Average Error: 2.0 → 0.3
Time: 37.6s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[{\left({\left(e^{\sqrt[3]{\mathsf{fma}\left(\log z - t, y, a \cdot \left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, \mathsf{fma}\left(1, z, b\right)\right)\right)\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(\log z - t, y, a \cdot \left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, \mathsf{fma}\left(1, z, b\right)\right)\right)\right)}\right)}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right)\right) - b\right)\right)}\right)} \cdot x\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
{\left({\left(e^{\sqrt[3]{\mathsf{fma}\left(\log z - t, y, a \cdot \left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, \mathsf{fma}\left(1, z, b\right)\right)\right)\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(\log z - t, y, a \cdot \left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, \mathsf{fma}\left(1, z, b\right)\right)\right)\right)}\right)}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right)\right) - b\right)\right)}\right)} \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r137922 = x;
        double r137923 = y;
        double r137924 = z;
        double r137925 = log(r137924);
        double r137926 = t;
        double r137927 = r137925 - r137926;
        double r137928 = r137923 * r137927;
        double r137929 = a;
        double r137930 = 1.0;
        double r137931 = r137930 - r137924;
        double r137932 = log(r137931);
        double r137933 = b;
        double r137934 = r137932 - r137933;
        double r137935 = r137929 * r137934;
        double r137936 = r137928 + r137935;
        double r137937 = exp(r137936);
        double r137938 = r137922 * r137937;
        return r137938;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r137939 = z;
        double r137940 = log(r137939);
        double r137941 = t;
        double r137942 = r137940 - r137941;
        double r137943 = y;
        double r137944 = a;
        double r137945 = 1.0;
        double r137946 = log(r137945);
        double r137947 = 0.5;
        double r137948 = 2.0;
        double r137949 = pow(r137939, r137948);
        double r137950 = pow(r137945, r137948);
        double r137951 = r137949 / r137950;
        double r137952 = b;
        double r137953 = fma(r137945, r137939, r137952);
        double r137954 = fma(r137947, r137951, r137953);
        double r137955 = r137946 - r137954;
        double r137956 = r137944 * r137955;
        double r137957 = fma(r137942, r137943, r137956);
        double r137958 = cbrt(r137957);
        double r137959 = exp(r137958);
        double r137960 = pow(r137959, r137958);
        double r137961 = r137945 * r137939;
        double r137962 = fma(r137947, r137951, r137961);
        double r137963 = r137946 - r137962;
        double r137964 = r137963 - r137952;
        double r137965 = r137944 * r137964;
        double r137966 = fma(r137943, r137942, r137965);
        double r137967 = cbrt(r137966);
        double r137968 = pow(r137960, r137967);
        double r137969 = x;
        double r137970 = r137968 * r137969;
        return r137970;
}

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

Derivation

  1. Initial program 2.0

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]

  2. Simplified1.7

    \[\leadsto \color{blue}{e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\log \left(1 - z\right) - b\right)\right)} \cdot x}\]

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

    \[\leadsto e^{\mathsf{fma}\left(y, \log z - t, a \cdot \left(\color{blue}{\left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right)\right)} - b\right)\right)} \cdot x\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.3

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

  7. Applied exp-prod0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1 z)) b))))))