Average Error: 0.1 → 0.1
Time: 5.5s
Precision: binary64
\[x \cdot \sin y + z \cdot \cos y \]
↓
\[\mathsf{fma}\left(\cos y, z, x \cdot \sin y\right) \]
x \cdot \sin y + z \cdot \cos y
↓
\mathsf{fma}\left(\cos y, z, x \cdot \sin y\right)

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

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. Taylor expanded in x around 0 0.1

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

  4. Simplified0.1

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

  5. Applied *-un-lft-identity_binary640.1

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

  6. Applied associate-*l*_binary640.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2022067 
(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))))