Average Error: 3.8 → 2.5
Time: 36.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.0089747297515344 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{x + e^{\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{elif}\;t \le 5.358377107920803 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \left(\left(\frac{\sqrt{t + a}}{\frac{t}{z}}\right)\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot 2.0} \cdot y}\\ \end{array}\]
\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}\;t \le -5.0089747297515344 \cdot 10^{-254}:\\
\;\;\;\;\frac{x}{x + e^{\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot 2.0} \cdot y}\\

\mathbf{elif}\;t \le 5.358377107920803 \cdot 10^{-277}:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \left(\left(\frac{\sqrt{t + a}}{\frac{t}{z}}\right)\right)\right)}, x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + e^{\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\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 r20687660 = x;
        double r20687661 = y;
        double r20687662 = 2.0;
        double r20687663 = z;
        double r20687664 = t;
        double r20687665 = a;
        double r20687666 = r20687664 + r20687665;
        double r20687667 = sqrt(r20687666);
        double r20687668 = r20687663 * r20687667;
        double r20687669 = r20687668 / r20687664;
        double r20687670 = b;
        double r20687671 = c;
        double r20687672 = r20687670 - r20687671;
        double r20687673 = 5.0;
        double r20687674 = 6.0;
        double r20687675 = r20687673 / r20687674;
        double r20687676 = r20687665 + r20687675;
        double r20687677 = 3.0;
        double r20687678 = r20687664 * r20687677;
        double r20687679 = r20687662 / r20687678;
        double r20687680 = r20687676 - r20687679;
        double r20687681 = r20687672 * r20687680;
        double r20687682 = r20687669 - r20687681;
        double r20687683 = r20687662 * r20687682;
        double r20687684 = exp(r20687683);
        double r20687685 = r20687661 * r20687684;
        double r20687686 = r20687660 + r20687685;
        double r20687687 = r20687660 / r20687686;
        return r20687687;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r20687688 = t;
        double r20687689 = -5.0089747297515344e-254;
        bool r20687690 = r20687688 <= r20687689;
        double r20687691 = x;
        double r20687692 = z;
        double r20687693 = a;
        double r20687694 = r20687688 + r20687693;
        double r20687695 = sqrt(r20687694);
        double r20687696 = r20687695 / r20687688;
        double r20687697 = b;
        double r20687698 = c;
        double r20687699 = r20687697 - r20687698;
        double r20687700 = -r20687699;
        double r20687701 = 5.0;
        double r20687702 = 6.0;
        double r20687703 = r20687701 / r20687702;
        double r20687704 = r20687693 + r20687703;
        double r20687705 = 2.0;
        double r20687706 = 3.0;
        double r20687707 = r20687688 * r20687706;
        double r20687708 = r20687705 / r20687707;
        double r20687709 = r20687704 - r20687708;
        double r20687710 = r20687700 * r20687709;
        double r20687711 = fma(r20687692, r20687696, r20687710);
        double r20687712 = r20687711 * r20687705;
        double r20687713 = exp(r20687712);
        double r20687714 = y;
        double r20687715 = r20687713 * r20687714;
        double r20687716 = r20687691 + r20687715;
        double r20687717 = r20687691 / r20687716;
        double r20687718 = 5.358377107920803e-277;
        bool r20687719 = r20687688 <= r20687718;
        double r20687720 = r20687698 - r20687697;
        double r20687721 = r20687705 / r20687688;
        double r20687722 = r20687721 / r20687706;
        double r20687723 = r20687722 - r20687693;
        double r20687724 = r20687703 - r20687723;
        double r20687725 = r20687688 / r20687692;
        double r20687726 = r20687695 / r20687725;
        double r20687727 = /* ERROR: no posit support in C */;
        double r20687728 = /* ERROR: no posit support in C */;
        double r20687729 = fma(r20687720, r20687724, r20687728);
        double r20687730 = r20687705 * r20687729;
        double r20687731 = exp(r20687730);
        double r20687732 = fma(r20687714, r20687731, r20687691);
        double r20687733 = r20687691 / r20687732;
        double r20687734 = r20687719 ? r20687733 : r20687717;
        double r20687735 = r20687690 ? r20687717 : r20687734;
        return r20687735;
}

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
Target3.1
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;t \lt -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3.0 \cdot t\right) \cdot \left(a - \frac{5.0}{6.0}\right)\right) - \left(\left(\frac{5.0}{6.0} + a\right) \cdot \left(3.0 \cdot t\right) - 2.0\right) \cdot \left(\left(a - \frac{5.0}{6.0}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3.0\right) \cdot \left(a - \frac{5.0}{6.0}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -5.0089747297515344e-254 or 5.358377107920803e-277 < t

    1. Initial program 3.1

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

    3. Applied *-un-lft-identity3.1

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

    4. Applied times-frac2.3

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

    5. Applied fma-neg1.4

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

    if -5.0089747297515344e-254 < t < 5.358377107920803e-277

    1. Initial program 12.0

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

    2. Simplified7.8

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

    4. Applied insert-posit1616.9

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.0089747297515344 \cdot 10^{-254}:\\ \;\;\;\;\frac{x}{x + e^{\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{elif}\;t \le 5.358377107920803 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \left(\left(\frac{\sqrt{t + a}}{\frac{t}{z}}\right)\right)\right)}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + e^{\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, \left(-\left(b - c\right)\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot 2.0} \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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)))))))))))