math.sin on complex, real part

Average Accuracy: 99.9% → 98.7%

Time: 11.3s

Precision: binary64

Cost: 13696

Math

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

↓

\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \]

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))
  (+ (+ 2.0 (* im im)) (* (pow im 4.0) 0.08333333333333333))))

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)) * ((2.0 + (im * im)) + (pow(im, 4.0) * 0.08333333333333333));
}

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)) * ((2.0d0 + (im * im)) + ((im ** 4.0d0) * 0.08333333333333333d0))
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)) * ((2.0 + (im * im)) + (Math.pow(im, 4.0) * 0.08333333333333333));
}

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)) * ((2.0 + (im * im)) + (math.pow(im, 4.0) * 0.08333333333333333))

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(0.5 * sin(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64((im ^ 4.0) * 0.08333333333333333)))
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)) * ((2.0 + (im * im)) + ((im ^ 4.0) * 0.08333333333333333));
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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

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

   [Start] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   sub0-neg [=>] 99.9	\[ \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Taylor expanded in im around 0 98.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

4. Simplified 98.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]
   Proof

   [Start] 98.7	\[ \left(0.5 \cdot \sin re\right) \cdot \left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right) \]
   associate-+r+ [=>] 98.7	\[ \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + {im}^{2}\right) + 0.08333333333333333 \cdot {im}^{4}\right)} \]
   unpow2 [=>] 98.7	\[ \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + \color{blue}{im \cdot im}\right) + 0.08333333333333333 \cdot {im}^{4}\right) \]
   *-commutative [=>] 98.7	\[ \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + \color{blue}{{im}^{4} \cdot 0.08333333333333333}\right) \]

5. Final simplification 98.7%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \]

Alternatives

Alternative 1
Accuracy	98.5%
Cost	13376

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

Alternative 2
Accuracy	98.5%
Cost	6976

\[\sin re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]

Alternative 3
Accuracy	98.0%
Cost	6464

\[\sin re \]

Alternative 4
Accuracy	50.6%
Cost	576

\[re \cdot \left(0.5 \cdot \left(im \cdot im\right) + 1\right) \]

Alternative 5
Accuracy	50.6%
Cost	576

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

Alternative 6
Accuracy	50.3%
Cost	64

\[re \]

Reproduce

herbie shell --seed 2023133 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))