Trigonometry A

Average Error: 0.1 → 0.1

Time: 10.8s

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
   (*.f64 (/.f64 (sin.f64 v) (fma.f64 e (cos.f64 v) 1)) e): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 v) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 e (cos.f64 v)) 1))) e): 1 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (sin.f64 v) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 e (cos.f64 v))))) e): 0 points increase in error, 0 points decrease in error
   (Rewrite=> *-commutative_binary64 (*.f64 e (/.f64 (sin.f64 v) (+.f64 1 (*.f64 e (cos.f64 v)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 e (sin.f64 v)) (+.f64 1 (*.f64 e (cos.f64 v))))): 4 points increase in error, 1 points decrease in error

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	6856

\[\begin{array}{l} t_0 := \sin v \cdot e\\ \mathbf{if}\;v \leq -3.6 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 5 \cdot 10^{-149}:\\ \;\;\;\;v \cdot \frac{e}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	28.0
Cost	1352

\[\begin{array}{l} t_0 := \frac{e}{\frac{1}{v} + -1}\\ \mathbf{if}\;v \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 3.2:\\ \;\;\;\;\frac{e}{v \cdot \left(0.16666666666666666 + e \cdot -0.3333333333333333\right) + \left(\frac{1}{v} + \frac{e}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	28.1
Cost	712

\[\begin{array}{l} t_0 := \frac{e}{\frac{1}{v} + -1}\\ \mathbf{if}\;v \leq -1.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 3.2:\\ \;\;\;\;v \cdot \frac{e}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	31.2
Cost	448

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

Alternative 7
Error	30.9
Cost	448

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

Alternative 8
Error	31.5
Cost	192

\[v \cdot e \]

Alternative 9
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2022321 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))