math.sin on complex, real part

Percentage Accurate: 100.0% → 99.6%

Time: 8.5s

Alternatives: 14

Speedup: 1.0×

• 49.3% 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: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im\_m \leq 2.2:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(e^{im\_m} + 3\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im_m 2.2) (* t_0 (fma im_m im_m 2.0)) (* t_0 (+ (exp im_m) 3.0)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im_m <= 2.2) {
		tmp = t_0 * fma(im_m, im_m, 2.0);
	} else {
		tmp = t_0 * (exp(im_m) + 3.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im_m <= 2.2)
		tmp = Float64(t_0 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(t_0 * Float64(exp(im_m) + 3.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im$95$m, 2.2], N[(t$95$0 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if im < 2.2000000000000002

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. unpow2 80.5%

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

      3. fma-define 80.5%

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

   7. Simplified 80.5%

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

   if 2.2000000000000002 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- im_m)) (exp im_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $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. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

Alternative 3: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im\_m \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{im\_m} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.041666666666666664 \cdot {im\_m}^{4}\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 4.0)
   (sin re)
   (if (<= im_m 1.15e+77)
     (* (+ (exp im_m) 3.0) (* 0.5 re))
     (* (sin re) (* 0.041666666666666664 (pow im_m 4.0))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4.0) {
		tmp = sin(re);
	} else if (im_m <= 1.15e+77) {
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	} else {
		tmp = sin(re) * (0.041666666666666664 * pow(im_m, 4.0));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.0d0) then
        tmp = sin(re)
    else if (im_m <= 1.15d+77) then
        tmp = (exp(im_m) + 3.0d0) * (0.5d0 * re)
    else
        tmp = sin(re) * (0.041666666666666664d0 * (im_m ** 4.0d0))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4.0) {
		tmp = Math.sin(re);
	} else if (im_m <= 1.15e+77) {
		tmp = (Math.exp(im_m) + 3.0) * (0.5 * re);
	} else {
		tmp = Math.sin(re) * (0.041666666666666664 * Math.pow(im_m, 4.0));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 4.0:
		tmp = math.sin(re)
	elif im_m <= 1.15e+77:
		tmp = (math.exp(im_m) + 3.0) * (0.5 * re)
	else:
		tmp = math.sin(re) * (0.041666666666666664 * math.pow(im_m, 4.0))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 4.0)
		tmp = sin(re);
	elseif (im_m <= 1.15e+77)
		tmp = Float64(Float64(exp(im_m) + 3.0) * Float64(0.5 * re));
	else
		tmp = Float64(sin(re) * Float64(0.041666666666666664 * (im_m ^ 4.0)));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 4.0)
		tmp = sin(re);
	elseif (im_m <= 1.15e+77)
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	else
		tmp = sin(re) * (0.041666666666666664 * (im_m ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 4.0], N[Sin[re], $MachinePrecision], If[LessEqual[im$95$m, 1.15e+77], N[(N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(0.041666666666666664 * N[Power[im$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   if 4 < im < 1.14999999999999997e77

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 54.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 54.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 54.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   if 1.14999999999999997e77 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 95.1%

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \sin re\right)} \]
   6. Step-by-step derivation

      1. +-commutative 95.1%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.041666666666666664 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \sin re\right) + \sin re} \]

      2. fma-define 95.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, 0.041666666666666664 \cdot \left({im}^{2} \cdot \sin re\right) + 0.5 \cdot \sin re, \sin re\right)} \]

      3. associate-*r* 95.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\left(0.041666666666666664 \cdot {im}^{2}\right) \cdot \sin re} + 0.5 \cdot \sin re, \sin re\right) \]

      4. distribute-rgt-out 95.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \color{blue}{\sin re \cdot \left(0.041666666666666664 \cdot {im}^{2} + 0.5\right)}, \sin re\right) \]

      5. *-commutative 95.1%

         \[\leadsto \mathsf{fma}\left({im}^{2}, \sin re \cdot \left(\color{blue}{{im}^{2} \cdot 0.041666666666666664} + 0.5\right), \sin re\right) \]

   7. Simplified 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({im}^{2}, \sin re \cdot \left({im}^{2} \cdot 0.041666666666666664 + 0.5\right), \sin re\right)} \]

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left(\sin re \cdot {im}^{4}\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.041666666666666664 \cdot \sin re\right) \cdot {im}^{4}} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\sin re \cdot 0.041666666666666664\right)} \cdot {im}^{4} \]

      4. associate-*l* 100.0%

         \[\leadsto \color{blue}{\sin re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\sin re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(0.041666666666666664 \cdot {im}^{4}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.15:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im\_m} + 3\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.15) (sin re) (* (* 0.5 (sin re)) (+ (exp im_m) 3.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.15) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * sin(re)) * (exp(im_m) + 3.0);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.15d0) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * sin(re)) * (exp(im_m) + 3.0d0)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.15) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(im_m) + 3.0);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.15:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(im_m) + 3.0)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.15)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(im_m) + 3.0));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.15)
		tmp = sin(re);
	else
		tmp = (0.5 * sin(re)) * (exp(im_m) + 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.15], N[Sin[re], $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1499999999999999

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   if 1.1499999999999999 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.15:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{im} + 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 4.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im\_m \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{im\_m} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im\_m \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 4.8)
   (sin re)
   (if (<= im_m 3.3e+100)
     (* (+ (exp im_m) 3.0) (* 0.5 re))
     (*
      (* 0.5 (sin re))
      (+
       4.0
       (* im_m (+ 1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666))))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = sin(re);
	} else if (im_m <= 3.3e+100) {
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.8d0) then
        tmp = sin(re)
    else if (im_m <= 3.3d+100) then
        tmp = (exp(im_m) + 3.0d0) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (4.0d0 + (im_m * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4.8) {
		tmp = Math.sin(re);
	} else if (im_m <= 3.3e+100) {
		tmp = (Math.exp(im_m) + 3.0) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 4.8:
		tmp = math.sin(re)
	elif im_m <= 3.3e+100:
		tmp = (math.exp(im_m) + 3.0) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 4.8)
		tmp = sin(re);
	elseif (im_m <= 3.3e+100)
		tmp = Float64(Float64(exp(im_m) + 3.0) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 4.8)
		tmp = sin(re);
	elseif (im_m <= 3.3e+100)
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 4.8], N[Sin[re], $MachinePrecision], If[LessEqual[im$95$m, 3.3e+100], N[(N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.79999999999999982

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   if 4.79999999999999982 < im < 3.3000000000000001e100

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 56.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 56.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 56.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   if 3.3000000000000001e100 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 98.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   9. Simplified 98.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.8:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 3.3 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 4.5:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im\_m \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im\_m} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im\_m \cdot \left(1 + 0.5 \cdot im\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 4.5)
   (sin re)
   (if (<= im_m 1.9e+154)
     (* (+ (exp im_m) 3.0) (* 0.5 re))
     (* (* 0.5 (sin re)) (+ 4.0 (* im_m (+ 1.0 (* 0.5 im_m))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4.5) {
		tmp = sin(re);
	} else if (im_m <= 1.9e+154) {
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	} else {
		tmp = (0.5 * sin(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 4.5d0) then
        tmp = sin(re)
    else if (im_m <= 1.9d+154) then
        tmp = (exp(im_m) + 3.0d0) * (0.5d0 * re)
    else
        tmp = (0.5d0 * sin(re)) * (4.0d0 + (im_m * (1.0d0 + (0.5d0 * im_m))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 4.5) {
		tmp = Math.sin(re);
	} else if (im_m <= 1.9e+154) {
		tmp = (Math.exp(im_m) + 3.0) * (0.5 * re);
	} else {
		tmp = (0.5 * Math.sin(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 4.5:
		tmp = math.sin(re)
	elif im_m <= 1.9e+154:
		tmp = (math.exp(im_m) + 3.0) * (0.5 * re)
	else:
		tmp = (0.5 * math.sin(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 4.5)
		tmp = sin(re);
	elseif (im_m <= 1.9e+154)
		tmp = Float64(Float64(exp(im_m) + 3.0) * Float64(0.5 * re));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(0.5 * im_m)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 4.5)
		tmp = sin(re);
	elseif (im_m <= 1.9e+154)
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	else
		tmp = (0.5 * sin(re)) * (4.0 + (im_m * (1.0 + (0.5 * im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 4.5], N[Sin[re], $MachinePrecision], If[LessEqual[im$95$m, 1.9e+154], N[(N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(0.5 * im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 4.5

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   if 4.5 < im < 1.8999999999999999e154

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 67.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 67.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 67.7%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   if 1.8999999999999999e154 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + \color{blue}{im \cdot 0.5}\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.5:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + 0.5 \cdot im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 3.3:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im\_m} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 3.3) (sin re) (* (+ (exp im_m) 3.0) (* 0.5 re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.3) {
		tmp = sin(re);
	} else {
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 3.3d0) then
        tmp = sin(re)
    else
        tmp = (exp(im_m) + 3.0d0) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 3.3) {
		tmp = Math.sin(re);
	} else {
		tmp = (Math.exp(im_m) + 3.0) * (0.5 * re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 3.3:
		tmp = math.sin(re)
	else:
		tmp = (math.exp(im_m) + 3.0) * (0.5 * re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 3.3)
		tmp = sin(re);
	else
		tmp = Float64(Float64(exp(im_m) + 3.0) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 3.3)
		tmp = sin(re);
	else
		tmp = (exp(im_m) + 3.0) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 3.3], N[Sin[re], $MachinePrecision], N[(N[(N[Exp[im$95$m], $MachinePrecision] + 3.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.2999999999999998

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   if 3.2999999999999998 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 79.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 79.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 79.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.3:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(e^{im} + 3\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 6 \cdot 10^{+20}:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(4 + im\_m \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 6e+20)
   (sin re)
   (*
    (* 0.5 re)
    (+ 4.0 (* im_m (+ 1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 6e+20) {
		tmp = sin(re);
	} else {
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 6d+20) then
        tmp = sin(re)
    else
        tmp = (0.5d0 * re) * (4.0d0 + (im_m * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 6e+20) {
		tmp = Math.sin(re);
	} else {
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 6e+20:
		tmp = math.sin(re)
	else:
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 6e+20)
		tmp = sin(re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 6e+20)
		tmp = sin(re);
	else
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 6e+20], N[Sin[re], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6e20

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 65.0%

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

   if 6e20 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 82.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 82.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 82.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   10. Taylor expanded in im around 0 76.8%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 80.8%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   12. Simplified 76.8%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 49.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 35000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im\_m \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im\_m + 4\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(2 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 35000000.0)
   re
   (if (<= im_m 1.9e+140)
     (* (* re (+ 0.5 (* -0.08333333333333333 (* re re)))) (+ im_m 4.0))
     (* re (+ 2.0 (* im_m (+ 0.5 (* im_m 0.25))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 35000000.0) {
		tmp = re;
	} else if (im_m <= 1.9e+140) {
		tmp = (re * (0.5 + (-0.08333333333333333 * (re * re)))) * (im_m + 4.0);
	} else {
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 35000000.0d0) then
        tmp = re
    else if (im_m <= 1.9d+140) then
        tmp = (re * (0.5d0 + ((-0.08333333333333333d0) * (re * re)))) * (im_m + 4.0d0)
    else
        tmp = re * (2.0d0 + (im_m * (0.5d0 + (im_m * 0.25d0))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 35000000.0) {
		tmp = re;
	} else if (im_m <= 1.9e+140) {
		tmp = (re * (0.5 + (-0.08333333333333333 * (re * re)))) * (im_m + 4.0);
	} else {
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 35000000.0:
		tmp = re
	elif im_m <= 1.9e+140:
		tmp = (re * (0.5 + (-0.08333333333333333 * (re * re)))) * (im_m + 4.0)
	else:
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 35000000.0)
		tmp = re;
	elseif (im_m <= 1.9e+140)
		tmp = Float64(Float64(re * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))) * Float64(im_m + 4.0));
	else
		tmp = Float64(re * Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.25)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 35000000.0)
		tmp = re;
	elseif (im_m <= 1.9e+140)
		tmp = (re * (0.5 + (-0.08333333333333333 * (re * re)))) * (im_m + 4.0);
	else
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 35000000.0], re, If[LessEqual[im$95$m, 1.9e+140], N[(N[(re * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im$95$m + 4.0), $MachinePrecision]), $MachinePrecision], N[(re * N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 3.5e7

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.0%

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

   6. Taylor expanded in re around 0 34.5%

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

   if 3.5e7 < im < 1.9e140

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 3.5%

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

   8. Taylor expanded in re around 0 33.6%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(4 + im\right) \]
   9. Step-by-step derivation

      1. unpow2 33.6%

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

   10. Applied egg-rr 33.6%

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

   if 1.9e140 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 90.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 90.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 90.5%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   10. Taylor expanded in im around 0 61.3%

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

   11. Taylor expanded in re around 0 85.9%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 35000000:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right) \cdot \left(im + 4\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.5% accurate, 14.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.15:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(4 + im\_m \cdot \left(1 + im\_m \cdot \left(0.5 + im\_m \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.15)
   re
   (*
    (* 0.5 re)
    (+ 4.0 (* im_m (+ 1.0 (* im_m (+ 0.5 (* im_m 0.16666666666666666)))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.15) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.15d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * (4.0d0 + (im_m * (1.0d0 + (im_m * (0.5d0 + (im_m * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.15) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.15:
		tmp = re
	else:
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.15)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(4.0 + Float64(im_m * Float64(1.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.15)
		tmp = re;
	else
		tmp = (0.5 * re) * (4.0 + (im_m * (1.0 + (im_m * (0.5 + (im_m * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.15], re, N[(N[(0.5 * re), $MachinePrecision] * N[(4.0 + N[(im$95$m * N[(1.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1499999999999999

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   6. Taylor expanded in re around 0 34.7%

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

   if 1.1499999999999999 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 79.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 79.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 79.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   10. Taylor expanded in im around 0 72.3%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + 0.16666666666666666 \cdot im\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative 76.2%

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(4 + im \cdot \left(1 + im \cdot \left(0.5 + \color{blue}{im \cdot 0.16666666666666666}\right)\right)\right) \]

   12. Simplified 72.3%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(4 + im \cdot \left(1 + im \cdot \left(0.5 + im \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 48.3% accurate, 19.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(2 + im\_m \cdot \left(0.5 + im\_m \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.1) re (* re (+ 2.0 (* im_m (+ 0.5 (* im_m 0.25)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.1) {
		tmp = re;
	} else {
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.1d0) then
        tmp = re
    else
        tmp = re * (2.0d0 + (im_m * (0.5d0 + (im_m * 0.25d0))))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.1) {
		tmp = re;
	} else {
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.1:
		tmp = re
	else:
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.1)
		tmp = re;
	else
		tmp = Float64(re * Float64(2.0 + Float64(im_m * Float64(0.5 + Float64(im_m * 0.25)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.1)
		tmp = re;
	else
		tmp = re * (2.0 + (im_m * (0.5 + (im_m * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.1], re, N[(re * N[(2.0 + N[(im$95$m * N[(0.5 + N[(im$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1000000000000001

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   6. Taylor expanded in re around 0 34.7%

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

   if 1.1000000000000001 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in re around 0 79.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(3 + e^{im}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 79.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   9. Simplified 79.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right) \cdot \left(3 + e^{im}\right)} \]

   10. Taylor expanded in im around 0 43.3%

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

   11. Taylor expanded in re around 0 58.5%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(2 + im \cdot \left(0.5 + im \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.9% accurate, 25.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 1.1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im\_m + 4\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.1) re (* (* 0.5 re) (+ im_m 4.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.1) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im_m + 4.0);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.1d0) then
        tmp = re
    else
        tmp = (0.5d0 * re) * (im_m + 4.0d0)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.1) {
		tmp = re;
	} else {
		tmp = (0.5 * re) * (im_m + 4.0);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.1:
		tmp = re
	else:
		tmp = (0.5 * re) * (im_m + 4.0)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.1)
		tmp = re;
	else
		tmp = Float64(Float64(0.5 * re) * Float64(im_m + 4.0));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.1)
		tmp = re;
	else
		tmp = (0.5 * re) * (im_m + 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.1], re, N[(N[(0.5 * re), $MachinePrecision] * N[(im$95$m + 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1000000000000001

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   6. Taylor expanded in re around 0 34.7%

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

   if 1.1000000000000001 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 4.4%

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

   8. Taylor expanded in re around 0 19.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(4 + im\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 19.0%

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

      2. *-commutative 19.0%

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

      3. +-commutative 19.0%

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

   10. Simplified 19.0%

       \[\leadsto \color{blue}{\left(re \cdot 0.5\right) \cdot \left(im + 4\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.1:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(im + 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.9% accurate, 30.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im\_m \leq 195:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;im\_m \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 195.0) re (* im_m (* 0.5 re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 195.0) {
		tmp = re;
	} else {
		tmp = im_m * (0.5 * re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 195.0d0) then
        tmp = re
    else
        tmp = im_m * (0.5d0 * re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 195.0) {
		tmp = re;
	} else {
		tmp = im_m * (0.5 * re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 195.0:
		tmp = re
	else:
		tmp = im_m * (0.5 * re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 195.0)
		tmp = re;
	else
		tmp = Float64(im_m * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 195.0)
		tmp = re;
	else
		tmp = im_m * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 195.0], re, N[(im$95$m * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 195

   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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 66.3%

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

   6. Taylor expanded in re around 0 34.7%

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

   if 195 < 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. cancel-sign-sub 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. sub-neg 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{3} + e^{im}\right) \]

   7. Taylor expanded in im around 0 4.4%

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

   8. Taylor expanded in im around inf 4.4%

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

   9. Taylor expanded in re around 0 19.0%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 195:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.1% accurate, 309.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 re)

C

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

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = re
end function

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m):
	return re

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 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. cancel-sign-sub 100.0%

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

   3. distribute-rgt-out-- 100.0%

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

   4. sub-neg 100.0%

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

   5. remove-double-neg 100.0%

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

   6. neg-sub0 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in im around 0 49.4%

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

6. Taylor expanded in re around 0 26.2%

   \[\leadsto \color{blue}{re} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))