Average Error: 3.9 → 2.2
Time: 9.8s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 8.85175204914312434 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\mathsf{fma}\left(a, b, -\mathsf{fma}\left(a, c, 0.83333333333333337 \cdot c\right)\right)\right)}}\\ \end{array}\]
\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}\;z \le 8.85175204914312434 \cdot 10^{213}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\mathsf{fma}\left(a, b, -\mathsf{fma}\left(a, c, 0.83333333333333337 \cdot c\right)\right)\right)}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r364689 = x;
        double r364690 = y;
        double r364691 = 2.0;
        double r364692 = z;
        double r364693 = t;
        double r364694 = a;
        double r364695 = r364693 + r364694;
        double r364696 = sqrt(r364695);
        double r364697 = r364692 * r364696;
        double r364698 = r364697 / r364693;
        double r364699 = b;
        double r364700 = c;
        double r364701 = r364699 - r364700;
        double r364702 = 5.0;
        double r364703 = 6.0;
        double r364704 = r364702 / r364703;
        double r364705 = r364694 + r364704;
        double r364706 = 3.0;
        double r364707 = r364693 * r364706;
        double r364708 = r364691 / r364707;
        double r364709 = r364705 - r364708;
        double r364710 = r364701 * r364709;
        double r364711 = r364698 - r364710;
        double r364712 = r364691 * r364711;
        double r364713 = exp(r364712);
        double r364714 = r364690 * r364713;
        double r364715 = r364689 + r364714;
        double r364716 = r364689 / r364715;
        return r364716;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r364717 = z;
        double r364718 = 8.851752049143124e+213;
        bool r364719 = r364717 <= r364718;
        double r364720 = x;
        double r364721 = y;
        double r364722 = 2.0;
        double r364723 = t;
        double r364724 = a;
        double r364725 = r364723 + r364724;
        double r364726 = sqrt(r364725);
        double r364727 = r364717 * r364726;
        double r364728 = 1.0;
        double r364729 = r364728 / r364723;
        double r364730 = b;
        double r364731 = c;
        double r364732 = r364730 - r364731;
        double r364733 = 5.0;
        double r364734 = 6.0;
        double r364735 = r364733 / r364734;
        double r364736 = r364724 + r364735;
        double r364737 = 3.0;
        double r364738 = r364723 * r364737;
        double r364739 = r364722 / r364738;
        double r364740 = r364736 - r364739;
        double r364741 = r364732 * r364740;
        double r364742 = -r364741;
        double r364743 = fma(r364727, r364729, r364742);
        double r364744 = r364722 * r364743;
        double r364745 = exp(r364744);
        double r364746 = r364721 * r364745;
        double r364747 = r364720 + r364746;
        double r364748 = r364720 / r364747;
        double r364749 = r364717 / r364728;
        double r364750 = r364726 / r364723;
        double r364751 = 0.8333333333333334;
        double r364752 = r364751 * r364731;
        double r364753 = fma(r364724, r364731, r364752);
        double r364754 = -r364753;
        double r364755 = fma(r364724, r364730, r364754);
        double r364756 = -r364755;
        double r364757 = fma(r364749, r364750, r364756);
        double r364758 = r364722 * r364757;
        double r364759 = exp(r364758);
        double r364760 = r364721 * r364759;
        double r364761 = r364720 + r364760;
        double r364762 = r364720 / r364761;
        double r364763 = r364719 ? r364748 : r364762;
        return r364763;
}

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

Original3.9
Target3.0
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 8.851752049143124e+213

    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 div-inv2.9

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

    4. Applied fma-neg1.8

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

    if 8.851752049143124e+213 < z

    1. Initial program 15.1

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

    3. Applied *-un-lft-identity15.1

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

    4. Applied times-frac8.7

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

    5. Applied fma-neg5.7

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

    6. Taylor expanded around inf 8.0

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\color{blue}{\left(a \cdot b - \left(a \cdot c + 0.83333333333333337 \cdot c\right)\right)}\right)}}\]

    7. Simplified7.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\color{blue}{\mathsf{fma}\left(a, b, -\mathsf{fma}\left(a, c, 0.83333333333333337 \cdot c\right)\right)}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 8.85175204914312434 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\mathsf{fma}\left(a, b, -\mathsf{fma}\left(a, c, 0.83333333333333337 \cdot c\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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)))))))))))