Average Error: 0.1 → 0.1
Time: 6.9s
Precision: binary64
Cost: 13376
\[0 \leq e \land e \leq 1\]
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
\[e \cdot \frac{\sin v}{1 + \cos v \cdot e} \]
(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))
(FPCore (e v) :precision binary64 (* e (/ (sin v) (+ 1.0 (* (cos v) e)))))
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 + (cos(v) * e)));
}
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 + (cos(v) * e)))
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) {
	return e * (Math.sin(v) / (1.0 + (Math.cos(v) * e)));
}
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 + (math.cos(v) * e)))
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(cos(v) * e))))
end
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 + (cos(v) * e)));
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_] := N[(e * N[(N[Sin[v], $MachinePrecision] / N[(1.0 + N[(N[Cos[v], $MachinePrecision] * e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\frac{e \cdot \sin v}{1 + e \cdot \cos v}
e \cdot \frac{\sin v}{1 + \cos v \cdot e}

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. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sin v}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot e} \]
    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): 0 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))))): 3 points increase in error, 3 points decrease in error
  3. Taylor expanded in v around inf 0.1

    \[\leadsto \color{blue}{\frac{\sin v}{1 + \cos v \cdot e}} \cdot e \]
  4. Final simplification0.1

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

Alternatives

Alternative 1
Error0.3
Cost13248
\[\frac{\sin v}{\cos v + \frac{1}{e}} \]
Alternative 2
Error1.4
Cost6592
\[\sin v \cdot e \]
Alternative 3
Error30.3
Cost1344
\[\frac{e}{v \cdot \left(e \cdot -0.5 + -0.16666666666666666 \cdot \left(-1 - e\right)\right) + \left(\frac{e}{v} + \frac{1}{v}\right)} \]
Alternative 4
Error31.3
Cost448
\[v \cdot \left(e - e \cdot e\right) \]
Alternative 5
Error31.0
Cost448
\[v \cdot \frac{e}{1 + e} \]
Alternative 6
Error31.6
Cost192
\[v \cdot e \]
Alternative 7
Error61.1
Cost64
\[v \]

Error

Reproduce

herbie shell --seed 2022328 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))