Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Average Error: 0.2% → 0.2%

Time: 9.7s

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (fma x (cos y) (* 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(x, cos(y), (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(x, cos(y), Float64(z * 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[(x * N[Cos[y], $MachinePrecision] + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

   [Start] 0.2	\[ x \cdot \cos y + z \cdot \sin y \]
   fma-def [=>] 0.2	\[ \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2%
Cost	13248

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

Alternative 2
Error	25.74%
Cost	7517

\[\begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := z \cdot \sin y\\ \mathbf{if}\;y \leq -8 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0048:\\ \;\;\;\;x + y \cdot z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+153} \lor \neg \left(y \leq 2.05 \cdot 10^{+251}\right) \land y \leq 5.8 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.65%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-100} \lor \neg \left(z \leq 3.5 \cdot 10^{-70}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 4
Error	25.46%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00033 \lor \neg \left(y \leq 8 \cdot 10^{-23}\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]

Alternative 5
Error	58.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-153}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-236}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	48.22%
Cost	320

\[x + y \cdot z \]

Alternative 7
Error	60.71%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023103 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))