Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B

Average Error: 0.1 → 0.1

Time: 9.3s

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))

↓

(FPCore (x y z) :precision binary64 (fma (cos y) z (* (sin y) x)))

C

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, (sin(y) * x));
}

Julia

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end

↓

function code(x, y, z)
	return fma(cos(y), z, Float64(sin(y) * x))
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[Cos[y], $MachinePrecision] * z + N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Taylor expanded in x around 0 0.1

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

3. Simplified 0.1

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

   [Start] 0.1	\[ \cos y \cdot z + \sin y \cdot x \]
   *-commutative [<=] 0.1	\[ \cos y \cdot z + \color{blue}{x \cdot \sin y} \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(\cos y, z, x \cdot \sin y\right)} \]
   *-commutative [=>] 0.1	\[ \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y \cdot x}\right) \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	16.7
Cost	7252

\[\begin{array}{l} t_0 := \cos y \cdot z\\ t_1 := \sin y \cdot x\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -0.0072:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-13}:\\ \;\;\;\;z + y \cdot x\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	9.4
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-133} \lor \neg \left(x \leq 9.2 \cdot 10^{-48}\right):\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]

Alternative 4
Error	16.1
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.008 \lor \neg \left(y \leq 4.5 \cdot 10^{-13}\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 5
Error	31.1
Cost	320

\[z + y \cdot x \]

Alternative 6
Error	39.6
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022356 
(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))))