Average Error: 0.1 → 0.1
Time: 5.8s
Precision: 64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
↓
\[\mathsf{fma}\left(z, 1 - \log t, x + y\right) + \left(a - 0.5\right) \cdot b\]
\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b
↓
\mathsf{fma}\left(z, 1 - \log t, x + y\right) + \left(a - 0.5\right) \cdot b
double f(double x, double y, double z, double t, double a, double b) {
        double r440945 = x;
        double r440946 = y;
        double r440947 = r440945 + r440946;
        double r440948 = z;
        double r440949 = r440947 + r440948;
        double r440950 = t;
        double r440951 = log(r440950);
        double r440952 = r440948 * r440951;
        double r440953 = r440949 - r440952;
        double r440954 = a;
        double r440955 = 0.5;
        double r440956 = r440954 - r440955;
        double r440957 = b;
        double r440958 = r440956 * r440957;
        double r440959 = r440953 + r440958;
        return r440959;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r440960 = z;
        double r440961 = 1.0;
        double r440962 = t;
        double r440963 = log(r440962);
        double r440964 = r440961 - r440963;
        double r440965 = x;
        double r440966 = y;
        double r440967 = r440965 + r440966;
        double r440968 = fma(r440960, r440964, r440967);
        double r440969 = a;
        double r440970 = 0.5;
        double r440971 = r440969 - r440970;
        double r440972 = b;
        double r440973 = r440971 * r440972;
        double r440974 = r440968 + r440973;
        return r440974;
}

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

Target

Original0.1
Target0.3
Herbie0.1
\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(\left(x + \left(z + y\right)\right) - \log t \cdot z\right)} + \left(a - 0.5\right) \cdot b\]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, 1 - \log t, x + y\right)} + \left(a - 0.5\right) \cdot b\]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(z, 1 - \log t, x + y\right) + \left(a - 0.5\right) \cdot b\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))