math.sin on complex, real part

Average Error: 0.0 → 0.0

Time: 19.3s

Precision: binary64

Cost: 19712

Math

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

↓

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

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (* 0.5 (* (sin re) (+ (exp (- im)) (exp 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 0.5 * (sin(re) * (exp(-im) + exp(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 = 0.5d0 * (sin(re) * (exp(-im) + exp(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 0.5 * (Math.sin(re) * (Math.exp(-im) + Math.exp(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 0.5 * (math.sin(re) * (math.exp(-im) + math.exp(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(0.5 * Float64(sin(re) * Float64(exp(Float64(-im)) + exp(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 = 0.5 * (sin(re) * (exp(-im) + exp(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[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

   [Start] 0.0	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   rational.json-simplify-2 [=>] 0.0	\[ \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]
   rational.json-simplify-43 [<=] 0.0	\[ \color{blue}{\sin re \cdot \left(\left(e^{0 - im} + e^{im}\right) \cdot 0.5\right)} \]
   rational.json-simplify-2 [<=] 0.0	\[ \sin re \cdot \color{blue}{\left(0.5 \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]
   rational.json-simplify-6 [<=] 0.0	\[ \sin re \cdot \left(0.5 \cdot \color{blue}{\left(1 \cdot \left(e^{0 - im} + e^{im}\right)\right)}\right) \]
   rational.json-simplify-2 [<=] 0.0	\[ \sin re \cdot \color{blue}{\left(\left(1 \cdot \left(e^{0 - im} + e^{im}\right)\right) \cdot 0.5\right)} \]
   rational.json-simplify-2 [=>] 0.0	\[ \sin re \cdot \left(\color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot 1\right)} \cdot 0.5\right) \]
   metadata-eval [<=] 0.0	\[ \sin re \cdot \left(\left(\left(e^{0 - im} + e^{im}\right) \cdot \color{blue}{\left(-1 \cdot -1\right)}\right) \cdot 0.5\right) \]
   rational.json-simplify-43 [<=] 0.0	\[ \sin re \cdot \left(\color{blue}{\left(-1 \cdot \left(\left(e^{0 - im} + e^{im}\right) \cdot -1\right)\right)} \cdot 0.5\right) \]
   rational.json-simplify-2 [<=] 0.0	\[ \sin re \cdot \left(\left(-1 \cdot \color{blue}{\left(-1 \cdot \left(e^{0 - im} + e^{im}\right)\right)}\right) \cdot 0.5\right) \]
   rational.json-simplify-43 [<=] 0.0	\[ \color{blue}{0.5 \cdot \left(\sin re \cdot \left(-1 \cdot \left(-1 \cdot \left(e^{0 - im} + e^{im}\right)\right)\right)\right)} \]
   rational.json-simplify-2 [=>] 0.0	\[ 0.5 \cdot \left(\sin re \cdot \left(-1 \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot -1\right)}\right)\right) \]
   rational.json-simplify-43 [=>] 0.0	\[ 0.5 \cdot \left(\sin re \cdot \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \left(-1 \cdot -1\right)\right)}\right) \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.8
Cost	13312

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

Alternative 2
Error	1.1
Cost	6720

\[0.5 \cdot \left(\sin re \cdot 2\right) \]

Alternative 3
Error	32.2
Cost	320

\[0.5 \cdot \left(re + re\right) \]

Reproduce

herbie shell --seed 2023064 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))