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

Average Accuracy: 99.8% → 99.8%

Time: 9.9s

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

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

↓

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

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(x, sin(y), (z * cos(y)));
}

Julia

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

↓

function code(x, y, z)
	return fma(x, sin(y), Float64(z * cos(y)))
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[(x * N[Sin[y], $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Simplified 99.8%

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

   [Start] 99.8	\[ x \cdot \sin y + z \cdot \cos y \]
   fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)} \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13248

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

Alternative 2
Accuracy	75.3%
Cost	7516

\[\begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.00031:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0125:\\ \;\;\;\;z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	75.3%
Cost	7516

\[\begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{+235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -0.00182:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	86.5%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-33} \lor \neg \left(x \leq 1.55 \cdot 10^{-56}\right):\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \cos y\\ \end{array} \]

Alternative 5
Accuracy	75.2%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00026 \lor \neg \left(y \leq 0.00015\right):\\ \;\;\;\;z \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;z + x \cdot y\\ \end{array} \]

Alternative 6
Accuracy	42.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{+224}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+178}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7
Accuracy	52.9%
Cost	320

\[z + x \cdot y \]

Alternative 8
Accuracy	38.9%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023160 
(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))))