Average Error: 0.1 → 0.2
Time: 13.6s
Precision: 64
\[x \cdot \sin y + z \cdot \cos y\]
↓
\[x \cdot \sin y + \left(\sqrt[3]{\cos y} \cdot z\right) \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\]
x \cdot \sin y + z \cdot \cos y
↓
x \cdot \sin y + \left(\sqrt[3]{\cos y} \cdot z\right) \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}
double f(double x, double y, double z) {
        double r3441745 = x;
        double r3441746 = y;
        double r3441747 = sin(r3441746);
        double r3441748 = r3441745 * r3441747;
        double r3441749 = z;
        double r3441750 = cos(r3441746);
        double r3441751 = r3441749 * r3441750;
        double r3441752 = r3441748 + r3441751;
        return r3441752;
}
↓
double f(double x, double y, double z) {
        double r3441753 = x;
        double r3441754 = y;
        double r3441755 = sin(r3441754);
        double r3441756 = r3441753 * r3441755;
        double r3441757 = cos(r3441754);
        double r3441758 = cbrt(r3441757);
        double r3441759 = z;
        double r3441760 = r3441758 * r3441759;
        double r3441761 = r3441757 * r3441757;
        double r3441762 = 0.3333333333333333;
        double r3441763 = pow(r3441761, r3441762);
        double r3441764 = r3441760 * r3441763;
        double r3441765 = r3441756 + r3441764;
        return r3441765;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Derivation

  1. Initial program 0.1

    \[x \cdot \sin y + z \cdot \cos y\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.4

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

  4. Applied associate-*r*0.4

    \[\leadsto x \cdot \sin y + \color{blue}{\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}}\]
  5. Using strategy rm

  6. Applied pow1/316.0

    \[\leadsto x \cdot \sin y + \left(z \cdot \left(\sqrt[3]{\cos y} \cdot \color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\cos y}\]

  7. Applied pow1/316.0

    \[\leadsto x \cdot \sin y + \left(z \cdot \left(\color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}} \cdot {\left(\cos y\right)}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{\cos y}\]

  8. Applied pow-prod-down0.2

    \[\leadsto x \cdot \sin y + \left(z \cdot \color{blue}{{\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\cos y}\]
  9. Using strategy rm

  10. Applied pow10.2

    \[\leadsto x \cdot \sin y + \left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos y}\right)}^{1}}\]

  11. Applied pow10.2

    \[\leadsto x \cdot \sin y + \color{blue}{{\left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right)}^{1}} \cdot {\left(\sqrt[3]{\cos y}\right)}^{1}\]

  12. Applied pow-prod-down0.2

    \[\leadsto x \cdot \sin y + \color{blue}{{\left(\left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y}\right)}^{1}}\]

  13. Simplified0.3

    \[\leadsto x \cdot \sin y + {\color{blue}{\left(\left(\sqrt[3]{\cos y} \cdot z\right) \cdot \sqrt[3]{\cos y \cdot \cos y}\right)}}^{1}\]
  14. Using strategy rm

  15. Applied pow1/30.2

    \[\leadsto x \cdot \sin y + {\left(\left(\sqrt[3]{\cos y} \cdot z\right) \cdot \color{blue}{{\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}}\right)}^{1}\]

  16. Final simplification0.2

    \[\leadsto x \cdot \sin y + \left(\sqrt[3]{\cos y} \cdot z\right) \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  (+ (* x (sin y)) (* z (cos y))))