Average Error: 0.1 → 0.1
Time: 2.6s
Precision: 64
\[\left(x \cdot y\right) \cdot \left(1 - y\right)\]
↓
\[\left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x\]
\left(x \cdot y\right) \cdot \left(1 - y\right)
↓
\left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x
double f(double x, double y) {
        double r25312 = x;
        double r25313 = y;
        double r25314 = r25312 * r25313;
        double r25315 = 1.0;
        double r25316 = r25315 - r25313;
        double r25317 = r25314 * r25316;
        return r25317;
}
↓
double f(double x, double y) {
        double r25318 = 1.0;
        double r25319 = 1.0;
        double r25320 = cbrt(r25319);
        double r25321 = 3.0;
        double r25322 = pow(r25320, r25321);
        double r25323 = r25318 * r25322;
        double r25324 = y;
        double r25325 = -r25324;
        double r25326 = r25323 + r25325;
        double r25327 = x;
        double r25328 = r25326 * r25327;
        double r25329 = r25328 * r25324;
        double r25330 = fma(r25325, r25318, r25324);
        double r25331 = r25330 * r25324;
        double r25332 = r25331 * r25327;
        double r25333 = r25329 + r25332;
        return r25333;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(x \cdot y\right) \cdot \left(1 - y\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.5

    \[\leadsto \left(x \cdot y\right) \cdot \left(1 - \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right)\]

  4. Applied add-cube-cbrt0.5

    \[\leadsto \left(x \cdot y\right) \cdot \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}} - \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)\]

  5. Applied prod-diff0.5

    \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{y}, \sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\right)}\]

  6. Applied distribute-lft-in0.5

    \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, -\sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \left(x \cdot y\right) \cdot \mathsf{fma}\left(-\sqrt[3]{y}, \sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}\]

  7. Simplified0.1

    \[\leadsto \color{blue}{\left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y} + \left(x \cdot y\right) \cdot \mathsf{fma}\left(-\sqrt[3]{y}, \sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\]

  8. Simplified0.1

    \[\leadsto \left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \color{blue}{\left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x}\]

  9. Final simplification0.1

    \[\leadsto \left(\left(1 \cdot {\left(\sqrt[3]{1}\right)}^{3} + \left(-y\right)\right) \cdot x\right) \cdot y + \left(\mathsf{fma}\left(-y, 1, y\right) \cdot y\right) \cdot x\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1 y)))