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

Average Accuracy: 99.8% → 99.8%

Time: 13.9s

Precision: binary64

Cost: 32896

Math

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

↓

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

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(Float64(fma(Float64(-z), sin(y), t_0) + 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[(N[((-z) * N[Sin[y], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{z \cdot \left(-\sin y\right) + \left(x \cdot \cos y + \mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right)\right)} \]
   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-def [<=] 99.8	\[ \color{blue}{\left(x \cdot \cos y + \left(-z \cdot \sin y\right)\right)} + \mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right) \]
   +-commutative [=>] 99.8	\[ \color{blue}{\left(\left(-z \cdot \sin y\right) + x \cdot \cos y\right)} + \mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right) \]
   associate-+l+ [=>] 99.8	\[ \color{blue}{\left(-z \cdot \sin y\right) + \left(x \cdot \cos y + \mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right)\right)} \]
   distribute-rgt-neg-in [=>] 99.8	\[ \color{blue}{z \cdot \left(-\sin y\right)} + \left(x \cdot \cos y + \mathsf{fma}\left(-z, \sin y, z \cdot \sin y\right)\right) \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13248

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

Alternative 2
Accuracy	74.7%
Cost	7184

\[\begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.0006:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 660:\\ \;\;\;\;x - z \cdot y\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	74.5%
Cost	7184

\[\begin{array}{l} t_0 := z \cdot \left(-\sin y\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 660:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	85.8%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+64} \lor \neg \left(x \leq 3 \cdot 10^{+113}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]

Alternative 5
Accuracy	74.6%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.036 \lor \neg \left(y \leq 660\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot y\\ \end{array} \]

Alternative 6
Accuracy	41.6%
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-165}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-151}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	51.6%
Cost	320

\[x - z \cdot y \]

Alternative 8
Accuracy	39.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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))))