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

Average Error: 0.1 → 0.1

Time: 10.9s

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 0.1

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

2. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	7.9
Cost	7625

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-6} \lor \neg \left(z \leq 1.25 \cdot 10^{-46}\right):\\ \;\;\;\;x - z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\cos y}} - \frac{z}{\frac{1}{y} + y \cdot 0.16666666666666666}\\ \end{array} \]

Alternative 3
Error	10.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-21} \lor \neg \left(x \leq 7.2 \cdot 10^{+246}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \sin y\\ \end{array} \]

Alternative 4
Error	16.0
Cost	6920

\[\begin{array}{l} \mathbf{if}\;y \leq -0.11:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{elif}\;y \leq 3.9:\\ \;\;\;\;\left(x + \left(x \cdot -0.5\right) \cdot \left(y \cdot y\right)\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-\sin y\right)\\ \end{array} \]

Alternative 5
Error	15.7
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 0.041\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(x \cdot -0.5\right) \cdot \left(y \cdot y\right)\right) - z \cdot y\\ \end{array} \]

Alternative 6
Error	38.8
Cost	520

\[\begin{array}{l} \mathbf{if}\;z \leq 10^{+226}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+263}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	30.4
Cost	320

\[x - z \cdot y \]

Alternative 8
Error	39.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023057 
(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))))