Average Error: 3.8 → 2.5
Time: 5.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]
\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}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z \cdot \sqrt{t + a}, \frac{1}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r44933 = x;
        double r44934 = y;
        double r44935 = 2.0;
        double r44936 = z;
        double r44937 = t;
        double r44938 = a;
        double r44939 = r44937 + r44938;
        double r44940 = sqrt(r44939);
        double r44941 = r44936 * r44940;
        double r44942 = r44941 / r44937;
        double r44943 = b;
        double r44944 = c;
        double r44945 = r44943 - r44944;
        double r44946 = 5.0;
        double r44947 = 6.0;
        double r44948 = r44946 / r44947;
        double r44949 = r44938 + r44948;
        double r44950 = 3.0;
        double r44951 = r44937 * r44950;
        double r44952 = r44935 / r44951;
        double r44953 = r44949 - r44952;
        double r44954 = r44945 * r44953;
        double r44955 = r44942 - r44954;
        double r44956 = r44935 * r44955;
        double r44957 = exp(r44956);
        double r44958 = r44934 * r44957;
        double r44959 = r44933 + r44958;
        double r44960 = r44933 / r44959;
        return r44960;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r44961 = x;
        double r44962 = y;
        double r44963 = 2.0;
        double r44964 = z;
        double r44965 = t;
        double r44966 = a;
        double r44967 = r44965 + r44966;
        double r44968 = sqrt(r44967);
        double r44969 = r44964 * r44968;
        double r44970 = 1.0;
        double r44971 = r44970 / r44965;
        double r44972 = 5.0;
        double r44973 = 6.0;
        double r44974 = r44972 / r44973;
        double r44975 = r44966 + r44974;
        double r44976 = 3.0;
        double r44977 = r44965 * r44976;
        double r44978 = r44963 / r44977;
        double r44979 = r44975 - r44978;
        double r44980 = b;
        double r44981 = c;
        double r44982 = r44980 - r44981;
        double r44983 = r44979 * r44982;
        double r44984 = -r44983;
        double r44985 = fma(r44969, r44971, r44984);
        double r44986 = -r44982;
        double r44987 = r44986 + r44982;
        double r44988 = r44979 * r44987;
        double r44989 = r44985 + r44988;
        double r44990 = r44963 * r44989;
        double r44991 = exp(r44990);
        double r44992 = r44962 * r44991;
        double r44993 = r44961 + r44992;
        double r44994 = r44961 / r44993;
        return r44994;
}

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

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

  3. Applied div-inv3.8

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

  4. Applied prod-diff22.3

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

  5. Simplified2.5

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

  6. Final simplification2.5

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

Reproduce

herbie shell --seed 2020027 +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)))))))))))