Average Error: 3.5 → 2.3
Time: 5.9s
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 r73936 = x;
        double r73937 = y;
        double r73938 = 2.0;
        double r73939 = z;
        double r73940 = t;
        double r73941 = a;
        double r73942 = r73940 + r73941;
        double r73943 = sqrt(r73942);
        double r73944 = r73939 * r73943;
        double r73945 = r73944 / r73940;
        double r73946 = b;
        double r73947 = c;
        double r73948 = r73946 - r73947;
        double r73949 = 5.0;
        double r73950 = 6.0;
        double r73951 = r73949 / r73950;
        double r73952 = r73941 + r73951;
        double r73953 = 3.0;
        double r73954 = r73940 * r73953;
        double r73955 = r73938 / r73954;
        double r73956 = r73952 - r73955;
        double r73957 = r73948 * r73956;
        double r73958 = r73945 - r73957;
        double r73959 = r73938 * r73958;
        double r73960 = exp(r73959);
        double r73961 = r73937 * r73960;
        double r73962 = r73936 + r73961;
        double r73963 = r73936 / r73962;
        return r73963;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r73964 = x;
        double r73965 = y;
        double r73966 = 2.0;
        double r73967 = z;
        double r73968 = t;
        double r73969 = a;
        double r73970 = r73968 + r73969;
        double r73971 = sqrt(r73970);
        double r73972 = r73967 * r73971;
        double r73973 = 1.0;
        double r73974 = r73973 / r73968;
        double r73975 = 5.0;
        double r73976 = 6.0;
        double r73977 = r73975 / r73976;
        double r73978 = r73969 + r73977;
        double r73979 = 3.0;
        double r73980 = r73968 * r73979;
        double r73981 = r73966 / r73980;
        double r73982 = r73978 - r73981;
        double r73983 = b;
        double r73984 = c;
        double r73985 = r73983 - r73984;
        double r73986 = r73982 * r73985;
        double r73987 = -r73986;
        double r73988 = fma(r73972, r73974, r73987);
        double r73989 = -r73985;
        double r73990 = r73989 + r73985;
        double r73991 = r73982 * r73990;
        double r73992 = r73988 + r73991;
        double r73993 = r73966 * r73992;
        double r73994 = exp(r73993);
        double r73995 = r73965 * r73994;
        double r73996 = r73964 + r73995;
        double r73997 = r73964 / r73996;
        return r73997;
}

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 3.5

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

  3. Applied div-inv3.5

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

  4. Applied prod-diff22.2

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

  5. Simplified2.3

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

  6. Final simplification2.3

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

Reproduce

herbie shell --seed 2020025 +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)))))))))))