Trigonometry A

Average Error: 0.1 → 0.1

Time: 12.2s

Precision: binary64

Cost: 19648

\[0 \leq e \land e \leq 1\]

Math

\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]

↓

\[e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

↓

(FPCore (e v) :precision binary64 (* e (/ (sin v) (fma e (cos v) 1.0))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

↓

double code(double e, double v) {
	return e * (sin(v) / fma(e, cos(v), 1.0));
}

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

↓

function code(e, v)
	return Float64(e * Float64(sin(v) / fma(e, cos(v), 1.0)))
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] / N[(e * N[Cos[v], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]
   Proof

   [Start] 0.1	\[ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   associate-*r/ [<=] 0.1	\[ \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]
   *-commutative [<=] 0.1	\[ \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]
   +-commutative [=>] 0.1	\[ \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \cdot e \]
   fma-def [=>] 0.1	\[ \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot e \]

3. Final simplification 0.1

   \[\leadsto e \cdot \frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \]

Alternatives

Alternative 1
Error	0.1
Cost	13376

\[\frac{\sin v \cdot e}{1 + e \cdot \cos v} \]

Alternative 2
Error	0.2
Cost	13248

\[\frac{\sin v}{\cos v + \frac{1}{e}} \]

Alternative 3
Error	0.9
Cost	6857

\[\begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-6} \lor \neg \left(v \leq 3.5 \cdot 10^{-51}\right):\\ \;\;\;\;\sin v \cdot e\\ \mathbf{else}:\\ \;\;\;\;e \cdot \frac{v}{e + 1}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	6848

\[\frac{\sin v \cdot e}{e + 1} \]

Alternative 5
Error	29.9
Cost	1472

\[e \cdot \frac{1}{v \cdot \left(e \cdot -0.5 + \left(e + 1\right) \cdot 0.16666666666666666\right) + \left(\frac{e}{v} + \frac{1}{v}\right)} \]

Alternative 6
Error	30.0
Cost	1344

\[\frac{e}{v \cdot \left(e \cdot -0.5 + \left(e + 1\right) \cdot 0.16666666666666666\right) + \left(\frac{e}{v} + \frac{1}{v}\right)} \]

Alternative 7
Error	30.0
Cost	1088

\[\frac{e}{\left(\frac{e}{v} + \frac{1}{v}\right) + v \cdot \left(e \cdot -0.5 + 0.16666666666666666\right)} \]

Alternative 8
Error	30.2
Cost	960

\[\frac{e}{\left(\frac{e}{v} + \frac{1}{v}\right) + -0.3333333333333333 \cdot \left(v \cdot e\right)} \]

Alternative 9
Error	30.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;e \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;e \cdot \frac{1}{\frac{1}{v} + v \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{\frac{e + 1}{v}}\\ \end{array} \]

Alternative 10
Error	30.0
Cost	708

\[\begin{array}{l} \mathbf{if}\;e \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{e}{\frac{1}{v} + v \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{\frac{e + 1}{v}}\\ \end{array} \]

Alternative 11
Error	30.9
Cost	448

\[\left(v \cdot e\right) \cdot \left(1 - e\right) \]

Alternative 12
Error	30.6
Cost	448

\[e \cdot \frac{v}{e + 1} \]

Alternative 13
Error	31.2
Cost	192

\[v \cdot e \]

Alternative 14
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023031 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))