Average Error: 3.7 → 3.3
Time: 35.1s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\ \end{array}\]
\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 \le -2.978001093890994 \cdot 10^{+55}:\\
\;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r22469647 = x;
        double r22469648 = y;
        double r22469649 = 2.0;
        double r22469650 = z;
        double r22469651 = t;
        double r22469652 = a;
        double r22469653 = r22469651 + r22469652;
        double r22469654 = sqrt(r22469653);
        double r22469655 = r22469650 * r22469654;
        double r22469656 = r22469655 / r22469651;
        double r22469657 = b;
        double r22469658 = c;
        double r22469659 = r22469657 - r22469658;
        double r22469660 = 5.0;
        double r22469661 = 6.0;
        double r22469662 = r22469660 / r22469661;
        double r22469663 = r22469652 + r22469662;
        double r22469664 = 3.0;
        double r22469665 = r22469651 * r22469664;
        double r22469666 = r22469649 / r22469665;
        double r22469667 = r22469663 - r22469666;
        double r22469668 = r22469659 * r22469667;
        double r22469669 = r22469656 - r22469668;
        double r22469670 = r22469649 * r22469669;
        double r22469671 = exp(r22469670);
        double r22469672 = r22469648 * r22469671;
        double r22469673 = r22469647 + r22469672;
        double r22469674 = r22469647 / r22469673;
        return r22469674;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r22469675 = b;
        double r22469676 = -2.978001093890994e+55;
        bool r22469677 = r22469675 <= r22469676;
        double r22469678 = x;
        double r22469679 = t;
        double r22469680 = a;
        double r22469681 = r22469679 + r22469680;
        double r22469682 = sqrt(r22469681);
        double r22469683 = z;
        double r22469684 = r22469682 * r22469683;
        double r22469685 = /* ERROR: no posit support in C */;
        double r22469686 = /* ERROR: no posit support in C */;
        double r22469687 = r22469686 / r22469679;
        double r22469688 = 5.0;
        double r22469689 = 6.0;
        double r22469690 = r22469688 / r22469689;
        double r22469691 = r22469690 + r22469680;
        double r22469692 = 2.0;
        double r22469693 = 3.0;
        double r22469694 = r22469693 * r22469679;
        double r22469695 = r22469692 / r22469694;
        double r22469696 = r22469691 - r22469695;
        double r22469697 = c;
        double r22469698 = r22469675 - r22469697;
        double r22469699 = r22469696 * r22469698;
        double r22469700 = r22469687 - r22469699;
        double r22469701 = r22469700 * r22469692;
        double r22469702 = exp(r22469701);
        double r22469703 = y;
        double r22469704 = r22469702 * r22469703;
        double r22469705 = r22469678 + r22469704;
        double r22469706 = r22469678 / r22469705;
        double r22469707 = r22469683 / r22469679;
        double r22469708 = r22469682 * r22469707;
        double r22469709 = r22469692 / r22469679;
        double r22469710 = r22469709 / r22469693;
        double r22469711 = r22469680 - r22469710;
        double r22469712 = r22469690 + r22469711;
        double r22469713 = r22469712 * r22469698;
        double r22469714 = r22469708 - r22469713;
        double r22469715 = exp(r22469714);
        double r22469716 = sqrt(r22469715);
        double r22469717 = log(r22469716);
        double r22469718 = r22469717 + r22469717;
        double r22469719 = r22469692 * r22469718;
        double r22469720 = exp(r22469719);
        double r22469721 = r22469703 * r22469720;
        double r22469722 = r22469678 + r22469721;
        double r22469723 = r22469678 / r22469722;
        double r22469724 = r22469677 ? r22469706 : r22469723;
        return r22469724;
}

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

Original3.7
Target3.0
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -2.978001093890994e+55

    1. Initial program 5.4

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

    3. Applied insert-posit165.7

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

    if -2.978001093890994e+55 < b

    1. Initial program 3.2

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

    3. Applied add-log-exp8.1

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

    4. Applied add-log-exp15.6

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

    5. Applied diff-log15.6

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

    6. Simplified2.6

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

    8. Applied add-sqr-sqrt2.6

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

    9. Applied log-prod2.6

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))))))))))