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

Average Accuracy: 99.8% → 99.8%

Time: 12.7s

Precision: binary64

Cost: 32896

Math

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

↓

\[\begin{array}{l} t_0 := z \cdot \sin y\\ \mathsf{fma}\left(-z, \sin y, t_0\right) + \left(x \cdot \cos y - t_0\right) \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (sin y))))
   (+ (fma (- z) (sin y) t_0) (- (* x (cos y)) t_0))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) - (z * sin(y));
}

↓

double code(double x, double y, double z) {
	double t_0 = z * sin(y);
	return fma(-z, sin(y), t_0) + ((x * cos(y)) - t_0);
}

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(z * sin(y))
	return Float64(fma(Float64(-z), sin(y), t_0) + Float64(Float64(x * cos(y)) - t_0))
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_] := Block[{t$95$0 = N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]}, N[(N[((-z) * N[Sin[y], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ x \cdot \cos y - z \cdot \sin y \]
   *-commutative [=>] 99.8	\[ x \cdot \cos y - \color{blue}{\sin y \cdot z} \]
   prod-diff [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(x, \cos y, -z \cdot \sin y\right) + \mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right)} \]
   fma-neg [<=] 99.8	\[ \color{blue}{\left(x \cdot \cos y - z \cdot \sin y\right)} + \mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right) \]
   +-commutative [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right) + \left(x \cdot \cos y - z \cdot \sin y\right)} \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13248

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

Alternative 2
Accuracy	75.2%
Cost	7580

\[\begin{array}{l} t_0 := x \cdot \cos y\\ t_1 := z \cdot \left(-\sin y\right)\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.18 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.005:\\ \;\;\;\;\left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - z \cdot y\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	86.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-118} \lor \neg \left(z \leq 2.05 \cdot 10^{-44}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]

Alternative 4
Accuracy	74.2%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \lor \neg \left(y \leq 0.019\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + -0.5 \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) - z \cdot y\\ \end{array} \]

Alternative 5
Accuracy	40.5%
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-240}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	51.9%
Cost	320

\[x - z \cdot y \]

Alternative 7
Accuracy	39.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(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))))