Average Error: 0.1 → 0.1
Time: 22.6s
Precision: 64
\[\left(x \cdot \log y - z\right) - y\]
↓
\[\left(\mathsf{fma}\left(\log \left(\sqrt[3]{y}\right), x + x, \log \left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right) \cdot x\right) - z\right) - y\]
\left(x \cdot \log y - z\right) - y
↓
\left(\mathsf{fma}\left(\log \left(\sqrt[3]{y}\right), x + x, \log \left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right) \cdot x\right) - z\right) - y
double f(double x, double y, double z) {
        double r1135078 = x;
        double r1135079 = y;
        double r1135080 = log(r1135079);
        double r1135081 = r1135078 * r1135080;
        double r1135082 = z;
        double r1135083 = r1135081 - r1135082;
        double r1135084 = r1135083 - r1135079;
        return r1135084;
}
↓
double f(double x, double y, double z) {
        double r1135085 = y;
        double r1135086 = cbrt(r1135085);
        double r1135087 = log(r1135086);
        double r1135088 = x;
        double r1135089 = r1135088 + r1135088;
        double r1135090 = sqrt(r1135085);
        double r1135091 = cbrt(r1135090);
        double r1135092 = r1135091 * r1135091;
        double r1135093 = log(r1135092);
        double r1135094 = r1135093 * r1135088;
        double r1135095 = fma(r1135087, r1135089, r1135094);
        double r1135096 = z;
        double r1135097 = r1135095 - r1135096;
        double r1135098 = r1135097 - r1135085;
        return r1135098;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  3. Applied add-cube-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-rgt-in0.1

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

  6. Simplified0.1

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

  8. Applied fma-def0.1

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

  10. Applied add-sqr-sqrt0.1

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

  11. Applied cbrt-prod0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  (- (- (* x (log y)) z) y))