Average Error: 0.1 → 0.2
Time: 4.7s
Precision: 64
\[x \cdot \sin y + z \cdot \cos y\]
↓
\[\mathsf{fma}\left(x, \sin y, \left({\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}} \cdot z\right) \cdot \sqrt[3]{\cos y}\right)\]
x \cdot \sin y + z \cdot \cos y
↓
\mathsf{fma}\left(x, \sin y, \left({\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}} \cdot z\right) \cdot \sqrt[3]{\cos y}\right)
double f(double x, double y, double z) {
        double r175622 = x;
        double r175623 = y;
        double r175624 = sin(r175623);
        double r175625 = r175622 * r175624;
        double r175626 = z;
        double r175627 = cos(r175623);
        double r175628 = r175626 * r175627;
        double r175629 = r175625 + r175628;
        return r175629;
}
↓
double f(double x, double y, double z) {
        double r175630 = x;
        double r175631 = y;
        double r175632 = sin(r175631);
        double r175633 = cos(r175631);
        double r175634 = 2.0;
        double r175635 = pow(r175633, r175634);
        double r175636 = 0.3333333333333333;
        double r175637 = pow(r175635, r175636);
        double r175638 = z;
        double r175639 = r175637 * r175638;
        double r175640 = cbrt(r175633);
        double r175641 = r175639 * r175640;
        double r175642 = fma(r175630, r175632, r175641);
        return r175642;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x \cdot \sin y + z \cdot \cos y\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.4

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

  5. Applied associate-*r*0.4

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

  7. Applied pow1/315.6

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

  8. Applied pow1/315.5

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

  9. Applied pow-prod-down0.2

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

  10. Simplified0.2

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot {\color{blue}{\left({\left(\cos y\right)}^{2}\right)}}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y}\right)\]
  11. Using strategy rm

  12. Applied *-commutative0.2

    \[\leadsto \mathsf{fma}\left(x, \sin y, \color{blue}{\left({\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}} \cdot z\right)} \cdot \sqrt[3]{\cos y}\right)\]

  13. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left({\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}} \cdot z\right) \cdot \sqrt[3]{\cos y}\right)\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))