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 r160370 = x;
        double r160371 = y;
        double r160372 = sin(r160371);
        double r160373 = r160370 * r160372;
        double r160374 = z;
        double r160375 = cos(r160371);
        double r160376 = r160374 * r160375;
        double r160377 = r160373 + r160376;
        return r160377;
}
↓
double f(double x, double y, double z) {
        double r160378 = x;
        double r160379 = y;
        double r160380 = sin(r160379);
        double r160381 = cos(r160379);
        double r160382 = 2.0;
        double r160383 = pow(r160381, r160382);
        double r160384 = 0.3333333333333333;
        double r160385 = pow(r160383, r160384);
        double r160386 = z;
        double r160387 = r160385 * r160386;
        double r160388 = cbrt(r160381);
        double r160389 = r160387 * r160388;
        double r160390 = fma(r160378, r160380, r160389);
        return r160390;
}

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))))