math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 14

Speedup: 1.0×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - 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]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - 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]

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

double code(double re, double im) {
	return sin(re) * fma(0.5, exp(im), (0.5 / exp(im)));
}

Julia

function code(re, im)
	return Float64(sin(re) * fma(0.5, exp(im), Float64(0.5 / exp(im))))
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(0.5 * N[Exp[im], $MachinePrecision] + N[(0.5 / N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

   10. sub-neg 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

   11. *-commutative 100.0%

       \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 87.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.35)
   (+ (sin re) (* 0.5 (* (sin re) (* im im))))
   (* (sin re) (+ 0.001953125 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	} else {
		tmp = sin(re) * (0.001953125 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.35d0) then
        tmp = sin(re) + (0.5d0 * (sin(re) * (im * im)))
    else
        tmp = sin(re) * (0.001953125d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = Math.sin(re) + (0.5 * (Math.sin(re) * (im * im)));
	} else {
		tmp = Math.sin(re) * (0.001953125 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.35:
		tmp = math.sin(re) + (0.5 * (math.sin(re) * (im * im)))
	else:
		tmp = math.sin(re) * (0.001953125 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.35)
		tmp = Float64(sin(re) + Float64(0.5 * Float64(sin(re) * Float64(im * im))));
	else
		tmp = Float64(sin(re) * Float64(0.001953125 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.35)
		tmp = sin(re) + (0.5 * (sin(re) * (im * im)));
	else
		tmp = sin(re) * (0.001953125 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.35], N[(N[Sin[re], $MachinePrecision] + N[(0.5 * N[(N[Sin[re], $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3500000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.4%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. unpow2 84.4%

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

   6. Simplified 84.4%

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

   if 1.3500000000000001 < im

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out 99.9%

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

      6. cancel-sign-sub 99.9%

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

      7. distribute-lft-neg-in 99.9%

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

      8. neg-mul-1 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 99.9%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 99.9%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. distribute-rgt-in 99.9%

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

      15. distribute-lft-in 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around inf 99.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\sin re + 0.5 \cdot \left(\sin re \cdot \left(im \cdot im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 4: 87.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.35)
   (* (* (sin re) 0.5) (+ (* im im) 2.0))
   (* (sin re) (+ 0.001953125 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = (sin(re) * 0.5) * ((im * im) + 2.0);
	} else {
		tmp = sin(re) * (0.001953125 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.35d0) then
        tmp = (sin(re) * 0.5d0) * ((im * im) + 2.0d0)
    else
        tmp = sin(re) * (0.001953125d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.35) {
		tmp = (Math.sin(re) * 0.5) * ((im * im) + 2.0);
	} else {
		tmp = Math.sin(re) * (0.001953125 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.35:
		tmp = (math.sin(re) * 0.5) * ((im * im) + 2.0)
	else:
		tmp = math.sin(re) * (0.001953125 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.35)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(sin(re) * Float64(0.001953125 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.35)
		tmp = (sin(re) * 0.5) * ((im * im) + 2.0);
	else
		tmp = sin(re) * (0.001953125 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.35], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.3500000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 84.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 84.4%

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

   6. Simplified 84.4%

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

   if 1.3500000000000001 < im

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out 99.9%

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

      6. cancel-sign-sub 99.9%

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

      7. distribute-lft-neg-in 99.9%

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

      8. neg-mul-1 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 99.9%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 99.9%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. distribute-rgt-in 99.9%

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

      15. distribute-lft-in 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.1%

      \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around inf 99.1%

      \[\leadsto \color{blue}{\sin re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.35:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 5: 84.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2 \lor \neg \left(im \leq 6.4 \cdot 10^{+150}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 6.2) (not (<= im 6.4e+150)))
   (* (* (sin re) 0.5) (+ (* im im) 2.0))
   (* re (+ 0.001953125 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 6.2) || !(im <= 6.4e+150)) {
		tmp = (sin(re) * 0.5) * ((im * im) + 2.0);
	} else {
		tmp = re * (0.001953125 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= 6.2d0) .or. (.not. (im <= 6.4d+150))) then
        tmp = (sin(re) * 0.5d0) * ((im * im) + 2.0d0)
    else
        tmp = re * (0.001953125d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= 6.2) || !(im <= 6.4e+150)) {
		tmp = (Math.sin(re) * 0.5) * ((im * im) + 2.0);
	} else {
		tmp = re * (0.001953125 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= 6.2) or not (im <= 6.4e+150):
		tmp = (math.sin(re) * 0.5) * ((im * im) + 2.0)
	else:
		tmp = re * (0.001953125 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 6.2) || !(im <= 6.4e+150))
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(Float64(im * im) + 2.0));
	else
		tmp = Float64(re * Float64(0.001953125 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= 6.2) || ~((im <= 6.4e+150)))
		tmp = (sin(re) * 0.5) * ((im * im) + 2.0);
	else
		tmp = re * (0.001953125 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 6.2], N[Not[LessEqual[im, 6.4e+150]], $MachinePrecision]], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.20000000000000018 or 6.40000000000000031e150 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 86.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. unpow2 86.2%

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

   6. Simplified 86.2%

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

   if 6.20000000000000018 < im < 6.40000000000000031e150

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out 99.9%

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

      6. cancel-sign-sub 99.9%

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

      7. distribute-lft-neg-in 99.9%

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

      8. neg-mul-1 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 99.9%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 99.9%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. distribute-rgt-in 99.9%

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

      15. distribute-lft-in 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 88.2%

      \[\leadsto \color{blue}{re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2 \lor \neg \left(im \leq 6.4 \cdot 10^{+150}\right):\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 6: 69.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.2) (sin re) (* re (+ 0.001953125 (* 0.5 (exp im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = sin(re);
	} else {
		tmp = re * (0.001953125 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.2d0) then
        tmp = sin(re)
    else
        tmp = re * (0.001953125d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (0.001953125 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.2:
		tmp = math.sin(re)
	else:
		tmp = re * (0.001953125 + (0.5 * math.exp(im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.2)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(0.001953125 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.2)
		tmp = sin(re);
	else
		tmp = re * (0.001953125 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.2], N[Sin[re], $MachinePrecision], N[(re * N[(0.001953125 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.20000000000000018

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 64.4%

      \[\leadsto \color{blue}{\sin re} \]

   if 6.20000000000000018 < im

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out 99.9%

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

      6. cancel-sign-sub 99.9%

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

      7. distribute-lft-neg-in 99.9%

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

      8. neg-mul-1 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 99.9%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 99.9%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 99.9%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 99.9%

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

      13. remove-double-neg 99.9%

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

      14. distribute-rgt-in 99.9%

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

      15. distribute-lft-in 99.9%

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

   3. Simplified 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \sin re \cdot \mathsf{fma}\left(0.5, e^{im}, \color{blue}{0.001953125}\right) \]

   6. Taylor expanded in re around 0 79.7%

      \[\leadsto \color{blue}{re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.001953125 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 7: 62.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 240000000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 240000000000.0) (sin re) (* re (+ 1.0 (* 0.5 (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 240000000000.0) {
		tmp = sin(re);
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 240000000000.0d0) then
        tmp = sin(re)
    else
        tmp = re * (1.0d0 + (0.5d0 * (im * im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 240000000000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (1.0 + (0.5 * (im * im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 240000000000.0:
		tmp = math.sin(re)
	else:
		tmp = re * (1.0 + (0.5 * (im * im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 240000000000.0)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 240000000000.0)
		tmp = sin(re);
	else
		tmp = re * (1.0 + (0.5 * (im * im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 240000000000.0], N[Sin[re], $MachinePrecision], N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.4e11

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 62.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 2.4e11 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 53.6%

      \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   5. Step-by-step derivation

      1. unpow2 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in im around 0 53.6%

      \[\leadsto \sin re + \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
   8. Step-by-step derivation

      1. unpow2 53.6%

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

      2. associate-*r* 39.0%

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

      3. associate-*r* 39.0%

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

      4. *-commutative 39.0%

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

      5. associate-*r* 39.0%

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

      6. *-commutative 39.0%

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

   9. Simplified 39.0%

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

   10. Taylor expanded in re around 0 43.2%

       \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 43.2%

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

   12. Simplified 43.2%

       \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 240000000000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 28.8% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1050:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1050.0)
   (* 0.5 (* re 2.0))
   (+ 0.08333333333333333 (/ (/ 0.25 re) re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1050.0) {
		tmp = 0.5 * (re * 2.0);
	} else {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1050.0d0) then
        tmp = 0.5d0 * (re * 2.0d0)
    else
        tmp = 0.08333333333333333d0 + ((0.25d0 / re) / re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1050.0) {
		tmp = 0.5 * (re * 2.0);
	} else {
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1050.0:
		tmp = 0.5 * (re * 2.0)
	else:
		tmp = 0.08333333333333333 + ((0.25 / re) / re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1050.0)
		tmp = Float64(0.5 * Float64(re * 2.0));
	else
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / re) / re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1050.0)
		tmp = 0.5 * (re * 2.0);
	else
		tmp = 0.08333333333333333 + ((0.25 / re) / re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1050.0], N[(0.5 * N[(re * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 + N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1050

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 64.1%

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

   5. Taylor expanded in im around 0 35.9%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{2}\right) \]

   if 1050 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 14.9%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 14.9%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

      2. metadata-eval 14.9%

         \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

      3. unpow2 14.9%

         \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

      4. associate-/r* 14.9%

         \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]

   8. Simplified 14.9%

      \[\leadsto \color{blue}{0.08333333333333333 + \frac{\frac{0.25}{re}}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1050:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 + \frac{\frac{0.25}{re}}{re}\\ \end{array} \]

Alternative 9: 47.0% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ re \cdot \left(1 + 0.5 \cdot \left(im \cdot im\right)\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re (+ 1.0 (* 0.5 (* im im)))))

C

double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (0.5d0 * (im * im)))
end function

Java

public static double code(double re, double im) {
	return re * (1.0 + (0.5 * (im * im)));
}

Python

def code(re, im):
	return re * (1.0 + (0.5 * (im * im)))

Julia

function code(re, im)
	return Float64(re * Float64(1.0 + Float64(0.5 * Float64(im * im))))
end

MATLAB

function tmp = code(re, im)
	tmp = re * (1.0 + (0.5 * (im * im)));
end

Wolfram

code[re_, im_] := N[(re * N[(1.0 + N[(0.5 * N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

   10. sub-neg 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

   11. *-commutative 100.0%

       \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 75.3%

   \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
5. Step-by-step derivation

   1. unpow2 75.3%

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

6. Simplified 75.3%

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

7. Taylor expanded in im around 0 75.3%

   \[\leadsto \sin re + \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
8. Step-by-step derivation

   1. unpow2 75.3%

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

   2. associate-*r* 68.0%

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

   3. associate-*r* 68.0%

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

   4. *-commutative 68.0%

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

   5. associate-*r* 68.0%

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

   6. *-commutative 68.0%

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

9. Simplified 68.0%

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

10. Taylor expanded in re around 0 50.8%

    \[\leadsto \color{blue}{re \cdot \left(1 + 0.5 \cdot {im}^{2}\right)} \]
11. Step-by-step derivation

    1. unpow2 50.8%

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

12. Simplified 50.8%

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

13. Final simplification 50.8%

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

Alternative 10: 28.8% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1060:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1060.0) (* 0.5 (* re 2.0)) (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1060.0) {
		tmp = 0.5 * (re * 2.0);
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1060.0d0) then
        tmp = 0.5d0 * (re * 2.0d0)
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1060.0) {
		tmp = 0.5 * (re * 2.0);
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1060.0:
		tmp = 0.5 * (re * 2.0)
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1060.0)
		tmp = Float64(0.5 * Float64(re * 2.0));
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1060.0)
		tmp = 0.5 * (re * 2.0);
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1060.0], N[(0.5 * N[(re * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1060

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 64.1%

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

   5. Taylor expanded in im around 0 35.9%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{2}\right) \]

   if 1060 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 14.8%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 14.8%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 14.8%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1060:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 11: 28.8% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 720.0) (* 0.5 (* re 2.0)) (/ (/ 0.25 re) re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = 0.5 * (re * 2.0);
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 720.0d0) then
        tmp = 0.5d0 * (re * 2.0d0)
    else
        tmp = (0.25d0 / re) / re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 720.0) {
		tmp = 0.5 * (re * 2.0);
	} else {
		tmp = (0.25 / re) / re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 720.0:
		tmp = 0.5 * (re * 2.0)
	else:
		tmp = (0.25 / re) / re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 720.0)
		tmp = Float64(0.5 * Float64(re * 2.0));
	else
		tmp = Float64(Float64(0.25 / re) / re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 720.0)
		tmp = 0.5 * (re * 2.0);
	else
		tmp = (0.25 / re) / re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 720.0], N[(0.5 * N[(re * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / re), $MachinePrecision] / re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 720

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-out 100.0%

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

      7. neg-mul-1 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

      8. associate-*r* 100.0%

         \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

      9. distribute-rgt-out-- 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

      10. sub-neg 100.0%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

      11. neg-sub0 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 64.1%

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

   5. Taylor expanded in im around 0 35.9%

      \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{2}\right) \]

   if 720 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. cancel-sign-sub 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. neg-mul-1 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

      10. sub-neg 100.0%

          \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

      12. neg-mul-1 100.0%

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

      13. remove-double-neg 100.0%

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

      14. distribute-rgt-in 100.0%

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

      15. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 14.8%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 14.8%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

      2. associate-/r* 14.8%

         \[\leadsto \color{blue}{\frac{\frac{0.25}{re}}{re}} \]

   8. Simplified 14.8%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{re}}{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 720:\\ \;\;\;\;0.5 \cdot \left(re \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{re}}{re}\\ \end{array} \]

Alternative 12: 26.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(re \cdot 2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* re 2.0)))

C

double code(double re, double im) {
	return 0.5 * (re * 2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (re * 2.0d0)
end function

Java

public static double code(double re, double im) {
	return 0.5 * (re * 2.0);
}

Python

def code(re, im):
	return 0.5 * (re * 2.0)

Julia

function code(re, im)
	return Float64(0.5 * Float64(re * 2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (re * 2.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(re * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-lft-in 100.0%

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

   2. *-commutative 100.0%

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

   3. cancel-sign-sub 100.0%

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

   4. distribute-lft-neg-out 100.0%

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

   5. *-commutative 100.0%

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

   6. distribute-rgt-neg-out 100.0%

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

   7. neg-mul-1 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - e^{im} \cdot \color{blue}{\left(-1 \cdot \left(0.5 \cdot \sin re\right)\right)} \]

   8. associate-*r* 100.0%

      \[\leadsto e^{0 - im} \cdot \left(0.5 \cdot \sin re\right) - \color{blue}{\left(e^{im} \cdot -1\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   9. distribute-rgt-out-- 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} - e^{im} \cdot -1\right)} \]

   10. sub-neg 100.0%

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(e^{0 - im} + \left(-e^{im} \cdot -1\right)\right)} \]

   11. neg-sub0 100.0%

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

   12. *-commutative 100.0%

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

   13. neg-mul-1 100.0%

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

   14. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in re around 0 67.8%

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

5. Taylor expanded in im around 0 28.1%

   \[\leadsto 0.5 \cdot \left(re \cdot \color{blue}{2}\right) \]

6. Final simplification 28.1%

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

Alternative 13: 4.4% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -2.0)

C

double code(double re, double im) {
	return -2.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -2.0d0
end function

Java

public static double code(double re, double im) {
	return -2.0;
}

Python

def code(re, im):
	return -2.0

Julia

function code(re, im)
	return -2.0
end

MATLAB

function tmp = code(re, im)
	tmp = -2.0;
end

Wolfram

code[re_, im_] := -2.0

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

   10. sub-neg 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

   11. *-commutative 100.0%

       \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in im around 0 75.3%

   \[\leadsto \color{blue}{\sin re + 0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
5. Step-by-step derivation

   1. unpow2 75.3%

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

6. Simplified 75.3%

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

7. Taylor expanded in im around 0 75.3%

   \[\leadsto \sin re + \color{blue}{0.5 \cdot \left({im}^{2} \cdot \sin re\right)} \]
8. Step-by-step derivation

   1. unpow2 75.3%

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

   2. associate-*r* 68.0%

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

   3. associate-*r* 68.0%

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

   4. *-commutative 68.0%

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

   5. associate-*r* 68.0%

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

   6. *-commutative 68.0%

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

9. Simplified 68.0%

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

11. Applied egg-rr 3.9%

    \[\leadsto \sin re + \color{blue}{-2} \]

12. Taylor expanded in re around 0 4.0%

    \[\leadsto \color{blue}{-2} \]

13. Final simplification 4.0%

    \[\leadsto -2 \]

Alternative 14: 4.3% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

double code(double re, double im) {
	return 0.08333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.08333333333333333d0
end function

Java

public static double code(double re, double im) {
	return 0.08333333333333333;
}

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

function tmp = code(re, im)
	tmp = 0.08333333333333333;
end

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*r* 100.0%

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

   4. associate-*r* 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. cancel-sign-sub 100.0%

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

   7. distribute-lft-neg-in 100.0%

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

   8. neg-mul-1 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right) \]

   9. *-commutative 100.0%

      \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 - \color{blue}{\left(e^{0 - im} \cdot 0.5\right) \cdot -1}\right) \]

   10. sub-neg 100.0%

       \[\leadsto \sin re \cdot \color{blue}{\left(e^{im} \cdot 0.5 + \left(-\left(e^{0 - im} \cdot 0.5\right) \cdot -1\right)\right)} \]

   11. *-commutative 100.0%

       \[\leadsto \sin re \cdot \left(e^{im} \cdot 0.5 + \left(-\color{blue}{-1 \cdot \left(e^{0 - im} \cdot 0.5\right)}\right)\right) \]

   12. neg-mul-1 100.0%

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

   13. remove-double-neg 100.0%

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

   14. distribute-rgt-in 100.0%

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

   15. distribute-lft-in 100.0%

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

3. Simplified 100.0%

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

5. Applied egg-rr 11.9%

   \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

6. Taylor expanded in re around 0 11.7%

   \[\leadsto \color{blue}{0.08333333333333333 + 0.25 \cdot \frac{1}{{re}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 11.7%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} \]

   2. metadata-eval 11.7%

      \[\leadsto 0.08333333333333333 + \frac{\color{blue}{0.25}}{{re}^{2}} \]

   3. unpow2 11.7%

      \[\leadsto 0.08333333333333333 + \frac{0.25}{\color{blue}{re \cdot re}} \]

   4. associate-/r* 11.7%

      \[\leadsto 0.08333333333333333 + \color{blue}{\frac{\frac{0.25}{re}}{re}} \]

8. Simplified 11.7%

   \[\leadsto \color{blue}{0.08333333333333333 + \frac{\frac{0.25}{re}}{re}} \]

9. Taylor expanded in re around inf 4.2%

   \[\leadsto \color{blue}{0.08333333333333333} \]

10. Final simplification 4.2%

    \[\leadsto 0.08333333333333333 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))