Average Error: 3.8 → 2.3
Time: 15.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.84581125504620826 \cdot 10^{-243} \lor \neg \left(t \le 4.3711936635722358 \cdot 10^{-236}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \log \left(e^{\mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]
\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}\;t \le -3.84581125504620826 \cdot 10^{-243} \lor \neg \left(t \le 4.3711936635722358 \cdot 10^{-236}\right):\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \log \left(e^{\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)}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\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(\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)}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r371623 = x;
        double r371624 = y;
        double r371625 = 2.0;
        double r371626 = z;
        double r371627 = t;
        double r371628 = a;
        double r371629 = r371627 + r371628;
        double r371630 = sqrt(r371629);
        double r371631 = r371626 * r371630;
        double r371632 = r371631 / r371627;
        double r371633 = b;
        double r371634 = c;
        double r371635 = r371633 - r371634;
        double r371636 = 5.0;
        double r371637 = 6.0;
        double r371638 = r371636 / r371637;
        double r371639 = r371628 + r371638;
        double r371640 = 3.0;
        double r371641 = r371627 * r371640;
        double r371642 = r371625 / r371641;
        double r371643 = r371639 - r371642;
        double r371644 = r371635 * r371643;
        double r371645 = r371632 - r371644;
        double r371646 = r371625 * r371645;
        double r371647 = exp(r371646);
        double r371648 = r371624 * r371647;
        double r371649 = r371623 + r371648;
        double r371650 = r371623 / r371649;
        return r371650;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r371651 = t;
        double r371652 = -3.845811255046208e-243;
        bool r371653 = r371651 <= r371652;
        double r371654 = 4.371193663572236e-236;
        bool r371655 = r371651 <= r371654;
        double r371656 = !r371655;
        bool r371657 = r371653 || r371656;
        double r371658 = x;
        double r371659 = y;
        double r371660 = 2.0;
        double r371661 = z;
        double r371662 = 1.0;
        double r371663 = r371661 / r371662;
        double r371664 = a;
        double r371665 = r371651 + r371664;
        double r371666 = sqrt(r371665);
        double r371667 = r371666 / r371651;
        double r371668 = b;
        double r371669 = c;
        double r371670 = r371668 - r371669;
        double r371671 = 5.0;
        double r371672 = 6.0;
        double r371673 = r371671 / r371672;
        double r371674 = r371664 + r371673;
        double r371675 = 3.0;
        double r371676 = r371651 * r371675;
        double r371677 = r371660 / r371676;
        double r371678 = r371674 - r371677;
        double r371679 = r371670 * r371678;
        double r371680 = -r371679;
        double r371681 = fma(r371663, r371667, r371680);
        double r371682 = exp(r371681);
        double r371683 = log(r371682);
        double r371684 = r371660 * r371683;
        double r371685 = exp(r371684);
        double r371686 = r371659 * r371685;
        double r371687 = r371658 + r371686;
        double r371688 = r371658 / r371687;
        double r371689 = r371661 * r371666;
        double r371690 = r371664 - r371673;
        double r371691 = r371690 * r371676;
        double r371692 = r371689 * r371691;
        double r371693 = r371664 * r371664;
        double r371694 = r371673 * r371673;
        double r371695 = r371693 - r371694;
        double r371696 = r371695 * r371676;
        double r371697 = r371690 * r371660;
        double r371698 = r371696 - r371697;
        double r371699 = r371670 * r371698;
        double r371700 = r371651 * r371699;
        double r371701 = r371692 - r371700;
        double r371702 = r371651 * r371691;
        double r371703 = r371701 / r371702;
        double r371704 = r371660 * r371703;
        double r371705 = exp(r371704);
        double r371706 = r371659 * r371705;
        double r371707 = r371658 + r371706;
        double r371708 = r371658 / r371707;
        double r371709 = r371657 ? r371688 : r371708;
        return r371709;
}

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

Original3.8
Target3.0
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.845811255046208e-243 or 4.371193663572236e-236 < t

    1. Initial program 3.2

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

    3. Applied *-un-lft-identity3.2

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

    4. Applied times-frac1.9

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

    5. Applied fma-neg1.1

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

    7. Applied add-log-exp1.1

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

    if -3.845811255046208e-243 < t < 4.371193663572236e-236

    1. Initial program 8.4

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

    3. Applied flip-+13.7

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

    4. Applied frac-sub13.7

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

    5. Applied associate-*r/13.7

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

    6. Applied frac-sub11.5

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.84581125504620826 \cdot 10^{-243} \lor \neg \left(t \le 4.3711936635722358 \cdot 10^{-236}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \log \left(e^{\mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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)))))))))))