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

Average Error: 0.1 → 0.1

Time: 1.3min

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	9.9
Cost	7572

\[\begin{array}{l} t_0 := \cos y \cdot x\\ t_1 := x + \begin{array}{l} \mathbf{if}\;z \ne 0:\\ \;\;\;\;\frac{-\sin y}{\frac{-1}{z}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot z\\ \end{array}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.9
Cost	7120

\[\begin{array}{l} t_0 := \sin y \cdot z\\ \mathbf{if}\;y \leq -2 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;\cos y \cdot x\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;y \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	16.3
Cost	6856

\[\begin{array}{l} t_0 := \cos y \cdot x\\ \mathbf{if}\;y \leq -0.205:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0076:\\ \;\;\;\;y \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	37.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-256}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	30.6
Cost	320

\[y \cdot z + x \]

Alternative 7
Error	38.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))