Average Error: 0.1 → 0.1
Time: 7.8s
Precision: binary64
Cost: 26624
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
↓
\[\begin{array}{l} t_0 := e \cdot \cos v\\ \frac{e \cdot \sin v}{1 - {t_0}^{2}} \cdot \left(1 - t_0\right) \end{array} \]
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
↓
(FPCore (e v)
 :precision binary64
 (let* ((t_0 (* e (cos v))))
   (* (/ (* e (sin v)) (- 1.0 (pow t_0 2.0))) (- 1.0 t_0))))
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 ((e * sin(v)) / (1.0 - pow(t_0, 2.0))) * (1.0 - t_0);
}

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 = ((e * sin(v)) / (1.0d0 - (t_0 ** 2.0d0))) * (1.0d0 - t_0)
end function

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 ((e * Math.sin(v)) / (1.0 - Math.pow(t_0, 2.0))) * (1.0 - t_0);
}

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 ((e * math.sin(v)) / (1.0 - math.pow(t_0, 2.0))) * (1.0 - t_0)

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(e * sin(v)) / Float64(1.0 - (t_0 ^ 2.0))) * Float64(1.0 - t_0))
end

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 = ((e * sin(v)) / (1.0 - (t_0 ^ 2.0))) * (1.0 - t_0);
end

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[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

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

  2. Applied egg-rr0.1

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

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error0.8
Cost13376
\[\sin v \cdot \left(e - e \cdot \left(e \cdot \cos v\right)\right) \]
Alternative 2
Error0.1
Cost13376
\[\sin v \cdot \frac{e}{1 + e \cdot \cos v} \]
Alternative 3
Error1.1
Cost6848
\[\sin v \cdot \left(e - e \cdot e\right) \]
Alternative 4
Error0.9
Cost6848
\[\frac{\sin v}{\frac{e + 1}{e}} \]
Alternative 5
Error0.8
Cost6848
\[\sin v \cdot \frac{e}{e + 1} \]
Alternative 6
Error1.5
Cost6592
\[e \cdot \sin v \]
Alternative 7
Error31.0
Cost1472
\[\frac{1}{\frac{v \cdot \left(e \cdot -0.5 + -0.16666666666666666 \cdot \left(-1 - e\right)\right) + \left(\frac{e}{v} + \frac{1}{v}\right)}{e}} \]
Alternative 8
Error30.3
Cost1472
\[e \cdot \frac{1}{v \cdot \left(e \cdot -0.5 + -0.16666666666666666 \cdot \left(-1 - e\right)\right) + \left(\frac{e}{v} + \frac{1}{v}\right)} \]
Alternative 9
Error31.1
Cost832
\[\frac{e \cdot v}{1 - e \cdot e} \cdot \left(1 - e\right) \]
Alternative 10
Error31.4
Cost448
\[v \cdot \left(e \cdot \left(1 - e\right)\right) \]
Alternative 11
Error31.1
Cost448
\[\frac{v}{1 + \frac{1}{e}} \]
Alternative 12
Error31.1
Cost448
\[v \cdot \frac{e}{e + 1} \]
Alternative 13
Error31.7
Cost192
\[e \cdot v \]
Alternative 14
Error61.1
Cost64
\[v \]

Error

Reproduce

herbie shell --seed 2022283 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))