Average Error: 0.3 → 0.3
Time: 40.8s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(y + x\right)\right)\]
\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, \left(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(y + x\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r253211 = x;
        double r253212 = y;
        double r253213 = r253211 + r253212;
        double r253214 = log(r253213);
        double r253215 = z;
        double r253216 = log(r253215);
        double r253217 = r253214 + r253216;
        double r253218 = t;
        double r253219 = r253217 - r253218;
        double r253220 = a;
        double r253221 = 0.5;
        double r253222 = r253220 - r253221;
        double r253223 = log(r253218);
        double r253224 = r253222 * r253223;
        double r253225 = r253219 + r253224;
        return r253225;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r253226 = t;
        double r253227 = log(r253226);
        double r253228 = a;
        double r253229 = 0.5;
        double r253230 = r253228 - r253229;
        double r253231 = z;
        double r253232 = cbrt(r253231);
        double r253233 = log(r253232);
        double r253234 = r253233 - r253226;
        double r253235 = r253232 * r253232;
        double r253236 = log(r253235);
        double r253237 = r253234 + r253236;
        double r253238 = y;
        double r253239 = x;
        double r253240 = r253238 + r253239;
        double r253241 = log(r253240);
        double r253242 = r253237 + r253241;
        double r253243 = fma(r253227, r253230, r253242);
        return r253243;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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. Simplified0.3

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

  4. Applied add-cube-cbrt0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \left(\log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} - t\right)\right)\]

  5. Applied log-prod0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \left(\color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)} - t\right)\right)\]

  6. Applied associate--l+0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)}\right)\]

  7. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(y + x\right)\right)\]

Reproduce

herbie shell --seed 2019179 +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))))