Average Error: 0.1 → 0.1
Time: 8.5s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)
double f(double x, double y, double z) {
        double r308241 = x;
        double r308242 = y;
        double r308243 = 0.5;
        double r308244 = r308242 + r308243;
        double r308245 = log(r308242);
        double r308246 = r308244 * r308245;
        double r308247 = r308241 - r308246;
        double r308248 = r308247 + r308242;
        double r308249 = z;
        double r308250 = r308248 - r308249;
        return r308250;
}
↓
double f(double x, double y, double z) {
        double r308251 = x;
        double r308252 = y;
        double r308253 = log(r308252);
        double r308254 = 0.5;
        double r308255 = r308252 + r308254;
        double r308256 = z;
        double r308257 = fma(r308253, r308255, r308256);
        double r308258 = r308257 - r308252;
        double r308259 = r308251 - r308258;
        return r308259;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 +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))