math.sin on complex, real part

Average Error: 0.1 → 0.1

Time: 14.5s

Precision: binary64

Cost: 19904

Math

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

↓

\[\frac{\frac{\sin re}{-2}}{\frac{e^{im}}{-1 - e^{im + im}}} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

↓

(FPCore (re im)
 :precision binary64
 (/ (/ (sin re) -2.0) (/ (exp im) (- -1.0 (exp (+ im im))))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

↓

double code(double re, double im) {
	return (sin(re) / -2.0) / (exp(im) / (-1.0 - exp((im + im))));
}

Fortran

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
    code = (sin(re) / (-2.0d0)) / (exp(im) / ((-1.0d0) - exp((im + im))))
end function

Java

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) {
	return (Math.sin(re) / -2.0) / (Math.exp(im) / (-1.0 - Math.exp((im + im))));
}

Python

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

↓

def code(re, im):
	return (math.sin(re) / -2.0) / (math.exp(im) / (-1.0 - math.exp((im + im))))

Julia

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

↓

function code(re, im)
	return Float64(Float64(sin(re) / -2.0) / Float64(exp(im) / Float64(-1.0 - exp(Float64(im + im)))))
end

MATLAB

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

↓

function tmp = code(re, im)
	tmp = (sin(re) / -2.0) / (exp(im) / (-1.0 - exp((im + im))));
end

Wolfram

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_] := N[(N[(N[Sin[re], $MachinePrecision] / -2.0), $MachinePrecision] / N[(N[Exp[im], $MachinePrecision] / N[(-1.0 - N[Exp[N[(im + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   rational.json-simplify-13 [=>] 0.1	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Applied egg-rr 0.1

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(1 + e^{im + im}\right) \cdot e^{-im}\right)} \]

4. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\frac{\frac{\sin re}{-2}}{\frac{-e^{im}}{1 + e^{im + im}}}} \]

5. Simplified 0.1

   \[\leadsto \color{blue}{\frac{\frac{\sin re}{-2}}{\frac{e^{im}}{-1 - e^{im + im}}}} \]
   Proof

   [Start] 0.1	\[ \frac{\frac{\sin re}{-2}}{\frac{-e^{im}}{1 + e^{im + im}}} \]
   rational.json-simplify-17 [=>] 0.1	\[ \frac{\frac{\sin re}{-2}}{\frac{-e^{im}}{\color{blue}{e^{im + im} - -1}}} \]
   rational.json-simplify-50 [<=] 0.1	\[ \frac{\frac{\sin re}{-2}}{\color{blue}{\frac{e^{im}}{-1 - e^{im + im}}}} \]

6. Final simplification 0.1

   \[\leadsto \frac{\frac{\sin re}{-2}}{\frac{e^{im}}{-1 - e^{im + im}}} \]

Alternatives

Alternative 1
Error	0.1
Cost	19712

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

Alternative 2
Error	0.8
Cost	13312

\[\left(0.5 \cdot \sin re\right) \cdot \left(2 + {im}^{2}\right) \]

Alternative 3
Error	1.2
Cost	6464

\[\sin re \]

Alternative 4
Error	31.2
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023074 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))