Average Error: 0.1 → 0.0
Time: 14.5s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \mathsf{fma}\left(y + 0.5, -\log y, e^{\log \left(\left(\left(-\log y\right) + \log y\right) \cdot \left(y + 0.5\right) + y\right)}\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \mathsf{fma}\left(y + 0.5, -\log y, e^{\log \left(\left(\left(-\log y\right) + \log y\right) \cdot \left(y + 0.5\right) + y\right)}\right)\right) - z
double f(double x, double y, double z) {
        double r199935 = x;
        double r199936 = y;
        double r199937 = 0.5;
        double r199938 = r199936 + r199937;
        double r199939 = log(r199936);
        double r199940 = r199938 * r199939;
        double r199941 = r199935 - r199940;
        double r199942 = r199941 + r199936;
        double r199943 = z;
        double r199944 = r199942 - r199943;
        return r199944;
}
↓
double f(double x, double y, double z) {
        double r199945 = x;
        double r199946 = y;
        double r199947 = 0.5;
        double r199948 = r199946 + r199947;
        double r199949 = log(r199946);
        double r199950 = -r199949;
        double r199951 = r199950 + r199949;
        double r199952 = r199951 * r199948;
        double r199953 = r199952 + r199946;
        double r199954 = log(r199953);
        double r199955 = exp(r199954);
        double r199956 = fma(r199948, r199950, r199955);
        double r199957 = r199945 + r199956;
        double r199958 = z;
        double r199959 = r199957 - r199958;
        return r199959;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.0
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

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 *-un-lft-identity0.1

    \[\leadsto \left(\left(\color{blue}{1 \cdot x} - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

  4. Applied prod-diff0.1

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

  5. Applied associate-+l+0.1

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

  6. Simplified0.2

    \[\leadsto \left(\mathsf{fma}\left(1, x, -\log y \cdot \left(y + 0.5\right)\right) + \color{blue}{\mathsf{fma}\left(-\log y, y + 0.5, \mathsf{fma}\left(y + 0.5, \log y, y\right)\right)}\right) - z\]
  7. Using strategy rm

  8. Applied fma-udef0.2

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

  9. Applied associate-+l+0.2

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

  10. Simplified0.1

    \[\leadsto \left(1 \cdot x + \color{blue}{\mathsf{fma}\left(y + 0.5, -\log y, \mathsf{fma}\left(-\log y, y + 0.5, \mathsf{fma}\left(y + 0.5, \log y, y\right)\right)\right)}\right) - z\]
  11. Using strategy rm

  12. Applied add-exp-log0.1

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"

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

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