Average Error: 4.1 → 1.4
Time: 22.1s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)\right)}, x\right)}\]
\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)}}
↓
\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r60955 = x;
        double r60956 = y;
        double r60957 = 2.0;
        double r60958 = z;
        double r60959 = t;
        double r60960 = a;
        double r60961 = r60959 + r60960;
        double r60962 = sqrt(r60961);
        double r60963 = r60958 * r60962;
        double r60964 = r60963 / r60959;
        double r60965 = b;
        double r60966 = c;
        double r60967 = r60965 - r60966;
        double r60968 = 5.0;
        double r60969 = 6.0;
        double r60970 = r60968 / r60969;
        double r60971 = r60960 + r60970;
        double r60972 = 3.0;
        double r60973 = r60959 * r60972;
        double r60974 = r60957 / r60973;
        double r60975 = r60971 - r60974;
        double r60976 = r60967 * r60975;
        double r60977 = r60964 - r60976;
        double r60978 = r60957 * r60977;
        double r60979 = exp(r60978);
        double r60980 = r60956 * r60979;
        double r60981 = r60955 + r60980;
        double r60982 = r60955 / r60981;
        return r60982;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r60983 = x;
        double r60984 = y;
        double r60985 = 2.0;
        double r60986 = exp(r60985);
        double r60987 = t;
        double r60988 = r60985 / r60987;
        double r60989 = 3.0;
        double r60990 = r60988 / r60989;
        double r60991 = a;
        double r60992 = 5.0;
        double r60993 = 6.0;
        double r60994 = r60992 / r60993;
        double r60995 = r60991 + r60994;
        double r60996 = r60990 - r60995;
        double r60997 = b;
        double r60998 = c;
        double r60999 = r60997 - r60998;
        double r61000 = z;
        double r61001 = cbrt(r61000);
        double r61002 = r61001 * r61001;
        double r61003 = cbrt(r60987);
        double r61004 = r61002 / r61003;
        double r61005 = r61001 / r61003;
        double r61006 = r60987 + r60991;
        double r61007 = sqrt(r61006);
        double r61008 = r61007 / r61003;
        double r61009 = r61005 * r61008;
        double r61010 = r61004 * r61009;
        double r61011 = fma(r60996, r60999, r61010);
        double r61012 = pow(r60986, r61011);
        double r61013 = fma(r60984, r61012, r60983);
        double r61014 = r60983 / r61013;
        return r61014;
}

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 4.1

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

  2. Simplified2.6

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

  4. Applied add-cube-cbrt2.6

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

  5. Applied times-frac1.5

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

  7. Applied add-cube-cbrt1.5

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

  8. Applied times-frac1.5

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

  9. Applied associate-*l*1.4

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

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))