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

Average Error: 0.2% → 0.19%

Time: 9.6s

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 0.2

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

2. Simplified 0.19

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

   [Start] 0.2	\[ x \cdot \sin y + z \cdot \cos y \]
   fma-def [=>] 0.19	\[ \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)} \]

3. Taylor expanded in x around 0 0.2

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

4. Simplified 0.19

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

   [Start] 0.2	\[ \cos y \cdot z + \sin y \cdot x \]
   *-commutative [<=] 0.2	\[ \cos y \cdot z + \color{blue}{x \cdot \sin y} \]
   fma-def [=>] 0.19	\[ \color{blue}{\mathsf{fma}\left(\cos y, z, x \cdot \sin y\right)} \]
   *-commutative [=>] 0.19	\[ \mathsf{fma}\left(\cos y, z, \color{blue}{\sin y \cdot x}\right) \]

5. Final simplification 0.19

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

Alternatives

Alternative 1
Error	0.2%
Cost	13248

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

Alternative 2
Error	15.92%
Cost	7250

\[\begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-33} \lor \neg \left(x \leq 1.65 \cdot 10^{-115} \lor \neg \left(x \leq 2.32 \cdot 10^{-29}\right) \land x \leq 6.5 \cdot 10^{+45}\right):\\ \;\;\;\;z + \sin y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\cos y \cdot z\\ \end{array} \]

Alternative 3
Error	25.51%
Cost	7120

\[\begin{array}{l} t_0 := \sin y \cdot x\\ t_1 := \cos y \cdot z\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.0285:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0052:\\ \;\;\;\;z + y \cdot x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	26.44%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0049 \lor \neg \left(y \leq 8400000000\right):\\ \;\;\;\;\cos y \cdot z\\ \mathbf{else}:\\ \;\;\;\;z + y \cdot x\\ \end{array} \]

Alternative 5
Error	59.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+75}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3.45 \cdot 10^{+177}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	49.09%
Cost	320

\[z + y \cdot x \]

Alternative 7
Error	62.16%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023089 
(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))))