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

Percentage Accurate: 99.8% → 99.8%

Time: 14.8s

Precision: binary64

Cost: 19584

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 99.8%

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

2. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin y, -z, x \cdot \cos y\right)} \]
   Step-by-step derivation

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

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	86.1%
Cost	13321

\[\begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+75} \lor \neg \left(x \leq 3.3 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin y, -z, x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	13248

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

Alternative 3
Accuracy	75.2%
Cost	7580

\[\begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.00046:\\ \;\;\;\;x - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	75.2%
Cost	7580

\[\begin{array}{l} t_0 := \sin y \cdot \left(-z\right)\\ t_1 := x \cdot \cos y\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.00032:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.00146:\\ \;\;\;\;\mathsf{fma}\left(-y, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	86.2%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+72} \lor \neg \left(x \leq 5.6 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - \sin y \cdot z\\ \end{array} \]

Alternative 6
Accuracy	74.1%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.023 \lor \neg \left(y \leq 3.2 \cdot 10^{+18}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot z\\ \end{array} \]

Alternative 7
Accuracy	41.3%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq 4.7 \cdot 10^{+165}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 8
Accuracy	52.9%
Cost	320

\[x - y \cdot z \]

Alternative 9
Accuracy	39.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023160 
(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))))