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

Average Error: 0.1 → 0.1

Time: 11.3s

Precision: binary64

Cost: 19584

Math

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

↓

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

FPCore

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

↓

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

C

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(cos(y), x, (z * -sin(y)));
}

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right) + \cos y \cdot x} \]

3. Simplified 0.1

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

   [Start] 0.1	\[ -1 \cdot \left(z \cdot \sin y\right) + \cos y \cdot x \]
   +-commutative [=>] 0.1	\[ \color{blue}{\cos y \cdot x + -1 \cdot \left(z \cdot \sin y\right)} \]
   mul-1-neg [=>] 0.1	\[ \cos y \cdot x + \color{blue}{\left(-z \cdot \sin y\right)} \]
   distribute-rgt-neg-out [<=] 0.1	\[ \cos y \cdot x + \color{blue}{z \cdot \left(-\sin y\right)} \]
   fma-udef [<=] 0.1	\[ \color{blue}{\mathsf{fma}\left(\cos y, x, z \cdot \left(-\sin y\right)\right)} \]
   distribute-rgt-neg-out [=>] 0.1	\[ \mathsf{fma}\left(\cos y, x, \color{blue}{-z \cdot \sin y}\right) \]
   distribute-lft-neg-in [=>] 0.1	\[ \mathsf{fma}\left(\cos y, x, \color{blue}{\left(-z\right) \cdot \sin y}\right) \]

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	7.7
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-27} \lor \neg \left(x \leq 2.9 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{x}{\frac{1}{\cos y}} - \frac{z}{\frac{1}{y} + y \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]

Alternative 3
Error	15.8
Cost	7184

\[\begin{array}{l} t_0 := \cos y \cdot x\\ t_1 := z \cdot \left(-\sin y\right)\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.3:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.66:\\ \;\;\;\;\left(x + -0.5 \cdot \left(y \cdot \left(y \cdot x\right)\right)\right) - y \cdot z\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	8.9
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-116} \lor \neg \left(z \leq 4 \cdot 10^{-53}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot x\\ \end{array} \]

Alternative 5
Error	16.1
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0022 \lor \neg \left(y \leq 0.66\right):\\ \;\;\;\;\cos y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5 \cdot \left(y \cdot \left(y \cdot x\right)\right)\right) - y \cdot z\\ \end{array} \]

Alternative 6
Error	37.1
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+185} \lor \neg \left(z \leq 1.1 \cdot 10^{+114}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	30.2
Cost	320

\[x - y \cdot z \]

Alternative 8
Error	39.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))