Trigonometry A

Average Error: 0.1 → 0.1

Time: 15.5s

Precision: binary64

Cost: 26624

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

Math

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

↓

\[\begin{array}{l} t_0 := e \cdot \cos v\\ \frac{\left(\sin v \cdot e\right) \cdot \left(1 - t_0\right)}{1 - {t_0}^{2}} \end{array} \]

FPCore

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

↓

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (* e (cos v))))
   (/ (* (* (sin v) e) (- 1.0 t_0)) (- 1.0 (pow t_0 2.0)))))

C

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

↓

double code(double e, double v) {
	double t_0 = e * cos(v);
	return ((sin(v) * e) * (1.0 - t_0)) / (1.0 - pow(t_0, 2.0));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

↓

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: t_0
    t_0 = e * cos(v)
    code = ((sin(v) * e) * (1.0d0 - t_0)) / (1.0d0 - (t_0 ** 2.0d0))
end function

Java

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

↓

public static double code(double e, double v) {
	double t_0 = e * Math.cos(v);
	return ((Math.sin(v) * e) * (1.0 - t_0)) / (1.0 - Math.pow(t_0, 2.0));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

↓

def code(e, v):
	t_0 = e * math.cos(v)
	return ((math.sin(v) * e) * (1.0 - t_0)) / (1.0 - math.pow(t_0, 2.0))

Julia

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

↓

function code(e, v)
	t_0 = Float64(e * cos(v))
	return Float64(Float64(Float64(sin(v) * e) * Float64(1.0 - t_0)) / Float64(1.0 - (t_0 ^ 2.0)))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

↓

function tmp = code(e, v)
	t_0 = e * cos(v);
	tmp = ((sin(v) * e) * (1.0 - t_0)) / (1.0 - (t_0 ^ 2.0));
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_] := Block[{t$95$0 = N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.1

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}} \]
   Proof

   [Start] 0.1	\[ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   associate-/l* [=>] 0.3	\[ \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}} \]

3. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{e}{1 - {\left(e \cdot \cos v\right)}^{2}} \cdot \left(\sin v \cdot \left(1 - e \cdot \cos v\right)\right)} \]

4. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\left(\sin v \cdot e\right) \cdot \left(1 - e \cdot \cos v\right)}{1 - {\left(e \cdot \cos v\right)}^{2}}} \]
   Proof

   [Start] 0.1	\[ \frac{e}{1 - {\left(e \cdot \cos v\right)}^{2}} \cdot \left(\sin v \cdot \left(1 - e \cdot \cos v\right)\right) \]
   associate-*r* [=>] 0.1	\[ \color{blue}{\left(\frac{e}{1 - {\left(e \cdot \cos v\right)}^{2}} \cdot \sin v\right) \cdot \left(1 - e \cdot \cos v\right)} \]
   associate-*l/ [=>] 0.1	\[ \color{blue}{\frac{e \cdot \sin v}{1 - {\left(e \cdot \cos v\right)}^{2}}} \cdot \left(1 - e \cdot \cos v\right) \]
   associate-*l/ [=>] 0.1	\[ \color{blue}{\frac{\left(e \cdot \sin v\right) \cdot \left(1 - e \cdot \cos v\right)}{1 - {\left(e \cdot \cos v\right)}^{2}}} \]
   *-commutative [<=] 0.1	\[ \frac{\color{blue}{\left(\sin v \cdot e\right)} \cdot \left(1 - e \cdot \cos v\right)}{1 - {\left(e \cdot \cos v\right)}^{2}} \]

5. Final simplification 0.1

   \[\leadsto \frac{\left(\sin v \cdot e\right) \cdot \left(1 - e \cdot \cos v\right)}{1 - {\left(e \cdot \cos v\right)}^{2}} \]

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	0.1
Cost	13376

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

Alternative 3
Error	0.2
Cost	13248

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

Alternative 4
Error	0.8
Cost	7232

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

Alternative 5
Error	1.0
Cost	6848

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

Alternative 6
Error	0.8
Cost	6848

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

Alternative 7
Error	1.4
Cost	6592

\[\sin v \cdot e \]

Alternative 8
Error	30.7
Cost	1216

\[\frac{e}{\left(v \cdot 0.16666666666666666 + -0.3333333333333333 \cdot \left(v \cdot e\right)\right) + \left(\frac{e}{v} + \frac{1}{v}\right)} \]

Alternative 9
Error	30.7
Cost	832

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

Alternative 10
Error	31.3
Cost	576

\[\frac{e}{v \cdot 0.16666666666666666 + \frac{1}{v}} \]

Alternative 11
Error	31.7
Cost	448

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

Alternative 12
Error	31.4
Cost	448

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

Alternative 13
Error	32.0
Cost	192

\[v \cdot e \]

Alternative 14
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023038 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))