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

Average Error: 0.1 → 0.1

Time: 10.3s

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 0.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)} \]
   Proof
   (fma.f64 x (sin.f64 y) (*.f64 z (cos.f64 y))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (sin.f64 y)) (*.f64 z (cos.f64 y)))): 2 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13248

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

Alternative 2
Error	16.3
Cost	7252

\[\begin{array}{l} t_0 := z \cdot \cos y\\ t_1 := x \cdot \sin y\\ \mathbf{if}\;y \leq -7.108449941537077 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -560.315978532113:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3707.5009248211663:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{elif}\;y \leq 4.6130015529212706 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1302365386005706 \cdot 10^{+292}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.9
Cost	6984

\[\begin{array}{l} t_0 := z \cdot \cos y\\ \mathbf{if}\;z \leq -5.798011671593216 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.907462836815851 \cdot 10^{+182}:\\ \;\;\;\;z + x \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	17.0
Cost	6856

\[\begin{array}{l} t_0 := x \cdot \sin y\\ \mathbf{if}\;y \leq -560.315978532113:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.388100587500896 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	30.6
Cost	6592

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

Alternative 6
Error	37.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.0640540526316818 \cdot 10^{-165}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.0760043640335563 \cdot 10^{-211}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 7
Error	30.6
Cost	320

\[z + x \cdot y \]

Alternative 8
Error	38.9
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022295 
(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))))