Average Error: 3.8 → 1.4
Time: 29.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}\right)}, x\right)}\]
\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)}}
↓
\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{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2149640 = x;
        double r2149641 = y;
        double r2149642 = 2.0;
        double r2149643 = z;
        double r2149644 = t;
        double r2149645 = a;
        double r2149646 = r2149644 + r2149645;
        double r2149647 = sqrt(r2149646);
        double r2149648 = r2149643 * r2149647;
        double r2149649 = r2149648 / r2149644;
        double r2149650 = b;
        double r2149651 = c;
        double r2149652 = r2149650 - r2149651;
        double r2149653 = 5.0;
        double r2149654 = 6.0;
        double r2149655 = r2149653 / r2149654;
        double r2149656 = r2149645 + r2149655;
        double r2149657 = 3.0;
        double r2149658 = r2149644 * r2149657;
        double r2149659 = r2149642 / r2149658;
        double r2149660 = r2149656 - r2149659;
        double r2149661 = r2149652 * r2149660;
        double r2149662 = r2149649 - r2149661;
        double r2149663 = r2149642 * r2149662;
        double r2149664 = exp(r2149663);
        double r2149665 = r2149641 * r2149664;
        double r2149666 = r2149640 + r2149665;
        double r2149667 = r2149640 / r2149666;
        return r2149667;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2149668 = x;
        double r2149669 = y;
        double r2149670 = 2.0;
        double r2149671 = c;
        double r2149672 = b;
        double r2149673 = r2149671 - r2149672;
        double r2149674 = 5.0;
        double r2149675 = 6.0;
        double r2149676 = r2149674 / r2149675;
        double r2149677 = t;
        double r2149678 = r2149670 / r2149677;
        double r2149679 = 3.0;
        double r2149680 = r2149678 / r2149679;
        double r2149681 = a;
        double r2149682 = r2149680 - r2149681;
        double r2149683 = r2149676 - r2149682;
        double r2149684 = z;
        double r2149685 = cbrt(r2149677);
        double r2149686 = r2149685 * r2149685;
        double r2149687 = r2149684 / r2149686;
        double r2149688 = r2149681 + r2149677;
        double r2149689 = sqrt(r2149688);
        double r2149690 = r2149689 / r2149685;
        double r2149691 = r2149687 * r2149690;
        double r2149692 = fma(r2149673, r2149683, r2149691);
        double r2149693 = r2149670 * r2149692;
        double r2149694 = exp(r2149693);
        double r2149695 = fma(r2149669, r2149694, r2149668);
        double r2149696 = r2149668 / r2149695;
        return r2149696;
}

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

Derivation

  1. Initial program 3.8

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

  2. Simplified2.1

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

  4. Applied add-cube-cbrt2.1

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

  5. Applied *-un-lft-identity2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{a + \color{blue}{1 \cdot t}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  6. Applied *-un-lft-identity2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{\color{blue}{1 \cdot a} + 1 \cdot t}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  7. Applied distribute-lft-out2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{\color{blue}{1 \cdot \left(a + t\right)}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  8. Applied sqrt-prod2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{a + t}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  9. Applied times-frac2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \color{blue}{\left(\frac{\sqrt{1}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}\right)}\right)}, x\right)}\]

  10. Applied associate-*r*1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\left(z \cdot \frac{\sqrt{1}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}}\right)}, x\right)}\]

  11. Simplified1.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))