Trigonometry A

Average Error: 0.1 → 0.1

Time: 9.5s

Precision: binary64

Cost: 13632

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

Math

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

↓

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

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)) 2.0) -2.0))))

C

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

↓

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (((e * cos(v)) + 2.0) + -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
    code = (e * sin(v)) / (1.0d0 + (((e * cos(v)) + 2.0d0) + (-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) {
	return (e * Math.sin(v)) / (1.0 + (((e * Math.cos(v)) + 2.0) + -2.0));
}

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)) + 2.0) + -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)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(Float64(Float64(e * cos(v)) + 2.0) + -2.0)))
end

MATLAB

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

↓

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (((e * cos(v)) + 2.0) + -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_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13632

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

Alternative 2
Error	0.3
Cost	13376

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

Alternative 3
Error	0.1
Cost	13376

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

Alternative 4
Error	0.2
Cost	13248

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

Alternative 5
Error	0.9
Cost	7104

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

Alternative 6
Error	0.8
Cost	6848

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

Alternative 7
Error	1.6
Cost	6592

\[e \cdot \sin v \]

Alternative 8
Error	30.0
Cost	1344

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

Alternative 9
Error	31.1
Cost	448

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

Alternative 10
Error	30.7
Cost	448

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

Alternative 11
Error	30.7
Cost	448

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

Alternative 12
Error	31.4
Cost	192

\[e \cdot v \]

Alternative 13
Error	61.1
Cost	64

\[v \]

Reproduce

herbie shell --seed 2023083 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))