math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-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-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. neg-sub0 100.0%

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

   6. 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%

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

Alternative 2: 77.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{if}\;im \leq 5800:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+135}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (fma im im 2.0))))
   (if (<= im 5800.0)
     t_0
     (if (<= im 5e+115)
       (log1p (expm1 re))
       (if (<= im 1.9e+135)
         (pow re -2.0)
         (if (<= im 8e+148) (* (fma im im 2.0) (* 0.5 re)) t_0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * fma(im, im, 2.0);
	double tmp;
	if (im <= 5800.0) {
		tmp = t_0;
	} else if (im <= 5e+115) {
		tmp = log1p(expm1(re));
	} else if (im <= 1.9e+135) {
		tmp = pow(re, -2.0);
	} else if (im <= 8e+148) {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0))
	tmp = 0.0
	if (im <= 5800.0)
		tmp = t_0;
	elseif (im <= 5e+115)
		tmp = log1p(expm1(re));
	elseif (im <= 1.9e+135)
		tmp = re ^ -2.0;
	elseif (im <= 8e+148)
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 5800.0], t$95$0, If[LessEqual[im, 5e+115], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 1.9e+135], N[Power[re, -2.0], $MachinePrecision], If[LessEqual[im, 8e+148], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 5800 or 8.0000000000000004e148 < 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. neg-sub0 100.0%

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

      6. 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 80.4%

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

   6. Simplified 80.4%

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

   if 5800 < im < 5.00000000000000008e115

   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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 10.0%

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

   6. Taylor expanded in re around 0 9.8%

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

   8. Applied egg-rr 42.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \]

   if 5.00000000000000008e115 < im < 1.9000000000000001e135

   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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 18.5%

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

   6. Taylor expanded in re around 0 18.1%

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

   8. Applied egg-rr 18.1%

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

   if 1.9000000000000001e135 < im < 8.0000000000000004e148

   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. neg-sub0 100.0%

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

      6. 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 8.2%

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

   6. Simplified 8.2%

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

   7. Taylor expanded in re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. associate-*l* 0.0%

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

      4. +-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. fma-udef 0.0%

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

   9. Simplified 0.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5800:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{+135}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{elif}\;im \leq 8 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]

Alternative 3: 83.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.0135 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im 0.0135) (not (<= im 1.35e+154)))
   (* (* 0.5 (sin re)) (fma im im 2.0))
   (* (+ (exp (- im)) (exp im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= 0.0135) || !(im <= 1.35e+154)) {
		tmp = (0.5 * sin(re)) * fma(im, im, 2.0);
	} else {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= 0.0135) || !(im <= 1.35e+154))
		tmp = Float64(Float64(0.5 * sin(re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, 0.0135], N[Not[LessEqual[im, 1.35e+154]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.0134999999999999998 or 1.35000000000000003e154 < 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. neg-sub0 100.0%

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

      6. 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 82.7%

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

   6. Simplified 82.7%

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

   if 0.0134999999999999998 < im < 1.35000000000000003e154

   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. neg-sub0 100.0%

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

      6. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 73.7%

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

   6. Simplified 73.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.0135 \lor \neg \left(im \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 4: 62.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.18 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 580.0)
   (sin re)
   (if (<= im 1.18e+107) (log1p (expm1 re)) (* (fma im im 2.0) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 580.0) {
		tmp = sin(re);
	} else if (im <= 1.18e+107) {
		tmp = log1p(expm1(re));
	} else {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 580.0)
		tmp = sin(re);
	elseif (im <= 1.18e+107)
		tmp = log1p(expm1(re));
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 580.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.18e+107], N[Log[1 + N[(Exp[re] - 1), $MachinePrecision]], $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 580

   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. neg-sub0 100.0%

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

      6. 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 64.1%

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

   if 580 < im < 1.18000000000000005e107

   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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 10.0%

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

   6. Taylor expanded in re around 0 9.7%

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

   8. Applied egg-rr 38.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)} \]

   if 1.18000000000000005e107 < 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. neg-sub0 100.0%

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

      6. 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 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in re around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. *-commutative 64.1%

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

      3. associate-*l* 64.1%

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

      4. +-commutative 64.1%

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

      5. unpow2 64.1%

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

      6. fma-udef 64.1%

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

   9. Simplified 64.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.18 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 5: 52.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+89} \lor \neg \left(im \leq 7.5 \cdot 10^{+115}\right) \land \left(im \leq 2.3 \cdot 10^{+163} \lor \neg \left(im \leq 3.8 \cdot 10^{+206}\right)\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 760.0)
   (sin re)
   (if (or (<= im 6.8e+89)
           (and (not (<= im 7.5e+115))
                (or (<= im 2.3e+163) (not (<= im 3.8e+206)))))
     (pow re -2.0)
     (* (+ re -0.020833333333333332) (+ re -0.020833333333333332)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = sin(re);
	} else if ((im <= 6.8e+89) || (!(im <= 7.5e+115) && ((im <= 2.3e+163) || !(im <= 3.8e+206)))) {
		tmp = pow(re, -2.0);
	} else {
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 760.0d0) then
        tmp = sin(re)
    else if ((im <= 6.8d+89) .or. (.not. (im <= 7.5d+115)) .and. (im <= 2.3d+163) .or. (.not. (im <= 3.8d+206))) then
        tmp = re ** (-2.0d0)
    else
        tmp = (re + (-0.020833333333333332d0)) * (re + (-0.020833333333333332d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = Math.sin(re);
	} else if ((im <= 6.8e+89) || (!(im <= 7.5e+115) && ((im <= 2.3e+163) || !(im <= 3.8e+206)))) {
		tmp = Math.pow(re, -2.0);
	} else {
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 760.0:
		tmp = math.sin(re)
	elif (im <= 6.8e+89) or (not (im <= 7.5e+115) and ((im <= 2.3e+163) or not (im <= 3.8e+206))):
		tmp = math.pow(re, -2.0)
	else:
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 760.0)
		tmp = sin(re);
	elseif ((im <= 6.8e+89) || (!(im <= 7.5e+115) && ((im <= 2.3e+163) || !(im <= 3.8e+206))))
		tmp = re ^ -2.0;
	else
		tmp = Float64(Float64(re + -0.020833333333333332) * Float64(re + -0.020833333333333332));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 760.0)
		tmp = sin(re);
	elseif ((im <= 6.8e+89) || (~((im <= 7.5e+115)) && ((im <= 2.3e+163) || ~((im <= 3.8e+206)))))
		tmp = re ^ -2.0;
	else
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 760.0], N[Sin[re], $MachinePrecision], If[Or[LessEqual[im, 6.8e+89], And[N[Not[LessEqual[im, 7.5e+115]], $MachinePrecision], Or[LessEqual[im, 2.3e+163], N[Not[LessEqual[im, 3.8e+206]], $MachinePrecision]]]], N[Power[re, -2.0], $MachinePrecision], N[(N[(re + -0.020833333333333332), $MachinePrecision] * N[(re + -0.020833333333333332), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 760

   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. neg-sub0 100.0%

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

      6. 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 64.1%

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

   if 760 < im < 6.8000000000000004e89 or 7.4999999999999997e115 < im < 2.30000000000000002e163 or 3.7999999999999999e206 < 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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 15.7%

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

   6. Taylor expanded in re around 0 15.6%

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

   8. Applied egg-rr 15.6%

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

   if 6.8000000000000004e89 < im < 7.4999999999999997e115 or 2.30000000000000002e163 < im < 3.7999999999999999e206

   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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 2.0%

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

   6. Taylor expanded in re around 0 2.0%

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

      1. associate-*r/ 2.0%

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

      2. metadata-eval 2.0%

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

   8. Simplified 2.0%

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

   10. Applied egg-rr 28.0%

       \[\leadsto \color{blue}{\left(-0.020833333333333332 + re\right) \cdot \left(-0.020833333333333332 + re\right)} \]
   11. Step-by-step derivation

       1. +-commutative 28.0%

          \[\leadsto \color{blue}{\left(re + -0.020833333333333332\right)} \cdot \left(-0.020833333333333332 + re\right) \]

       2. +-commutative 28.0%

          \[\leadsto \left(re + -0.020833333333333332\right) \cdot \color{blue}{\left(re + -0.020833333333333332\right)} \]

   12. Simplified 28.0%

       \[\leadsto \color{blue}{\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.8 \cdot 10^{+89} \lor \neg \left(im \leq 7.5 \cdot 10^{+115}\right) \land \left(im \leq 2.3 \cdot 10^{+163} \lor \neg \left(im \leq 3.8 \cdot 10^{+206}\right)\right):\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)\\ \end{array} \]

Alternative 6: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+82}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 600.0)
   (sin re)
   (if (<= im 6.5e+82)
     (+ 0.08333333333333333 (/ 0.25 (pow re 2.0)))
     (* (fma im im 2.0) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 600.0) {
		tmp = sin(re);
	} else if (im <= 6.5e+82) {
		tmp = 0.08333333333333333 + (0.25 / pow(re, 2.0));
	} else {
		tmp = fma(im, im, 2.0) * (0.5 * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 600.0)
		tmp = sin(re);
	elseif (im <= 6.5e+82)
		tmp = Float64(0.08333333333333333 + Float64(0.25 / (re ^ 2.0)));
	else
		tmp = Float64(fma(im, im, 2.0) * Float64(0.5 * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 600.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.5e+82], N[(0.08333333333333333 + N[(0.25 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 600

   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. neg-sub0 100.0%

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

      6. 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 64.1%

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

   if 600 < im < 6.5000000000000003e82

   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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 11.4%

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

   6. Taylor expanded in re around 0 11.4%

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

      1. associate-*r/ 11.4%

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

      2. metadata-eval 11.4%

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

   8. Simplified 11.4%

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

   if 6.5000000000000003e82 < 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. neg-sub0 100.0%

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

      6. 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 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in re around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*l* 60.8%

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

      4. +-commutative 60.8%

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

      5. unpow2 60.8%

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

      6. fma-udef 60.8%

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

   9. Simplified 60.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 600:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+82}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 7: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 980:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+81}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 980.0)
   (sin re)
   (if (<= im 1.35e+81) (pow re -2.0) (* (* 0.5 re) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 980.0) {
		tmp = sin(re);
	} else if (im <= 1.35e+81) {
		tmp = pow(re, -2.0);
	} else {
		tmp = (0.5 * re) * pow(im, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 980.0d0) then
        tmp = sin(re)
    else if (im <= 1.35d+81) then
        tmp = re ** (-2.0d0)
    else
        tmp = (0.5d0 * re) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 980.0) {
		tmp = Math.sin(re);
	} else if (im <= 1.35e+81) {
		tmp = Math.pow(re, -2.0);
	} else {
		tmp = (0.5 * re) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 980.0:
		tmp = math.sin(re)
	elif im <= 1.35e+81:
		tmp = math.pow(re, -2.0)
	else:
		tmp = (0.5 * re) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 980.0)
		tmp = sin(re);
	elseif (im <= 1.35e+81)
		tmp = re ^ -2.0;
	else
		tmp = Float64(Float64(0.5 * re) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 980.0)
		tmp = sin(re);
	elseif (im <= 1.35e+81)
		tmp = re ^ -2.0;
	else
		tmp = (0.5 * re) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 980.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 1.35e+81], N[Power[re, -2.0], $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 980

   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. neg-sub0 100.0%

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

      6. 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 64.1%

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

   if 980 < im < 1.35e81

   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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 11.4%

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

   6. Taylor expanded in re around 0 11.2%

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

   8. Applied egg-rr 11.2%

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

   if 1.35e81 < 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. neg-sub0 100.0%

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

      6. 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 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in re around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*l* 60.8%

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

      4. +-commutative 60.8%

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

      5. unpow2 60.8%

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

      6. fma-udef 60.8%

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

   9. Simplified 60.8%

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

   10. Taylor expanded in im around 0 60.8%

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

   11. Taylor expanded in im around inf 60.8%

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

       1. *-commutative 60.8%

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

       2. associate-*r* 60.8%

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

   13. Simplified 60.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 980:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+81}:\\ \;\;\;\;{re}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \]

Alternative 8: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+82}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 550.0)
   (sin re)
   (if (<= im 8.2e+82)
     (+ 0.08333333333333333 (/ 0.25 (pow re 2.0)))
     (* (* 0.5 re) (pow im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = sin(re);
	} else if (im <= 8.2e+82) {
		tmp = 0.08333333333333333 + (0.25 / pow(re, 2.0));
	} else {
		tmp = (0.5 * re) * pow(im, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 550.0d0) then
        tmp = sin(re)
    else if (im <= 8.2d+82) then
        tmp = 0.08333333333333333d0 + (0.25d0 / (re ** 2.0d0))
    else
        tmp = (0.5d0 * re) * (im ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 550.0) {
		tmp = Math.sin(re);
	} else if (im <= 8.2e+82) {
		tmp = 0.08333333333333333 + (0.25 / Math.pow(re, 2.0));
	} else {
		tmp = (0.5 * re) * Math.pow(im, 2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 550.0:
		tmp = math.sin(re)
	elif im <= 8.2e+82:
		tmp = 0.08333333333333333 + (0.25 / math.pow(re, 2.0))
	else:
		tmp = (0.5 * re) * math.pow(im, 2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 550.0)
		tmp = sin(re);
	elseif (im <= 8.2e+82)
		tmp = Float64(0.08333333333333333 + Float64(0.25 / (re ^ 2.0)));
	else
		tmp = Float64(Float64(0.5 * re) * (im ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 550.0)
		tmp = sin(re);
	elseif (im <= 8.2e+82)
		tmp = 0.08333333333333333 + (0.25 / (re ^ 2.0));
	else
		tmp = (0.5 * re) * (im ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 550.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 8.2e+82], N[(0.08333333333333333 + N[(0.25 / N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[Power[im, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 550

   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. neg-sub0 100.0%

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

      6. 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 64.1%

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

   if 550 < im < 8.1999999999999999e82

   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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 11.4%

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

   6. Taylor expanded in re around 0 11.4%

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

      1. associate-*r/ 11.4%

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

      2. metadata-eval 11.4%

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

   8. Simplified 11.4%

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

   if 8.1999999999999999e82 < 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. neg-sub0 100.0%

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

      6. 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 68.5%

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

   6. Simplified 68.5%

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

   7. Taylor expanded in re around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*l* 60.8%

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

      4. +-commutative 60.8%

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

      5. unpow2 60.8%

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

      6. fma-udef 60.8%

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

   9. Simplified 60.8%

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

   10. Taylor expanded in im around 0 60.8%

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

   11. Taylor expanded in im around inf 60.8%

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

       1. *-commutative 60.8%

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

       2. associate-*r* 60.8%

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

   13. Simplified 60.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 550:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{+82}:\\ \;\;\;\;0.08333333333333333 + \frac{0.25}{{re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot {im}^{2}\\ \end{array} \]

Alternative 9: 53.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 240:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 240.0)
   (sin re)
   (* (+ re -0.020833333333333332) (+ re -0.020833333333333332))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 240.0) {
		tmp = sin(re);
	} else {
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 240.0d0) then
        tmp = sin(re)
    else
        tmp = (re + (-0.020833333333333332d0)) * (re + (-0.020833333333333332d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 240.0) {
		tmp = Math.sin(re);
	} else {
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 240.0:
		tmp = math.sin(re)
	else:
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 240.0)
		tmp = sin(re);
	else
		tmp = Float64(Float64(re + -0.020833333333333332) * Float64(re + -0.020833333333333332));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 240.0)
		tmp = sin(re);
	else
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 240.0], N[Sin[re], $MachinePrecision], N[(N[(re + -0.020833333333333332), $MachinePrecision] * N[(re + -0.020833333333333332), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 240

   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. neg-sub0 100.0%

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

      6. 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 64.1%

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

   if 240 < 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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 12.5%

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

   6. Taylor expanded in re around 0 12.5%

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

      1. associate-*r/ 12.5%

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

      2. metadata-eval 12.5%

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

   8. Simplified 12.5%

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

   10. Applied egg-rr 14.1%

       \[\leadsto \color{blue}{\left(-0.020833333333333332 + re\right) \cdot \left(-0.020833333333333332 + re\right)} \]
   11. Step-by-step derivation

       1. +-commutative 14.1%

          \[\leadsto \color{blue}{\left(re + -0.020833333333333332\right)} \cdot \left(-0.020833333333333332 + re\right) \]

       2. +-commutative 14.1%

          \[\leadsto \left(re + -0.020833333333333332\right) \cdot \color{blue}{\left(re + -0.020833333333333332\right)} \]

   12. Simplified 14.1%

       \[\leadsto \color{blue}{\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 240:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)\\ \end{array} \]

Alternative 10: 29.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 260.0)
   re
   (* (+ re -0.020833333333333332) (+ re -0.020833333333333332))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 260.0) {
		tmp = re;
	} else {
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 260.0d0) then
        tmp = re
    else
        tmp = (re + (-0.020833333333333332d0)) * (re + (-0.020833333333333332d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 260.0) {
		tmp = re;
	} else {
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 260.0:
		tmp = re
	else:
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 260.0)
		tmp = re;
	else
		tmp = Float64(Float64(re + -0.020833333333333332) * Float64(re + -0.020833333333333332));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 260.0)
		tmp = re;
	else
		tmp = (re + -0.020833333333333332) * (re + -0.020833333333333332);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 260.0], re, N[(N[(re + -0.020833333333333332), $MachinePrecision] * N[(re + -0.020833333333333332), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 260

   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. neg-sub0 100.0%

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

      6. 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 78.9%

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

   6. Simplified 78.9%

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

   7. Taylor expanded in re around 0 50.0%

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

      1. *-commutative 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*l* 50.0%

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

      4. +-commutative 50.0%

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

      5. unpow2 50.0%

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

      6. fma-udef 50.0%

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

   9. Simplified 50.0%

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

   10. Taylor expanded in im around 0 35.0%

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

   if 260 < 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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 12.5%

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

   6. Taylor expanded in re around 0 12.5%

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

      1. associate-*r/ 12.5%

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

      2. metadata-eval 12.5%

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

   8. Simplified 12.5%

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

   10. Applied egg-rr 14.1%

       \[\leadsto \color{blue}{\left(-0.020833333333333332 + re\right) \cdot \left(-0.020833333333333332 + re\right)} \]
   11. Step-by-step derivation

       1. +-commutative 14.1%

          \[\leadsto \color{blue}{\left(re + -0.020833333333333332\right)} \cdot \left(-0.020833333333333332 + re\right) \]

       2. +-commutative 14.1%

          \[\leadsto \left(re + -0.020833333333333332\right) \cdot \color{blue}{\left(re + -0.020833333333333332\right)} \]

   12. Simplified 14.1%

       \[\leadsto \color{blue}{\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 260:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(re + -0.020833333333333332\right) \cdot \left(re + -0.020833333333333332\right)\\ \end{array} \]

Alternative 11: 29.1% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6200:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 6200.0) re (* re re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6200.0) {
		tmp = re;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6200.0d0) then
        tmp = re
    else
        tmp = re * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6200.0) {
		tmp = re;
	} else {
		tmp = re * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6200.0:
		tmp = re
	else:
		tmp = re * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6200.0)
		tmp = re;
	else
		tmp = Float64(re * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6200.0)
		tmp = re;
	else
		tmp = re * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6200.0], re, N[(re * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6200

   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. neg-sub0 100.0%

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

      6. 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 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in re around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. *-commutative 49.8%

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

      3. associate-*l* 49.8%

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

      4. +-commutative 49.8%

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

      5. unpow2 49.8%

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

      6. fma-udef 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in im around 0 34.8%

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

   if 6200 < 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. neg-sub0 100.0%

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

      6. 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%

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

   5. Applied egg-rr 12.7%

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

   6. Taylor expanded in re around 0 12.5%

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

   8. Applied egg-rr 14.1%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 6200:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot re\\ \end{array} \]

Alternative 12: 4.2% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 0.08333333333333333)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.08333333333333333

Julia

function code(re, im)
	return 0.08333333333333333
end

MATLAB

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

Wolfram

code[re_, im_] := 0.08333333333333333

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. 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. neg-sub0 100.0%

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

   6. 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%

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

5. Applied egg-rr 8.2%

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

6. Taylor expanded in re around 0 8.1%

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

   1. associate-*r/ 8.1%

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

   2. metadata-eval 8.1%

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

8. Simplified 8.1%

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

9. Taylor expanded in re around inf 3.5%

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

10. Final simplification 3.5%

    \[\leadsto 0.08333333333333333 \]

Alternative 13: 4.8% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0.9795918367346939 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.9795918367346939)

C

double code(double re, double im) {
	return 0.9795918367346939;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.9795918367346939d0
end function

Java

public static double code(double re, double im) {
	return 0.9795918367346939;
}

Python

def code(re, im):
	return 0.9795918367346939

Julia

function code(re, im)
	return 0.9795918367346939
end

MATLAB

function tmp = code(re, im)
	tmp = 0.9795918367346939;
end

Wolfram

code[re_, im_] := 0.9795918367346939

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. neg-sub0 100.0%

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

   6. 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%

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

5. Applied egg-rr 8.2%

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

6. Taylor expanded in re around 0 8.1%

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

   1. associate-*r/ 8.1%

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

   2. metadata-eval 8.1%

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

8. Simplified 8.1%

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

10. Applied egg-rr 3.8%

    \[\leadsto \color{blue}{\frac{re + 0.0005787037037037037}{re + \left(0.006944444444444444 - -0.020833333333333332 \cdot re\right)}} \]
11. Step-by-step derivation

    1. +-commutative 3.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{\color{blue}{\left(0.006944444444444444 - -0.020833333333333332 \cdot re\right) + re}} \]

    2. associate-+l- 3.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{\color{blue}{0.006944444444444444 - \left(-0.020833333333333332 \cdot re - re\right)}} \]

    3. *-lft-identity 3.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{0.006944444444444444 - \left(-0.020833333333333332 \cdot re - \color{blue}{1 \cdot re}\right)} \]

    4. distribute-rgt-out-- 3.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{0.006944444444444444 - \color{blue}{re \cdot \left(-0.020833333333333332 - 1\right)}} \]

    5. metadata-eval 3.8%

       \[\leadsto \frac{re + 0.0005787037037037037}{0.006944444444444444 - re \cdot \color{blue}{-1.0208333333333333}} \]

12. Simplified 3.8%

    \[\leadsto \color{blue}{\frac{re + 0.0005787037037037037}{0.006944444444444444 - re \cdot -1.0208333333333333}} \]

13. Taylor expanded in re around inf 3.8%

    \[\leadsto \color{blue}{0.9795918367346939} \]

14. Final simplification 3.8%

    \[\leadsto 0.9795918367346939 \]

Alternative 14: 26.1% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

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. 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. neg-sub0 100.0%

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

   6. 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 71.2%

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

6. Simplified 71.2%

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

7. Taylor expanded in re around 0 48.6%

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

   1. *-commutative 48.6%

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

   2. *-commutative 48.6%

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

   3. associate-*l* 48.6%

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

   4. +-commutative 48.6%

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

   5. unpow2 48.6%

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

   6. fma-udef 48.6%

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

9. Simplified 48.6%

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

10. Taylor expanded in im around 0 26.8%

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

11. Final simplification 26.8%

    \[\leadsto re \]

Reproduce

herbie shell --seed 2023322 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))