math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 14

Speedup: 1.0×

• 50.2% 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

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (* (sin re) (fma 0.5 (exp im) (/ 0.5 (exp im)))))

C

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

Julia

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

Wolfram

NOTE: im should be positive before calling this function
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%

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

   4. associate-*r* 100.0%

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

   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

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (* (* (sin re) 0.5) (+ (exp im) (exp (- im)))))

C

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

Fortran

NOTE: im should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
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: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 6.2)
   (* (* (sin re) 0.5) (fma im im 2.0))
   (if (<= im 2.7e+154)
     (* re (+ 0.5 (* 0.5 (exp im))))
     (* (sin re) (+ 1.0 (* im (+ 0.5 (* im 0.25))))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 6.2) {
		tmp = (sin(re) * 0.5) * fma(im, im, 2.0);
	} else if (im <= 2.7e+154) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = sin(re) * (1.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 6.2)
		tmp = Float64(Float64(sin(re) * 0.5) * fma(im, im, 2.0));
	elseif (im <= 2.7e+154)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 6.2], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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-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 82.5%

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

      1. +-commutative 82.5%

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

      2. unpow2 82.5%

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

      3. fma-def 82.5%

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

   6. Simplified 82.5%

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

   if 6.20000000000000018 < im < 2.70000000000000006e154

   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%

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

      4. associate-*r* 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in re around 0 74.2%

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

   if 2.70000000000000006e154 < 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%

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

      4. associate-*r* 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\left(1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \cdot \sin re \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \left(1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)}\right) \cdot \sin re \]

      2. *-commutative 100.0%

         \[\leadsto \left(1 + \left(\color{blue}{im \cdot 0.5} + 0.25 \cdot {im}^{2}\right)\right) \cdot \sin re \]

      3. *-commutative 100.0%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{{im}^{2} \cdot 0.25}\right)\right) \cdot \sin re \]

      4. unpow2 100.0%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{\left(im \cdot im\right)} \cdot 0.25\right)\right) \cdot \sin re \]

      5. associate-*l* 100.0%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{im \cdot \left(im \cdot 0.25\right)}\right)\right) \cdot \sin re \]

      6. distribute-lft-out 100.0%

         \[\leadsto \left(1 + \color{blue}{im \cdot \left(0.5 + im \cdot 0.25\right)}\right) \cdot \sin re \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \cdot \sin re \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6.2:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.5× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* (sin re) (+ 0.5 (* 0.5 (exp im)))))

C

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

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (0.5d0 + (0.5d0 * exp(im)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(0.5 + 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%

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

   4. associate-*r* 100.0%

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

   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 74.5%

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

5. Taylor expanded in re around inf 74.5%

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

   1. *-commutative 74.5%

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

7. Simplified 74.5%

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

8. Final simplification 74.5%

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

Alternative 5: 92.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 2.7)
   (sin re)
   (if (<= im 2.7e+154)
     (* re (+ 0.5 (* 0.5 (exp im))))
     (* (sin re) (+ 1.0 (* im (+ 0.5 (* im 0.25))))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 2.7) {
		tmp = sin(re);
	} else if (im <= 2.7e+154) {
		tmp = re * (0.5 + (0.5 * exp(im)));
	} else {
		tmp = sin(re) * (1.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.7d0) then
        tmp = sin(re)
    else if (im <= 2.7d+154) then
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    else
        tmp = sin(re) * (1.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 2.7) {
		tmp = Math.sin(re);
	} else if (im <= 2.7e+154) {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	} else {
		tmp = Math.sin(re) * (1.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 2.7:
		tmp = math.sin(re)
	elif im <= 2.7e+154:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	else:
		tmp = math.sin(re) * (1.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 2.7)
		tmp = sin(re);
	elseif (im <= 2.7e+154)
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.7)
		tmp = sin(re);
	elseif (im <= 2.7e+154)
		tmp = re * (0.5 + (0.5 * exp(im)));
	else
		tmp = sin(re) * (1.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 2.7], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.7e+154], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 2.7000000000000002

   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%

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

      4. associate-*r* 100.0%

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

      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 65.7%

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

   if 2.7000000000000002 < im < 2.70000000000000006e154

   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%

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

      4. associate-*r* 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in re around 0 74.2%

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

   if 2.70000000000000006e154 < 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%

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

      4. associate-*r* 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{\left(1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \cdot \sin re \]
   9. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \left(1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)}\right) \cdot \sin re \]

      2. *-commutative 100.0%

         \[\leadsto \left(1 + \left(\color{blue}{im \cdot 0.5} + 0.25 \cdot {im}^{2}\right)\right) \cdot \sin re \]

      3. *-commutative 100.0%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{{im}^{2} \cdot 0.25}\right)\right) \cdot \sin re \]

      4. unpow2 100.0%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{\left(im \cdot im\right)} \cdot 0.25\right)\right) \cdot \sin re \]

      5. associate-*l* 100.0%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{im \cdot \left(im \cdot 0.25\right)}\right)\right) \cdot \sin re \]

      6. distribute-lft-out 100.0%

         \[\leadsto \left(1 + \color{blue}{im \cdot \left(0.5 + im \cdot 0.25\right)}\right) \cdot \sin re \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \cdot \sin re \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.7:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 6: 72.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 400.0) (sin re) (* re (* 0.5 (fma im im 2.0)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 400.0) {
		tmp = sin(re);
	} else {
		tmp = re * (0.5 * fma(im, im, 2.0));
	}
	return tmp;
}

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 400.0)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(0.5 * fma(im, im, 2.0)));
	end
	return tmp
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 400.0], N[Sin[re], $MachinePrecision], N[(re * N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 400

   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%

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

      4. associate-*r* 100.0%

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

      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 65.7%

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

   if 400 < 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 56.9%

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

      1. +-commutative 56.9%

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

      2. unpow2 56.9%

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

      3. fma-def 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in re around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. *-commutative 47.2%

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

      3. associate-*l* 47.2%

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

      4. *-commutative 47.2%

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

      5. +-commutative 47.2%

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

      6. unpow2 47.2%

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

      7. fma-def 47.2%

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

   9. Simplified 47.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\right)\\ \end{array} \]

Alternative 7: 86.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 4:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 4.0) (sin re) (* re (+ 0.5 (* 0.5 (exp im))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 4.0) {
		tmp = sin(re);
	} else {
		tmp = re * (0.5 + (0.5 * exp(im)));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 4.0d0) then
        tmp = sin(re)
    else
        tmp = re * (0.5d0 + (0.5d0 * exp(im)))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 4.0) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (0.5 + (0.5 * Math.exp(im)));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 4.0:
		tmp = math.sin(re)
	else:
		tmp = re * (0.5 + (0.5 * math.exp(im)))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 4.0)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(0.5 + Float64(0.5 * exp(im))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 4.0)
		tmp = sin(re);
	else
		tmp = re * (0.5 + (0.5 * exp(im)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 4.0], N[Sin[re], $MachinePrecision], N[(re * N[(0.5 + N[(0.5 * N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4

   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%

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

      4. associate-*r* 100.0%

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

      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 65.7%

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

   if 4 < 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%

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

      4. associate-*r* 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in re around 0 73.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 + 0.5 \cdot e^{im}\right)\\ \end{array} \]

Alternative 8: 72.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 660.0) (sin re) (* re (* 0.5 (pow im 2.0)))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = sin(re);
	} else {
		tmp = re * (0.5 * pow(im, 2.0));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 660.0d0) then
        tmp = sin(re)
    else
        tmp = re * (0.5d0 * (im ** 2.0d0))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 660.0) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (0.5 * Math.pow(im, 2.0));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 660.0:
		tmp = math.sin(re)
	else:
		tmp = re * (0.5 * math.pow(im, 2.0))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 660.0)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(0.5 * (im ^ 2.0)));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 660.0)
		tmp = sin(re);
	else
		tmp = re * (0.5 * (im ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 660.0], N[Sin[re], $MachinePrecision], N[(re * N[(0.5 * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 660

   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%

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

      4. associate-*r* 100.0%

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

      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 65.7%

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

   if 660 < 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 56.9%

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

      1. +-commutative 56.9%

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

      2. unpow2 56.9%

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

      3. fma-def 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in re around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. *-commutative 47.2%

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

      3. associate-*l* 47.2%

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

      4. *-commutative 47.2%

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

      5. +-commutative 47.2%

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

      6. unpow2 47.2%

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

      7. fma-def 47.2%

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

   9. Simplified 47.2%

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

   10. Taylor expanded in im around inf 47.2%

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

       1. associate-*r* 47.2%

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

   12. Simplified 47.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 660:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot {im}^{2}\right)\\ \end{array} \]

Alternative 9: 72.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 2300000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im)
 :precision binary64
 (if (<= im 2300000.0) (sin re) (* re (+ 1.0 (* im (+ 0.5 (* im 0.25)))))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 2300000.0) {
		tmp = sin(re);
	} else {
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2300000.0d0) then
        tmp = sin(re)
    else
        tmp = re * (1.0d0 + (im * (0.5d0 + (im * 0.25d0))))
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 2300000.0) {
		tmp = Math.sin(re);
	} else {
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 2300000.0:
		tmp = math.sin(re)
	else:
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))))
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 2300000.0)
		tmp = sin(re);
	else
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2300000.0)
		tmp = sin(re);
	else
		tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 2300000.0], N[Sin[re], $MachinePrecision], N[(re * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.3e6

   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%

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

      4. associate-*r* 100.0%

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

      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 65.7%

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

   if 2.3e6 < 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%

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

      4. associate-*r* 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in re around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in im around 0 56.9%

      \[\leadsto \color{blue}{\left(1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \cdot \sin re \]
   9. Step-by-step derivation

      1. +-commutative 56.9%

         \[\leadsto \left(1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)}\right) \cdot \sin re \]

      2. *-commutative 56.9%

         \[\leadsto \left(1 + \left(\color{blue}{im \cdot 0.5} + 0.25 \cdot {im}^{2}\right)\right) \cdot \sin re \]

      3. *-commutative 56.9%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{{im}^{2} \cdot 0.25}\right)\right) \cdot \sin re \]

      4. unpow2 56.9%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{\left(im \cdot im\right)} \cdot 0.25\right)\right) \cdot \sin re \]

      5. associate-*l* 56.9%

         \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{im \cdot \left(im \cdot 0.25\right)}\right)\right) \cdot \sin re \]

      6. distribute-lft-out 56.9%

         \[\leadsto \left(1 + \color{blue}{im \cdot \left(0.5 + im \cdot 0.25\right)}\right) \cdot \sin re \]

   10. Simplified 56.9%

       \[\leadsto \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \cdot \sin re \]

   11. Taylor expanded in re around 0 47.2%

       \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2300000:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]

Alternative 10: 48.3% accurate, 28.1× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right) \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* re (+ 1.0 (* im (+ 0.5 (* im 0.25))))))

C

im = abs(im);
double code(double re, double im) {
	return re * (1.0 + (im * (0.5 + (im * 0.25))));
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (im * (0.5d0 + (im * 0.25d0))))
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return re * (1.0 + (im * (0.5 + (im * 0.25))));
}

Python

im = abs(im)
def code(re, im):
	return re * (1.0 + (im * (0.5 + (im * 0.25))))

Julia

im = abs(im)
function code(re, im)
	return Float64(re * Float64(1.0 + Float64(im * Float64(0.5 + Float64(im * 0.25)))))
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = re * (1.0 + (im * (0.5 + (im * 0.25))));
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(re * N[(1.0 + N[(im * N[(0.5 + N[(im * 0.25), $MachinePrecision]), $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%

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

   4. associate-*r* 100.0%

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

   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 74.5%

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

5. Taylor expanded in re around inf 74.5%

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

   1. *-commutative 74.5%

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

7. Simplified 74.5%

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

8. Taylor expanded in im around 0 74.8%

   \[\leadsto \color{blue}{\left(1 + \left(0.25 \cdot {im}^{2} + 0.5 \cdot im\right)\right)} \cdot \sin re \]
9. Step-by-step derivation

   1. +-commutative 74.8%

      \[\leadsto \left(1 + \color{blue}{\left(0.5 \cdot im + 0.25 \cdot {im}^{2}\right)}\right) \cdot \sin re \]

   2. *-commutative 74.8%

      \[\leadsto \left(1 + \left(\color{blue}{im \cdot 0.5} + 0.25 \cdot {im}^{2}\right)\right) \cdot \sin re \]

   3. *-commutative 74.8%

      \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{{im}^{2} \cdot 0.25}\right)\right) \cdot \sin re \]

   4. unpow2 74.8%

      \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{\left(im \cdot im\right)} \cdot 0.25\right)\right) \cdot \sin re \]

   5. associate-*l* 74.8%

      \[\leadsto \left(1 + \left(im \cdot 0.5 + \color{blue}{im \cdot \left(im \cdot 0.25\right)}\right)\right) \cdot \sin re \]

   6. distribute-lft-out 74.8%

      \[\leadsto \left(1 + \color{blue}{im \cdot \left(0.5 + im \cdot 0.25\right)}\right) \cdot \sin re \]

10. Simplified 74.8%

    \[\leadsto \color{blue}{\left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)} \cdot \sin re \]

11. Taylor expanded in re around 0 49.6%

    \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(0.5 + 0.25 \cdot im\right)\right)} \]

12. Final simplification 49.6%

    \[\leadsto re \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.25\right)\right) \]

Alternative 11: 33.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot im\right)\\ \end{array} \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (if (<= im 1.02e-5) re (* re (* 0.5 im))))

C

im = abs(im);
double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-5) {
		tmp = re;
	} else {
		tmp = re * (0.5 * im);
	}
	return tmp;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.02d-5) then
        tmp = re
    else
        tmp = re * (0.5d0 * im)
    end if
    code = tmp
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	double tmp;
	if (im <= 1.02e-5) {
		tmp = re;
	} else {
		tmp = re * (0.5 * im);
	}
	return tmp;
}

Python

im = abs(im)
def code(re, im):
	tmp = 0
	if im <= 1.02e-5:
		tmp = re
	else:
		tmp = re * (0.5 * im)
	return tmp

Julia

im = abs(im)
function code(re, im)
	tmp = 0.0
	if (im <= 1.02e-5)
		tmp = re;
	else
		tmp = Float64(re * Float64(0.5 * im));
	end
	return tmp
end

MATLAB

im = abs(im)
function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.02e-5)
		tmp = re;
	else
		tmp = re * (0.5 * im);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := If[LessEqual[im, 1.02e-5], re, N[(re * N[(0.5 * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.0200000000000001e-5

   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%

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

      4. associate-*r* 100.0%

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

      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 65.0%

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

   5. Taylor expanded in im around 0 64.2%

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

   6. Taylor expanded in re around 0 37.7%

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

   7. Taylor expanded in im around 0 34.9%

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

   if 1.0200000000000001e-5 < 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%

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

      4. associate-*r* 100.0%

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

      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 0 100.0%

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

   5. Taylor expanded in im around 0 4.2%

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

   6. Taylor expanded in re around 0 11.3%

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

   7. Taylor expanded in im around inf 11.3%

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

      1. associate-*r* 11.3%

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

   9. Simplified 11.3%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.02 \cdot 10^{-5}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(0.5 \cdot im\right)\\ \end{array} \]

Alternative 12: 32.9% accurate, 44.1× speedup

Math

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

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 (* re (+ 1.0 (* 0.5 im))))

C

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

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * (1.0d0 + (0.5d0 * im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := N[(re * N[(1.0 + N[(0.5 * 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-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%

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

   4. associate-*r* 100.0%

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

   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 74.5%

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

5. Taylor expanded in im around 0 48.0%

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

6. Taylor expanded in re around 0 30.6%

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

7. Final simplification 30.6%

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

Alternative 13: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ 0 \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 0.0)

C

im = abs(im);
double code(double re, double im) {
	return 0.0;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return 0.0;
}

Python

im = abs(im)
def code(re, im):
	return 0.0

Julia

im = abs(im)
function code(re, im)
	return 0.0
end

MATLAB

im = abs(im)
function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := 0.0

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%

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

   4. associate-*r* 100.0%

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

   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 2.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sin re \cdot -0.5, \sin re \cdot -0.5\right)} \]
6. Step-by-step derivation

   1. fma-udef 2.9%

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

   2. neg-mul-1 2.9%

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

   3. +-commutative 2.9%

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

   4. sub-neg 2.9%

      \[\leadsto \color{blue}{\sin re \cdot -0.5 - \sin re \cdot -0.5} \]

   5. +-inverses 2.9%

      \[\leadsto \color{blue}{0} \]

7. Simplified 2.9%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 2.9%

   \[\leadsto 0 \]

Alternative 14: 26.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} im = |im|\\ \\ re \end{array} \]

FPCore

NOTE: im should be positive before calling this function
(FPCore (re im) :precision binary64 re)

C

im = abs(im);
double code(double re, double im) {
	return re;
}

Fortran

NOTE: im should be positive before calling this function
real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re
end function

Java

im = Math.abs(im);
public static double code(double re, double im) {
	return re;
}

Python

im = abs(im)
def code(re, im):
	return re

Julia

im = abs(im)
function code(re, im)
	return re
end

MATLAB

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

Wolfram

NOTE: im should be positive before calling this function
code[re_, im_] := re

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%

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

   4. associate-*r* 100.0%

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

   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 74.5%

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

5. Taylor expanded in im around 0 48.0%

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

6. Taylor expanded in re around 0 30.6%

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

7. Taylor expanded in im around 0 26.2%

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

8. Final simplification 26.2%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2024024 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))