?

Average Error: 0.08% → 0.07%
Time: 11.1s
Precision: binary64
Cost: 26304

?

\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
↓
\[\begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \frac{t_0}{e^{im}} + t_0 \cdot e^{im} \end{array} \]
(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))
↓
(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re)))) (+ (/ t_0 (exp im)) (* t_0 (exp im)))))
double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}
↓
double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	return (t_0 / exp(im)) + (t_0 * exp(im));
}

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function
↓
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    t_0 = 0.5d0 * sin(re)
    code = (t_0 / exp(im)) + (t_0 * exp(im))
end function

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}
↓
public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	return (t_0 / Math.exp(im)) + (t_0 * Math.exp(im));
}

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))
↓
def code(re, im):
	t_0 = 0.5 * math.sin(re)
	return (t_0 / math.exp(im)) + (t_0 * math.exp(im))

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end
↓
function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	return Float64(Float64(t_0 / exp(im)) + Float64(t_0 * exp(im)))
end

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end
↓
function tmp = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = (t_0 / exp(im)) + (t_0 * exp(im));
end

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[Exp[im], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)
↓
\begin{array}{l}
t_0 := 0.5 \cdot \sin re\\
\frac{t_0}{e^{im}} + t_0 \cdot e^{im}
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.08

    \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

  2. Simplified0.08

    \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]
    Proof

    [Start]0.08

    \[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

    sub0-neg [=>]0.08

    \[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

  3. Applied egg-rr0.08

    \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot e^{-im} + \left(0.5 \cdot \sin re\right) \cdot e^{im}} \]

  4. Applied egg-rr0.07

    \[\leadsto \color{blue}{\frac{0.5 \cdot \sin re}{e^{im}}} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

  5. Final simplification0.07

    \[\leadsto \frac{0.5 \cdot \sin re}{e^{im}} + \left(0.5 \cdot \sin re\right) \cdot e^{im} \]

Alternatives

Alternative 1
Error0.08%
Cost19712
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + e^{-im}\right) \]
Alternative 2
Error1.14%
Cost13696
\[\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + \left(0.041666666666666664 \cdot {im}^{4} + 1\right)\right) \]
Alternative 3
Error1.34%
Cost13376
\[\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right) \]
Alternative 4
Error1.35%
Cost13248
\[\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]
Alternative 5
Error1.34%
Cost6976
\[\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]
Alternative 6
Error1.93%
Cost6464
\[\sin re \]
Alternative 7
Error50.65%
Cost576
\[0.5 \cdot \left(re \cdot \left(im \cdot im + 2\right)\right) \]
Alternative 8
Error50.65%
Cost576
\[re + \left(im \cdot im\right) \cdot \left(0.5 \cdot re\right) \]
Alternative 9
Error50.95%
Cost64
\[re \]

Error

Reproduce?

herbie shell --seed 2023090 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))