Average Error: 0.1 → 0.1
Time: 6.1s
Precision: binary64
\[x \cdot \cos y + z \cdot \sin y \]
↓
\[\mathsf{fma}\left(\sin y, z, {1}^{0.3333333333333333} \cdot \left(x \cdot \cos y\right)\right) \]
x \cdot \cos y + z \cdot \sin y
↓
\mathsf{fma}\left(\sin y, z, {1}^{0.3333333333333333} \cdot \left(x \cdot \cos y\right)\right)

(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z)
 :precision binary64
 (fma (sin y) z (* (pow 1.0 0.3333333333333333) (* x (cos y)))))
double code(double x, double y, double z) {
	return (x * cos(y)) + (z * sin(y));
}
↓
double code(double x, double y, double z) {
	return fma(sin(y), z, (pow(1.0, 0.3333333333333333) * (x * cos(y))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{\cos y \cdot x + \sin y \cdot z} \]

  4. Simplified0.1

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

  5. Applied add-cube-cbrt_binary640.4

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

  6. Applied associate-*l*_binary640.4

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

  7. Taylor expanded in y around inf 0.2

    \[\leadsto \mathsf{fma}\left(\sin y, z, \color{blue}{{\left({\cos y}^{2}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{\cos y} \cdot x\right)\right) \]

  8. Applied *-un-lft-identity_binary640.2

    \[\leadsto \mathsf{fma}\left(\sin y, z, {\left({\color{blue}{\left(1 \cdot \cos y\right)}}^{2}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\cos y} \cdot x\right)\right) \]

  9. Applied unpow-prod-down_binary640.2

    \[\leadsto \mathsf{fma}\left(\sin y, z, {\color{blue}{\left({1}^{2} \cdot {\cos y}^{2}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\cos y} \cdot x\right)\right) \]

  10. Applied unpow-prod-down_binary640.2

    \[\leadsto \mathsf{fma}\left(\sin y, z, \color{blue}{\left({\left({1}^{2}\right)}^{0.3333333333333333} \cdot {\left({\cos y}^{2}\right)}^{0.3333333333333333}\right)} \cdot \left(\sqrt[3]{\cos y} \cdot x\right)\right) \]

  11. Applied associate-*l*_binary640.2

    \[\leadsto \mathsf{fma}\left(\sin y, z, \color{blue}{{\left({1}^{2}\right)}^{0.3333333333333333} \cdot \left({\left({\cos y}^{2}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\cos y} \cdot x\right)\right)}\right) \]

  12. Simplified0.1

    \[\leadsto \mathsf{fma}\left(\sin y, z, {\left({1}^{2}\right)}^{0.3333333333333333} \cdot \color{blue}{\left(x \cdot \cos y\right)}\right) \]

  13. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\sin y, z, {1}^{0.3333333333333333} \cdot \left(x \cdot \cos y\right)\right) \]

Reproduce

herbie shell --seed 2022104 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))