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

Average Error: 3.4 → 2.4

Time: 23.2s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

↓

\[\frac{x}{x + y \cdot e^{\left(\frac{\sqrt{a + t}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} - \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right) \cdot 2.0}}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r7718446 = x;
        double r7718447 = y;
        double r7718448 = 2.0;
        double r7718449 = z;
        double r7718450 = t;
        double r7718451 = a;
        double r7718452 = r7718450 + r7718451;
        double r7718453 = sqrt(r7718452);
        double r7718454 = r7718449 * r7718453;
        double r7718455 = r7718454 / r7718450;
        double r7718456 = b;
        double r7718457 = c;
        double r7718458 = r7718456 - r7718457;
        double r7718459 = 5.0;
        double r7718460 = 6.0;
        double r7718461 = r7718459 / r7718460;
        double r7718462 = r7718451 + r7718461;
        double r7718463 = 3.0;
        double r7718464 = r7718450 * r7718463;
        double r7718465 = r7718448 / r7718464;
        double r7718466 = r7718462 - r7718465;
        double r7718467 = r7718458 * r7718466;
        double r7718468 = r7718455 - r7718467;
        double r7718469 = r7718448 * r7718468;
        double r7718470 = exp(r7718469);
        double r7718471 = r7718447 * r7718470;
        double r7718472 = r7718446 + r7718471;
        double r7718473 = r7718446 / r7718472;
        return r7718473;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r7718474 = x;
        double r7718475 = y;
        double r7718476 = a;
        double r7718477 = t;
        double r7718478 = r7718476 + r7718477;
        double r7718479 = sqrt(r7718478);
        double r7718480 = cbrt(r7718477);
        double r7718481 = r7718479 / r7718480;
        double r7718482 = z;
        double r7718483 = r7718480 * r7718480;
        double r7718484 = r7718482 / r7718483;
        double r7718485 = r7718481 * r7718484;
        double r7718486 = 5.0;
        double r7718487 = 6.0;
        double r7718488 = r7718486 / r7718487;
        double r7718489 = r7718476 + r7718488;
        double r7718490 = 2.0;
        double r7718491 = 3.0;
        double r7718492 = r7718477 * r7718491;
        double r7718493 = r7718490 / r7718492;
        double r7718494 = r7718489 - r7718493;
        double r7718495 = b;
        double r7718496 = c;
        double r7718497 = r7718495 - r7718496;
        double r7718498 = r7718494 * r7718497;
        double r7718499 = r7718485 - r7718498;
        double r7718500 = r7718499 * r7718490;
        double r7718501 = exp(r7718500);
        double r7718502 = r7718475 * r7718501;
        double r7718503 = r7718474 + r7718502;
        double r7718504 = r7718474 / r7718503;
        return r7718504;
}

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	3.4
Target	2.8
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

1. Initial program 3.4

   \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
2. Using strategy rm

3. Applied add-cube-cbrt 3.4

   \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

4. Applied times-frac 2.4

   \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

5. Final simplification 2.4

   \[\leadsto \frac{x}{x + y \cdot e^{\left(\frac{\sqrt{a + t}}{\sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} - \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right) \cdot \left(b - c\right)\right) \cdot 2.0}}\]

Reproduce

herbie shell --seed 2019156 
(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)))))))))))