Average Error: 0.1 → 0.1
Time: 6.9s
Precision: binary64
\[x \cdot \cos y + z \cdot \sin y \]
↓
\[\mathsf{fma}\left(\sin y, z, \cos y \cdot x\right) \]
x \cdot \cos y + z \cdot \sin y
↓
\mathsf{fma}\left(\sin y, z, \cos y \cdot x\right)

(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z) :precision binary64 (fma (sin y) z (* (cos y) x)))
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, (cos(y) * x));
}

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_binary640.9

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

  6. Applied pow1/3_binary6434.1

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

  7. Applied pow1/3_binary6434.4

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

  8. Applied pow1/3_binary6434.5

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

  9. Applied pow-sqr_binary6434.5

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

  10. Applied pow-prod-up_binary640.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2021329 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))