Average Error: 0.1 → 0.1
Time: 36.4s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \mathsf{fma}\left(-\log y, y + 0.5, y\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \mathsf{fma}\left(-\log y, y + 0.5, y\right)\right) - z
double f(double x, double y, double z) {
        double r250949 = x;
        double r250950 = y;
        double r250951 = 0.5;
        double r250952 = r250950 + r250951;
        double r250953 = log(r250950);
        double r250954 = r250952 * r250953;
        double r250955 = r250949 - r250954;
        double r250956 = r250955 + r250950;
        double r250957 = z;
        double r250958 = r250956 - r250957;
        return r250958;
}
↓
double f(double x, double y, double z) {
        double r250959 = x;
        double r250960 = y;
        double r250961 = log(r250960);
        double r250962 = -r250961;
        double r250963 = 0.5;
        double r250964 = r250960 + r250963;
        double r250965 = fma(r250962, r250964, r250960);
        double r250966 = r250959 + r250965;
        double r250967 = z;
        double r250968 = r250966 - r250967;
        return r250968;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
  2. Using strategy rm

  3. Applied sub-neg0.1

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

  4. Applied associate-+l+0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))