Average Error: 4.2 → 1.6
Time: 22.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}, x\right)}\]
\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, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \frac{z}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r59925 = x;
        double r59926 = y;
        double r59927 = 2.0;
        double r59928 = z;
        double r59929 = t;
        double r59930 = a;
        double r59931 = r59929 + r59930;
        double r59932 = sqrt(r59931);
        double r59933 = r59928 * r59932;
        double r59934 = r59933 / r59929;
        double r59935 = b;
        double r59936 = c;
        double r59937 = r59935 - r59936;
        double r59938 = 5.0;
        double r59939 = 6.0;
        double r59940 = r59938 / r59939;
        double r59941 = r59930 + r59940;
        double r59942 = 3.0;
        double r59943 = r59929 * r59942;
        double r59944 = r59927 / r59943;
        double r59945 = r59941 - r59944;
        double r59946 = r59937 * r59945;
        double r59947 = r59934 - r59946;
        double r59948 = r59927 * r59947;
        double r59949 = exp(r59948);
        double r59950 = r59926 * r59949;
        double r59951 = r59925 + r59950;
        double r59952 = r59925 / r59951;
        return r59952;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r59953 = x;
        double r59954 = y;
        double r59955 = 2.0;
        double r59956 = c;
        double r59957 = b;
        double r59958 = r59956 - r59957;
        double r59959 = 5.0;
        double r59960 = 6.0;
        double r59961 = r59959 / r59960;
        double r59962 = a;
        double r59963 = t;
        double r59964 = r59955 / r59963;
        double r59965 = 3.0;
        double r59966 = r59964 / r59965;
        double r59967 = r59962 - r59966;
        double r59968 = r59961 + r59967;
        double r59969 = z;
        double r59970 = cbrt(r59963);
        double r59971 = r59969 / r59970;
        double r59972 = r59963 + r59962;
        double r59973 = sqrt(r59972);
        double r59974 = r59970 * r59970;
        double r59975 = r59973 / r59974;
        double r59976 = r59971 * r59975;
        double r59977 = fma(r59958, r59968, r59976);
        double r59978 = r59955 * r59977;
        double r59979 = exp(r59978);
        double r59980 = fma(r59954, r59979, r59953);
        double r59981 = r59953 / r59980;
        return r59981;
}

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 4.2

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

  2. Simplified2.0

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

  4. Applied add-cube-cbrt2.0

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

  5. Applied *-un-lft-identity2.0

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

  6. Applied times-frac2.0

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

  7. Applied associate-*r*1.6

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

  8. Simplified1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019196 +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)))))))))))