Trigonometry A

Average Error: 0.2% → 0.21%

Time: 9.9s

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 0.2

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

2. Simplified 0.21

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

   [Start] 0.2	\[ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   associate-*r/ [<=] 0.21	\[ \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]
   *-commutative [<=] 0.21	\[ \color{blue}{\frac{\sin v}{1 + e \cdot \cos v} \cdot e} \]
   +-commutative [=>] 0.21	\[ \frac{\sin v}{\color{blue}{e \cdot \cos v + 1}} \cdot e \]
   fma-def [=>] 0.21	\[ \frac{\sin v}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot e \]

3. Final simplification 0.21

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

Alternatives

Alternative 1
Error	0.2%
Cost	13376

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

Alternative 2
Error	0.38%
Cost	13248

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

Alternative 3
Error	1.2%
Cost	6848

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

Alternative 4
Error	2.33%
Cost	6592

\[\sin v \cdot e \]

Alternative 5
Error	43.69%
Cost	576

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

Alternative 6
Error	48.08%
Cost	448

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

Alternative 7
Error	47.52%
Cost	448

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

Alternative 8
Error	48.64%
Cost	192

\[v \cdot e \]

Alternative 9
Error	95.45%
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023102 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))