Trigonometry A

Average Error: 0.1 → 0.1

Time: 9.8s

Precision: binary64

Cost: 13376

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

Math

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

↓

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

FPCore

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

↓

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

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) / (1.0 + (e * cos(v))));
}

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
    code = e * (sin(v) / (1.0d0 + (e * cos(v))))
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) {
	return e * (Math.sin(v) / (1.0 + (e * Math.cos(v))));
}

Python

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

↓

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

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) / Float64(1.0 + Float64(e * cos(v)))))
end

MATLAB

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

↓

function tmp = code(e, v)
	tmp = e * (sin(v) / (1.0 + (e * cos(v))));
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[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   rational.json-simplify-2 [=>] 0.1	\[ \frac{\color{blue}{\sin v \cdot e}}{1 + e \cdot \cos v} \]
   rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{e \cdot \frac{\sin v}{1 + e \cdot \cos v}} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.7
Cost	6848

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

Alternative 2
Error	1.4
Cost	6592

\[\sin v \cdot e \]

Alternative 3
Error	30.5
Cost	1344

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

Alternative 4
Error	30.8
Cost	1216

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

Alternative 5
Error	31.2
Cost	448

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

Alternative 6
Error	31.9
Cost	192

\[v \cdot e \]

Alternative 7
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023067 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))