Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 23.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r427376 = x;
        double r427377 = y;
        double r427378 = r427376 + r427377;
        double r427379 = log(r427378);
        double r427380 = z;
        double r427381 = log(r427380);
        double r427382 = r427379 + r427381;
        double r427383 = t;
        double r427384 = r427382 - r427383;
        double r427385 = a;
        double r427386 = 0.5;
        double r427387 = r427385 - r427386;
        double r427388 = log(r427383);
        double r427389 = r427387 * r427388;
        double r427390 = r427384 + r427389;
        return r427390;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r427391 = a;
        double r427392 = 0.5;
        double r427393 = r427391 - r427392;
        double r427394 = t;
        double r427395 = sqrt(r427394);
        double r427396 = log(r427395);
        double r427397 = x;
        double r427398 = y;
        double r427399 = r427397 + r427398;
        double r427400 = log(r427399);
        double r427401 = z;
        double r427402 = log(r427401);
        double r427403 = r427402 - r427394;
        double r427404 = r427400 + r427403;
        double r427405 = fma(r427393, r427396, r427404);
        double r427406 = r427396 * r427393;
        double r427407 = r427405 + r427406;
        return r427407;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 0.3

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

3. Applied add-sqr-sqrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-rgt-in 0.3

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

6. Applied associate-+r+ 0.3

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

7. Simplified 0.3

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

8. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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