Average Error: 0.1 → 0.1
Time: 6.1s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(x + \mathsf{fma}\left(-\left(y + 0.5\right), \log y, y\right)\right) - z
double f(double x, double y, double z) {
        double r314218 = x;
        double r314219 = y;
        double r314220 = 0.5;
        double r314221 = r314219 + r314220;
        double r314222 = log(r314219);
        double r314223 = r314221 * r314222;
        double r314224 = r314218 - r314223;
        double r314225 = r314224 + r314219;
        double r314226 = z;
        double r314227 = r314225 - r314226;
        return r314227;
}
↓
double f(double x, double y, double z) {
        double r314228 = x;
        double r314229 = y;
        double r314230 = 0.5;
        double r314231 = r314229 + r314230;
        double r314232 = -r314231;
        double r314233 = log(r314229);
        double r314234 = fma(r314232, r314233, r314229);
        double r314235 = r314228 + r314234;
        double r314236 = z;
        double r314237 = r314235 - r314236;
        return r314237;
}

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

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020089 +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))