Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.2 → 0.2

Time: 34.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r225280 = x;
        double r225281 = y;
        double r225282 = r225280 + r225281;
        double r225283 = log(r225282);
        double r225284 = z;
        double r225285 = log(r225284);
        double r225286 = r225283 + r225285;
        double r225287 = t;
        double r225288 = r225286 - r225287;
        double r225289 = a;
        double r225290 = 0.5;
        double r225291 = r225289 - r225290;
        double r225292 = log(r225287);
        double r225293 = r225291 * r225292;
        double r225294 = r225288 + r225293;
        return r225294;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r225295 = t;
        double r225296 = log(r225295);
        double r225297 = a;
        double r225298 = 0.5;
        double r225299 = r225297 - r225298;
        double r225300 = y;
        double r225301 = x;
        double r225302 = r225300 + r225301;
        double r225303 = log(r225302);
        double r225304 = z;
        double r225305 = log(r225304);
        double r225306 = r225305 - r225295;
        double r225307 = r225303 + r225306;
        double r225308 = fma(r225296, r225299, r225307);
        return r225308;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

3. Final simplification 0.2

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

Reproduce

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

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

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