Average Error: 0.1 → 0.1
Time: 6.4s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t
double f(double x, double y, double z, double t) {
        double r89190 = x;
        double r89191 = y;
        double r89192 = log(r89191);
        double r89193 = r89190 * r89192;
        double r89194 = r89193 - r89191;
        double r89195 = z;
        double r89196 = r89194 - r89195;
        double r89197 = t;
        double r89198 = log(r89197);
        double r89199 = r89196 + r89198;
        return r89199;
}
↓
double f(double x, double y, double z, double t) {
        double r89200 = x;
        double r89201 = 2.0;
        double r89202 = y;
        double r89203 = cbrt(r89202);
        double r89204 = log(r89203);
        double r89205 = r89201 * r89204;
        double r89206 = r89200 * r89204;
        double r89207 = fma(r89200, r89205, r89206);
        double r89208 = r89207 - r89202;
        double r89209 = z;
        double r89210 = r89208 - r89209;
        double r89211 = t;
        double r89212 = log(r89211);
        double r89213 = r89210 + r89212;
        return r89213;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

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

  3. Applied add-cube-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Simplified0.1

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]
  7. Using strategy rm

  8. Applied fma-def0.1

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

  9. Final simplification0.1

    \[\leadsto \left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))