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

Average Error: 0.1 → 0.1

Time: 10.2s

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

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

↓

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

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

Julia

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

↓

function code(x, y, z)
	return fma(x, sin(y), Float64(z * cos(y)))
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[(x * N[Sin[y], $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	16.4
Cost	7500

\[\begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.82:\\ \;\;\;\;x \cdot y + z \cdot \left(\left(\cos y + 1\right) + -1\right)\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.4
Cost	7252

\[\begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{+182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6.9 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.17:\\ \;\;\;\;z + x \cdot y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	16.0
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -6.9 \cdot 10^{-6} \lor \neg \left(y \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 5
Error	38.6
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+136}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6
Error	31.0
Cost	320

\[z + x \cdot y \]

Alternative 7
Error	39.1
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023060 
(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))))