Average Error: 0.3 → 0.3
Time: 31.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(x + y\right) - \left(\left(t - \log z\right) - \left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \left(\frac{1}{3} \cdot \log t\right)\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(x + y\right) - \left(\left(t - \log z\right) - \left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \left(\frac{1}{3} \cdot \log t\right)\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r67102 = x;
        double r67103 = y;
        double r67104 = r67102 + r67103;
        double r67105 = log(r67104);
        double r67106 = z;
        double r67107 = log(r67106);
        double r67108 = r67105 + r67107;
        double r67109 = t;
        double r67110 = r67108 - r67109;
        double r67111 = a;
        double r67112 = 0.5;
        double r67113 = r67111 - r67112;
        double r67114 = log(r67109);
        double r67115 = r67113 * r67114;
        double r67116 = r67110 + r67115;
        return r67116;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r67117 = x;
        double r67118 = y;
        double r67119 = r67117 + r67118;
        double r67120 = log(r67119);
        double r67121 = t;
        double r67122 = z;
        double r67123 = log(r67122);
        double r67124 = r67121 - r67123;
        double r67125 = a;
        double r67126 = 0.5;
        double r67127 = r67125 - r67126;
        double r67128 = 2.0;
        double r67129 = cbrt(r67121);
        double r67130 = log(r67129);
        double r67131 = r67128 * r67130;
        double r67132 = r67127 * r67131;
        double r67133 = 0.3333333333333333;
        double r67134 = log(r67121);
        double r67135 = r67133 * r67134;
        double r67136 = r67127 * r67135;
        double r67137 = r67132 + r67136;
        double r67138 = r67124 - r67137;
        double r67139 = r67120 - r67138;
        return r67139;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied distribute-lft-in0.3

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

  7. Simplified0.3

    \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right)} + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\right)\right)\]

  8. Taylor expanded around inf 0.3

    \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right) + \color{blue}{\left(a \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right) - 0.5 \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)\right)}\right)\right)\]

  9. Simplified0.3

    \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right) + \color{blue}{\left(a - 0.5\right) \cdot \left(-\frac{-1}{3} \cdot \log t\right)}\right)\right)\]

  10. Final simplification0.3

    \[\leadsto \log \left(x + y\right) - \left(\left(t - \log z\right) - \left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \left(\frac{1}{3} \cdot \log t\right)\right)\right)\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))