Average Error: 4.0 → 4.6
Time: 26.5s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;b - c \le -1.3562954072489081 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{x + e^{\left(\left(\left(\frac{\sqrt{a + t}}{t}\right)\right) \cdot z - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{elif}\;b - c \le 2.13457590843716 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{a + t}}{t} \cdot z - \left(\left(\frac{5.0}{6.0} + a\right) - \left(\left(\frac{2.0}{3.0 \cdot t}\right)\right)\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\left(\left(\left(\frac{\sqrt{a + t}}{t}\right)\right) \cdot z - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right) \cdot 2.0} \cdot y}\\ \end{array}\]
\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)}}
↓
\begin{array}{l}
\mathbf{if}\;b - c \le -1.3562954072489081 \cdot 10^{+104}:\\
\;\;\;\;\frac{x}{x + e^{\left(\left(\left(\frac{\sqrt{a + t}}{t}\right)\right) \cdot z - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right) \cdot 2.0} \cdot y}\\

\mathbf{elif}\;b - c \le 2.13457590843716 \cdot 10^{-161}:\\
\;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{a + t}}{t} \cdot z - \left(\left(\frac{5.0}{6.0} + a\right) - \left(\left(\frac{2.0}{3.0 \cdot t}\right)\right)\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r19816458 = x;
        double r19816459 = y;
        double r19816460 = 2.0;
        double r19816461 = z;
        double r19816462 = t;
        double r19816463 = a;
        double r19816464 = r19816462 + r19816463;
        double r19816465 = sqrt(r19816464);
        double r19816466 = r19816461 * r19816465;
        double r19816467 = r19816466 / r19816462;
        double r19816468 = b;
        double r19816469 = c;
        double r19816470 = r19816468 - r19816469;
        double r19816471 = 5.0;
        double r19816472 = 6.0;
        double r19816473 = r19816471 / r19816472;
        double r19816474 = r19816463 + r19816473;
        double r19816475 = 3.0;
        double r19816476 = r19816462 * r19816475;
        double r19816477 = r19816460 / r19816476;
        double r19816478 = r19816474 - r19816477;
        double r19816479 = r19816470 * r19816478;
        double r19816480 = r19816467 - r19816479;
        double r19816481 = r19816460 * r19816480;
        double r19816482 = exp(r19816481);
        double r19816483 = r19816459 * r19816482;
        double r19816484 = r19816458 + r19816483;
        double r19816485 = r19816458 / r19816484;
        return r19816485;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r19816486 = b;
        double r19816487 = c;
        double r19816488 = r19816486 - r19816487;
        double r19816489 = -1.3562954072489081e+104;
        bool r19816490 = r19816488 <= r19816489;
        double r19816491 = x;
        double r19816492 = a;
        double r19816493 = t;
        double r19816494 = r19816492 + r19816493;
        double r19816495 = sqrt(r19816494);
        double r19816496 = r19816495 / r19816493;
        double r19816497 = /* ERROR: no posit support in C */;
        double r19816498 = /* ERROR: no posit support in C */;
        double r19816499 = z;
        double r19816500 = r19816498 * r19816499;
        double r19816501 = 5.0;
        double r19816502 = 6.0;
        double r19816503 = r19816501 / r19816502;
        double r19816504 = r19816503 + r19816492;
        double r19816505 = 2.0;
        double r19816506 = 3.0;
        double r19816507 = r19816506 * r19816493;
        double r19816508 = r19816505 / r19816507;
        double r19816509 = r19816504 - r19816508;
        double r19816510 = r19816488 * r19816509;
        double r19816511 = r19816500 - r19816510;
        double r19816512 = r19816511 * r19816505;
        double r19816513 = exp(r19816512);
        double r19816514 = y;
        double r19816515 = r19816513 * r19816514;
        double r19816516 = r19816491 + r19816515;
        double r19816517 = r19816491 / r19816516;
        double r19816518 = 2.13457590843716e-161;
        bool r19816519 = r19816488 <= r19816518;
        double r19816520 = r19816496 * r19816499;
        double r19816521 = /* ERROR: no posit support in C */;
        double r19816522 = /* ERROR: no posit support in C */;
        double r19816523 = r19816504 - r19816522;
        double r19816524 = r19816523 * r19816488;
        double r19816525 = r19816520 - r19816524;
        double r19816526 = r19816525 * r19816505;
        double r19816527 = exp(r19816526);
        double r19816528 = r19816527 * r19816514;
        double r19816529 = r19816491 + r19816528;
        double r19816530 = r19816491 / r19816529;
        double r19816531 = r19816519 ? r19816530 : r19816517;
        double r19816532 = r19816490 ? r19816517 : r19816531;
        return r19816532;
}

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

Original4.0
Target3.2
Herbie4.6
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- b c) < -1.3562954072489081e+104 or 2.13457590843716e-161 < (- b c)

    1. Initial program 5.0

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

    3. Applied *-un-lft-identity5.0

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    4. Applied times-frac4.4

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    5. Simplified4.4

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

    7. Applied insert-posit164.8

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

    if -1.3562954072489081e+104 < (- b c) < 2.13457590843716e-161

    1. Initial program 1.2

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

    3. Applied *-un-lft-identity1.2

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    4. Applied times-frac1.3

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    5. Simplified1.3

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

    7. Applied insert-posit164.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(z \cdot \frac{\sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \color{blue}{\left(\left(\frac{2.0}{t \cdot 3.0}\right)\right)}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b - c \le -1.3562954072489081 \cdot 10^{+104}:\\ \;\;\;\;\frac{x}{x + e^{\left(\left(\left(\frac{\sqrt{a + t}}{t}\right)\right) \cdot z - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{elif}\;b - c \le 2.13457590843716 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\sqrt{a + t}}{t} \cdot z - \left(\left(\frac{5.0}{6.0} + a\right) - \left(\left(\frac{2.0}{3.0 \cdot t}\right)\right)\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\left(\left(\left(\frac{\sqrt{a + t}}{t}\right)\right) \cdot z - \left(b - c\right) \cdot \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right)\right) \cdot 2.0} \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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)))))))))))