r23446576 = 6.0;
        double r23446577 = r23446575 / r23446576;
        double r23446578 = r23446567 + r23446577;
        double r23446579 = 3.0;
        double r23446580 = r23446566 * r23446579;
        double r23446581 = r23446564 / r23446580;
        double r23446582 = r23446578 - r23446581;
        double r23446583 = r23446574 * r23446582;
        double r23446584 = r23446571 - r23446583;
        double r23446585 = r23446564 * r23446584;
        double r23446586 = exp(r23446585);
        double r23446587 = r23446563 * r23446586;
        double r23446588 = r23446562 + r23446587;
        double r23446589 = r23446562 / r23446588;
        return r23446589;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r23446590 = t;
        double r23446591 = -4.382061142785191e-96;
        bool r23446592 = r23446590 <= r23446591;
        double r23446593 = x;
        double r23446594 = z;
        double r23446595 = cbrt(r23446590);
        double r23446596 = r23446595 * r23446595;
        double r23446597 = r23446594 / r23446596;
        double r23446598 = a;
        double r23446599 = r23446590 + r23446598;
        double r23446600 = sqrt(r23446599);
        double r23446601 = r23446600 / r23446595;
        double r23446602 = r23446597 * r23446601;
        double r23446603 = 5.0;
        double r23446604 = 6.0;
        double r23446605 = r23446603 / r23446604;
        double r23446606 = r23446605 + r23446598;
        double r23446607 = 2.0;
        double r23446608 = 3.0;
        double r23446609 = r23446590 * r23446608;
        double r23446610 = r23446607 / r23446609;
        double r23446611 = r23446606 - r23446610;
        double r23446612 = b;
        double r23446613 = c;
        double r23446614 = r23446612 - r23446613;
        double r23446615 = r23446611 * r23446614;
        double r23446616 = r23446602 - r23446615;
        double r23446617 = r23446616 * r23446607;
        double r23446618 = exp(r23446617);
        double r23446619 = y;
        double r23446620 = r23446618 * r23446619;
        double r23446621 = r23446620 + r23446593;
        double r23446622 = r23446593 / r23446621;
        double r23446623 = 4.339752338022244e-171;
        bool r23446624 = r23446590 <= r23446623;
        double r23446625 = r23446600 * r23446594;
        double r23446626 = r23446598 - r23446605;
        double r23446627 = r23446626 * r23446609;
        double r23446628 = r23446625 * r23446627;
        double r23446629 = r23446606 * r23446627;
        double r23446630 = r23446626 * r23446607;
        double r23446631 = r23446629 - r23446630;
        double r23446632 = r23446631 * r23446614;
        double r23446633 = r23446590 * r23446632;
        double r23446634 = r23446628 - r23446633;
        double r23446635 = r23446627 * r23446590;
        double r23446636 = r23446634 / r23446635;
        double r23446637 = r23446636 * r23446607;
        double r23446638 = exp(r23446637);
        double r23446639 = r23446619 * r23446638;
        double r23446640 = r23446639 + r23446593;
        double r23446641 = r23446593 / r23446640;
        double r23446642 = r23446624 ? r23446641 : r23446622;
        double r23446643 = r23446592 ? r23446622 : r23446642;
        return r23446643;
}

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	3.6
Target	2.6
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if t < -4.382061142785191e-96 or 4.339752338022244e-171 < t

   1. Initial program 2.2

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

   3. Applied add-cube-cbrt 2.2

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

   4. Applied times-frac 0.8

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

   if -4.382061142785191e-96 < t < 4.339752338022244e-171

   1. Initial program 7.6

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

   3. Applied flip-+ 10.7

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

   4. Applied frac-sub 10.7

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

   5. Applied associate-*r/ 10.8

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

   6. Applied frac-sub 7.7

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

   8. Applied difference-of-squares 7.7

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

   9. Applied associate-*l* 4.6

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.382061142785191097588738309999822145457 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x}\\ \mathbf{elif}\;t \le 4.339752338022243962882841013808555445154 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y \cdot e^{\frac{\left(\sqrt{t + a} \cdot z\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(\left(\frac{5}{6} + a\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - \left(a - \frac{5}{6}\right) \cdot 2\right) \cdot \left(b - c\right)\right)}{\left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) \cdot t} \cdot 2} + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}} - \left(\left(\frac{5}{6} + a\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) \cdot 2} \cdot y + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))