math.sin on complex, imaginary part

Percentage Accurate: 54.8% → 99.6%

Time: 7.2s

Alternatives: 10

Speedup: 2.8×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * cos(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 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\log \left(e^{\left(-im\right) \cdot \cos re}\right)\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 (- INFINITY))
     (log (exp (* (- im) (cos re))))
     (if (<= t_0 4e-5)
       (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
       (* (* (cos re) 0.5) t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = log(exp((-im * cos(re))));
	} else if (t_0 <= 4e-5) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (cos(re) * 0.5) * t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.log(Math.exp((-im * Math.cos(re))));
	} else if (t_0 <= 4e-5) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = (Math.cos(re) * 0.5) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.log(math.exp((-im * math.cos(re))))
	elif t_0 <= 4e-5:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = (math.cos(re) * 0.5) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = log(exp(Float64(Float64(-im) * cos(re))));
	elseif (t_0 <= 4e-5)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = Float64(Float64(cos(re) * 0.5) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = log(exp((-im * cos(re))));
	elseif (t_0 <= 4e-5)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = (cos(re) * 0.5) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[Log[N[Exp[N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e-5], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.2%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 5.2%

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

      2. neg-mul-1 5.2%

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

   6. Simplified 5.2%

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

      1. distribute-lft-neg-out 5.2%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. add-sqr-sqrt 5.2%

         \[\leadsto -\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re \]

      3. sqrt-unprod 56.8%

         \[\leadsto -\color{blue}{\sqrt{im \cdot im}} \cdot \cos re \]

      4. sqr-neg 56.8%

         \[\leadsto -\sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re \]

      5. sqrt-unprod 0.0%

         \[\leadsto -\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re \]

      6. add-sqr-sqrt 0.3%

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

      7. log1p-expm1-u 0.0%

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

      8. log1p-udef 0.0%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)\right)} \]

      9. neg-log 0.0%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re\right)}\right) \]

      11. sqrt-unprod 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re\right)}\right) \]

      12. sqr-neg 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\sqrt{\color{blue}{im \cdot im}} \cdot \cos re\right)}\right) \]

      13. sqrt-unprod 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re\right)}\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im} \cdot \cos re\right)}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \cos re\right)}\right)} \]

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{1}{e^{im \cdot \cos re}}\right)} \]
   10. Step-by-step derivation

       1. rec-exp 100.0%

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

       2. distribute-lft-neg-in 100.0%

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

   11. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 4.00000000000000033e-5

   1. Initial program 7.8%

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

      1. neg-sub0 7.8%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 99.8%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 99.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   if 4.00000000000000033e-5 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty:\\ \;\;\;\;\log \left(e^{\left(-im\right) \cdot \cos re}\right)\\ \mathbf{elif}\;e^{-im} - e^{im} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -700 \lor \neg \left(im \leq 720\right):\\ \;\;\;\;\log \left(e^{\left(-im\right) \cdot \cos re}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -700.0) (not (<= im 720.0)))
   (log (exp (* (- im) (cos re))))
   (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -700.0) || !(im <= 720.0)) {
		tmp = log(exp((-im * cos(re))));
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-700.0d0)) .or. (.not. (im <= 720.0d0))) then
        tmp = log(exp((-im * cos(re))))
    else
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -700.0) || !(im <= 720.0)) {
		tmp = Math.log(Math.exp((-im * Math.cos(re))));
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -700.0) or not (im <= 720.0):
		tmp = math.log(math.exp((-im * math.cos(re))))
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -700.0) || !(im <= 720.0))
		tmp = log(exp(Float64(Float64(-im) * cos(re))));
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -700.0) || ~((im <= 720.0)))
		tmp = log(exp((-im * cos(re))));
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -700.0], N[Not[LessEqual[im, 720.0]], $MachinePrecision]], N[Log[N[Exp[N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -700 or 720 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 5.3%

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

      2. neg-mul-1 5.3%

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

   6. Simplified 5.3%

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

      1. distribute-lft-neg-out 5.3%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. add-sqr-sqrt 2.7%

         \[\leadsto -\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re \]

      3. sqrt-unprod 29.6%

         \[\leadsto -\color{blue}{\sqrt{im \cdot im}} \cdot \cos re \]

      4. sqr-neg 29.6%

         \[\leadsto -\sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re \]

      5. sqrt-unprod 0.2%

         \[\leadsto -\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re \]

      6. add-sqr-sqrt 0.3%

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

      7. log1p-expm1-u 0.0%

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

      8. log1p-udef 0.0%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)\right)} \]

      9. neg-log 0.0%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re\right)}\right) \]

      11. sqrt-unprod 51.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re\right)}\right) \]

      12. sqr-neg 51.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\sqrt{\color{blue}{im \cdot im}} \cdot \cos re\right)}\right) \]

      13. sqrt-unprod 51.9%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re\right)}\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im} \cdot \cos re\right)}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \cos re\right)}\right)} \]

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{1}{e^{im \cdot \cos re}}\right)} \]
   10. Step-by-step derivation

       1. rec-exp 100.0%

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

       2. distribute-lft-neg-in 100.0%

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

   11. Simplified 100.0%

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

   if -700 < im < 720

   1. Initial program 9.2%

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

      1. neg-sub0 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in im around 0 98.9%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 98.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 98.9%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 98.9%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 98.9%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 98.9%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 98.9%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -700 \lor \neg \left(im \leq 720\right):\\ \;\;\;\;\log \left(e^{\left(-im\right) \cdot \cos re}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(e^{im}\right)\\ t_1 := -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -2.4 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (log (exp im))))
        (t_1 (* -0.16666666666666666 (* (cos re) (pow im 3.0)))))
   (if (<= im -2.4e+141)
     t_1
     (if (<= im -210000000000.0)
       t_0
       (if (<= im 700.0) (* (- im) (cos re)) (if (<= im 5e+95) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = -log(exp(im));
	double t_1 = -0.16666666666666666 * (cos(re) * pow(im, 3.0));
	double tmp;
	if (im <= -2.4e+141) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 700.0) {
		tmp = -im * cos(re);
	} else if (im <= 5e+95) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -log(exp(im))
    t_1 = (-0.16666666666666666d0) * (cos(re) * (im ** 3.0d0))
    if (im <= (-2.4d+141)) then
        tmp = t_1
    else if (im <= (-210000000000.0d0)) then
        tmp = t_0
    else if (im <= 700.0d0) then
        tmp = -im * cos(re)
    else if (im <= 5d+95) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -Math.log(Math.exp(im));
	double t_1 = -0.16666666666666666 * (Math.cos(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -2.4e+141) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 700.0) {
		tmp = -im * Math.cos(re);
	} else if (im <= 5e+95) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -math.log(math.exp(im))
	t_1 = -0.16666666666666666 * (math.cos(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -2.4e+141:
		tmp = t_1
	elif im <= -210000000000.0:
		tmp = t_0
	elif im <= 700.0:
		tmp = -im * math.cos(re)
	elif im <= 5e+95:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-log(exp(im)))
	t_1 = Float64(-0.16666666666666666 * Float64(cos(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -2.4e+141)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 700.0)
		tmp = Float64(Float64(-im) * cos(re));
	elseif (im <= 5e+95)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -log(exp(im));
	t_1 = -0.16666666666666666 * (cos(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -2.4e+141)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 700.0)
		tmp = -im * cos(re);
	elseif (im <= 5e+95)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Log[N[Exp[im], $MachinePrecision]], $MachinePrecision])}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.4e+141], t$95$1, If[LessEqual[im, -210000000000.0], t$95$0, If[LessEqual[im, 700.0], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+95], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.39999999999999997e141 or 5.00000000000000025e95 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 98.8%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 98.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 98.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 98.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 98.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 98.8%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in im around inf 98.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]

   if -2.39999999999999997e141 < im < -2.1e11 or 700 < im < 5.00000000000000025e95

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.7%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 3.7%

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

      2. neg-mul-1 3.7%

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

   6. Simplified 3.7%

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

      1. distribute-lft-neg-out 3.7%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. add-sqr-sqrt 1.5%

         \[\leadsto -\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re \]

      3. sqrt-unprod 1.8%

         \[\leadsto -\color{blue}{\sqrt{im \cdot im}} \cdot \cos re \]

      4. sqr-neg 1.8%

         \[\leadsto -\sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re \]

      5. sqrt-unprod 0.3%

         \[\leadsto -\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re \]

      6. add-sqr-sqrt 0.5%

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

      7. log1p-expm1-u 0.0%

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

      8. log1p-udef 0.0%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)\right)} \]

      9. neg-log 0.0%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re\right)}\right) \]

      11. sqrt-unprod 42.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re\right)}\right) \]

      12. sqr-neg 42.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\sqrt{\color{blue}{im \cdot im}} \cdot \cos re\right)}\right) \]

      13. sqrt-unprod 42.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re\right)}\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im} \cdot \cos re\right)}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \cos re\right)}\right)} \]

   9. Taylor expanded in re around 0 78.8%

      \[\leadsto \color{blue}{\log \left(\frac{1}{e^{im}}\right)} \]
   10. Step-by-step derivation

       1. log-rec 78.8%

          \[\leadsto \color{blue}{-\log \left(e^{im}\right)} \]

   11. Simplified 78.8%

       \[\leadsto \color{blue}{-\log \left(e^{im}\right)} \]

   if -2.1e11 < im < 700

   1. Initial program 10.7%

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

      1. neg-sub0 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 96.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 96.8%

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

      2. neg-mul-1 96.8%

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

   6. Simplified 96.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4 \cdot 10^{+141}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;-\log \left(e^{im}\right)\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+95}:\\ \;\;\;\;-\log \left(e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 4: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(e^{im}\right)\\ t_1 := -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\\ \mathbf{if}\;im \leq -2.4 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (log (exp im))))
        (t_1 (* -0.16666666666666666 (* (cos re) (pow im 3.0)))))
   (if (<= im -2.4e+141)
     t_1
     (if (<= im -210000000000.0)
       t_0
       (if (<= im 700.0)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 5e+95) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = -log(exp(im));
	double t_1 = -0.16666666666666666 * (cos(re) * pow(im, 3.0));
	double tmp;
	if (im <= -2.4e+141) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 700.0) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 5e+95) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -log(exp(im))
    t_1 = (-0.16666666666666666d0) * (cos(re) * (im ** 3.0d0))
    if (im <= (-2.4d+141)) then
        tmp = t_1
    else if (im <= (-210000000000.0d0)) then
        tmp = t_0
    else if (im <= 700.0d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 5d+95) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -Math.log(Math.exp(im));
	double t_1 = -0.16666666666666666 * (Math.cos(re) * Math.pow(im, 3.0));
	double tmp;
	if (im <= -2.4e+141) {
		tmp = t_1;
	} else if (im <= -210000000000.0) {
		tmp = t_0;
	} else if (im <= 700.0) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 5e+95) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -math.log(math.exp(im))
	t_1 = -0.16666666666666666 * (math.cos(re) * math.pow(im, 3.0))
	tmp = 0
	if im <= -2.4e+141:
		tmp = t_1
	elif im <= -210000000000.0:
		tmp = t_0
	elif im <= 700.0:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 5e+95:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-log(exp(im)))
	t_1 = Float64(-0.16666666666666666 * Float64(cos(re) * (im ^ 3.0)))
	tmp = 0.0
	if (im <= -2.4e+141)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 700.0)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 5e+95)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -log(exp(im));
	t_1 = -0.16666666666666666 * (cos(re) * (im ^ 3.0));
	tmp = 0.0;
	if (im <= -2.4e+141)
		tmp = t_1;
	elseif (im <= -210000000000.0)
		tmp = t_0;
	elseif (im <= 700.0)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 5e+95)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = (-N[Log[N[Exp[im], $MachinePrecision]], $MachinePrecision])}, Block[{t$95$1 = N[(-0.16666666666666666 * N[(N[Cos[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.4e+141], t$95$1, If[LessEqual[im, -210000000000.0], t$95$0, If[LessEqual[im, 700.0], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e+95], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.39999999999999997e141 or 5.00000000000000025e95 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 98.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 98.8%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 98.8%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 98.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 98.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 98.8%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 98.8%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in im around inf 98.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]

   if -2.39999999999999997e141 < im < -2.1e11 or 700 < im < 5.00000000000000025e95

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.7%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 3.7%

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

      2. neg-mul-1 3.7%

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

   6. Simplified 3.7%

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

      1. distribute-lft-neg-out 3.7%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. add-sqr-sqrt 1.5%

         \[\leadsto -\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re \]

      3. sqrt-unprod 1.8%

         \[\leadsto -\color{blue}{\sqrt{im \cdot im}} \cdot \cos re \]

      4. sqr-neg 1.8%

         \[\leadsto -\sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re \]

      5. sqrt-unprod 0.3%

         \[\leadsto -\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re \]

      6. add-sqr-sqrt 0.5%

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

      7. log1p-expm1-u 0.0%

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

      8. log1p-udef 0.0%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)\right)} \]

      9. neg-log 0.0%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re\right)}\right) \]

      11. sqrt-unprod 42.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re\right)}\right) \]

      12. sqr-neg 42.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\sqrt{\color{blue}{im \cdot im}} \cdot \cos re\right)}\right) \]

      13. sqrt-unprod 42.3%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re\right)}\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im} \cdot \cos re\right)}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \cos re\right)}\right)} \]

   9. Taylor expanded in re around 0 78.8%

      \[\leadsto \color{blue}{\log \left(\frac{1}{e^{im}}\right)} \]
   10. Step-by-step derivation

       1. log-rec 78.8%

          \[\leadsto \color{blue}{-\log \left(e^{im}\right)} \]

   11. Simplified 78.8%

       \[\leadsto \color{blue}{-\log \left(e^{im}\right)} \]

   if -2.1e11 < im < 700

   1. Initial program 10.7%

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

      1. neg-sub0 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 97.3%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 97.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 97.3%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 97.3%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 97.3%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 97.3%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 97.3%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 97.3%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4 \cdot 10^{+141}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\\ \mathbf{elif}\;im \leq -210000000000:\\ \;\;\;\;-\log \left(e^{im}\right)\\ \mathbf{elif}\;im \leq 700:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+95}:\\ \;\;\;\;-\log \left(e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\\ \end{array} \]

Alternative 5: 86.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -210000000000 \lor \neg \left(im \leq 700\right):\\ \;\;\;\;-\log \left(e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -210000000000.0) (not (<= im 700.0)))
   (- (log (exp im)))
   (* (- im) (cos re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -210000000000.0) || !(im <= 700.0)) {
		tmp = -log(exp(im));
	} else {
		tmp = -im * cos(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 <= (-210000000000.0d0)) .or. (.not. (im <= 700.0d0))) then
        tmp = -log(exp(im))
    else
        tmp = -im * cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -210000000000.0) || !(im <= 700.0)) {
		tmp = -Math.log(Math.exp(im));
	} else {
		tmp = -im * Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -210000000000.0) or not (im <= 700.0):
		tmp = -math.log(math.exp(im))
	else:
		tmp = -im * math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -210000000000.0) || !(im <= 700.0))
		tmp = Float64(-log(exp(im)));
	else
		tmp = Float64(Float64(-im) * cos(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -210000000000.0) || ~((im <= 700.0)))
		tmp = -log(exp(im));
	else
		tmp = -im * cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -210000000000.0], N[Not[LessEqual[im, 700.0]], $MachinePrecision]], (-N[Log[N[Exp[im], $MachinePrecision]], $MachinePrecision]), N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.1e11 or 700 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.3%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 5.3%

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

      2. neg-mul-1 5.3%

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

   6. Simplified 5.3%

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

      1. distribute-lft-neg-out 5.3%

         \[\leadsto \color{blue}{-im \cdot \cos re} \]

      2. add-sqr-sqrt 2.8%

         \[\leadsto -\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re \]

      3. sqrt-unprod 30.0%

         \[\leadsto -\color{blue}{\sqrt{im \cdot im}} \cdot \cos re \]

      4. sqr-neg 30.0%

         \[\leadsto -\sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re \]

      5. sqrt-unprod 0.1%

         \[\leadsto -\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re \]

      6. add-sqr-sqrt 0.3%

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

      7. log1p-expm1-u 0.0%

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

      8. log1p-udef 0.0%

         \[\leadsto -\color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)\right)} \]

      9. neg-log 0.0%

         \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(\left(-im\right) \cdot \cos re\right)}\right)} \]

      10. add-sqr-sqrt 0.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{-im} \cdot \sqrt{-im}\right)} \cdot \cos re\right)}\right) \]

      11. sqrt-unprod 52.7%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}} \cdot \cos re\right)}\right) \]

      12. sqr-neg 52.7%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\sqrt{\color{blue}{im \cdot im}} \cdot \cos re\right)}\right) \]

      13. sqrt-unprod 52.7%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)} \cdot \cos re\right)}\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \log \left(\frac{1}{1 + \mathsf{expm1}\left(\color{blue}{im} \cdot \cos re\right)}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 + \mathsf{expm1}\left(im \cdot \cos re\right)}\right)} \]

   9. Taylor expanded in re around 0 72.5%

      \[\leadsto \color{blue}{\log \left(\frac{1}{e^{im}}\right)} \]
   10. Step-by-step derivation

       1. log-rec 72.5%

          \[\leadsto \color{blue}{-\log \left(e^{im}\right)} \]

   11. Simplified 72.5%

       \[\leadsto \color{blue}{-\log \left(e^{im}\right)} \]

   if -2.1e11 < im < 700

   1. Initial program 10.7%

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

      1. neg-sub0 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in im around 0 96.8%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 96.8%

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

      2. neg-mul-1 96.8%

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

   6. Simplified 96.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -210000000000 \lor \neg \left(im \leq 700\right):\\ \;\;\;\;-\log \left(e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \end{array} \]

Alternative 6: 75.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -4.2 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -700:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1650000000:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -4.2e+101)
     t_0
     (if (<= im -700.0)
       (* (pow re 2.0) (* im 0.5))
       (if (<= im 1650000000.0) (* (- im) (cos re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -4.2e+101) {
		tmp = t_0;
	} else if (im <= -700.0) {
		tmp = pow(re, 2.0) * (im * 0.5);
	} else if (im <= 1650000000.0) {
		tmp = -im * cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-4.2d+101)) then
        tmp = t_0
    else if (im <= (-700.0d0)) then
        tmp = (re ** 2.0d0) * (im * 0.5d0)
    else if (im <= 1650000000.0d0) then
        tmp = -im * cos(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -4.2e+101) {
		tmp = t_0;
	} else if (im <= -700.0) {
		tmp = Math.pow(re, 2.0) * (im * 0.5);
	} else if (im <= 1650000000.0) {
		tmp = -im * Math.cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -4.2e+101:
		tmp = t_0
	elif im <= -700.0:
		tmp = math.pow(re, 2.0) * (im * 0.5)
	elif im <= 1650000000.0:
		tmp = -im * math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -4.2e+101)
		tmp = t_0;
	elseif (im <= -700.0)
		tmp = Float64((re ^ 2.0) * Float64(im * 0.5));
	elseif (im <= 1650000000.0)
		tmp = Float64(Float64(-im) * cos(re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -4.2e+101)
		tmp = t_0;
	elseif (im <= -700.0)
		tmp = (re ^ 2.0) * (im * 0.5);
	elseif (im <= 1650000000.0)
		tmp = -im * cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -4.2e+101], t$95$0, If[LessEqual[im, -700.0], N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1650000000.0], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.2e101 or 1.65e9 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 79.4%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 79.4%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 79.4%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 79.4%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 79.4%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 79.4%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 79.4%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 57.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -4.2e101 < im < -700

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.6%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 3.6%

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

      2. neg-mul-1 3.6%

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

   6. Simplified 3.6%

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

   7. Taylor expanded in re around 0 32.4%

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

      1. mul-1-neg 32.4%

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

      2. +-commutative 32.4%

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

      3. *-commutative 32.4%

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

      4. associate-*l* 32.4%

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

      5. mul-1-neg 32.4%

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

      6. *-commutative 32.4%

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

      7. distribute-lft-out 32.4%

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

   9. Simplified 32.4%

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

   10. Taylor expanded in re around inf 31.3%

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

       1. associate-*r* 31.3%

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

       2. *-commutative 31.3%

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

       3. *-commutative 31.3%

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

   12. Simplified 31.3%

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

   if -700 < im < 1.65e9

   1. Initial program 9.2%

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

      1. neg-sub0 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in im around 0 98.3%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 98.3%

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

      2. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.2 \cdot 10^{+101}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -700:\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;im \leq 1650000000:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 7: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -480:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{2} + -1\right)\\ \mathbf{elif}\;im \leq 1650000000:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -4.5e+101)
     t_0
     (if (<= im -480.0)
       (* im (+ (* 0.5 (pow re 2.0)) -1.0))
       (if (<= im 1650000000.0) (* (- im) (cos re)) t_0)))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -4.5e+101) {
		tmp = t_0;
	} else if (im <= -480.0) {
		tmp = im * ((0.5 * pow(re, 2.0)) + -1.0);
	} else if (im <= 1650000000.0) {
		tmp = -im * cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-4.5d+101)) then
        tmp = t_0
    else if (im <= (-480.0d0)) then
        tmp = im * ((0.5d0 * (re ** 2.0d0)) + (-1.0d0))
    else if (im <= 1650000000.0d0) then
        tmp = -im * cos(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -4.5e+101) {
		tmp = t_0;
	} else if (im <= -480.0) {
		tmp = im * ((0.5 * Math.pow(re, 2.0)) + -1.0);
	} else if (im <= 1650000000.0) {
		tmp = -im * Math.cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -4.5e+101:
		tmp = t_0
	elif im <= -480.0:
		tmp = im * ((0.5 * math.pow(re, 2.0)) + -1.0)
	elif im <= 1650000000.0:
		tmp = -im * math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -4.5e+101)
		tmp = t_0;
	elseif (im <= -480.0)
		tmp = Float64(im * Float64(Float64(0.5 * (re ^ 2.0)) + -1.0));
	elseif (im <= 1650000000.0)
		tmp = Float64(Float64(-im) * cos(re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -4.5e+101)
		tmp = t_0;
	elseif (im <= -480.0)
		tmp = im * ((0.5 * (re ^ 2.0)) + -1.0);
	elseif (im <= 1650000000.0)
		tmp = -im * cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -4.5e+101], t$95$0, If[LessEqual[im, -480.0], N[(im * N[(N[(0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1650000000.0], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.5000000000000002e101 or 1.65e9 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 79.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 79.4%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + -1 \cdot \left(im \cdot \cos re\right)} \]

      2. mul-1-neg 79.4%

         \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) + \color{blue}{\left(-im \cdot \cos re\right)} \]

      3. unsub-neg 79.4%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \cos re\right) - im \cdot \cos re} \]

      4. associate-*r* 79.4%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re} - im \cdot \cos re \]

      5. distribute-rgt-out-- 79.4%

         \[\leadsto \color{blue}{\cos re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      6. *-commutative 79.4%

         \[\leadsto \cos re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   6. Simplified 79.4%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 57.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]

   if -4.5000000000000002e101 < im < -480

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 3.6%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 3.6%

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

      2. neg-mul-1 3.6%

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

   6. Simplified 3.6%

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

   7. Taylor expanded in re around 0 32.4%

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

      1. mul-1-neg 32.4%

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

      2. +-commutative 32.4%

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

      3. *-commutative 32.4%

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

      4. associate-*l* 32.4%

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

      5. mul-1-neg 32.4%

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

      6. *-commutative 32.4%

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

      7. distribute-lft-out 32.4%

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

   9. Simplified 32.4%

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

   if -480 < im < 1.65e9

   1. Initial program 9.2%

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

      1. neg-sub0 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in im around 0 98.3%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 98.3%

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

      2. neg-mul-1 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+101}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{elif}\;im \leq -480:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{2} + -1\right)\\ \mathbf{elif}\;im \leq 1650000000:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 8: 58.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -650 \lor \neg \left(im \leq 1.7 \cdot 10^{+50}\right):\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -650.0) (not (<= im 1.7e+50)))
   (* (pow re 2.0) (* im 0.5))
   (* (- im) (cos re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -650.0) || !(im <= 1.7e+50)) {
		tmp = pow(re, 2.0) * (im * 0.5);
	} else {
		tmp = -im * cos(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 <= (-650.0d0)) .or. (.not. (im <= 1.7d+50))) then
        tmp = (re ** 2.0d0) * (im * 0.5d0)
    else
        tmp = -im * cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -650.0) || !(im <= 1.7e+50)) {
		tmp = Math.pow(re, 2.0) * (im * 0.5);
	} else {
		tmp = -im * Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -650.0) or not (im <= 1.7e+50):
		tmp = math.pow(re, 2.0) * (im * 0.5)
	else:
		tmp = -im * math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -650.0) || !(im <= 1.7e+50))
		tmp = Float64((re ^ 2.0) * Float64(im * 0.5));
	else
		tmp = Float64(Float64(-im) * cos(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -650.0) || ~((im <= 1.7e+50)))
		tmp = (re ^ 2.0) * (im * 0.5);
	else
		tmp = -im * cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -650.0], N[Not[LessEqual[im, 1.7e+50]], $MachinePrecision]], N[(N[Power[re, 2.0], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -650 or 1.6999999999999999e50 < im

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 5.4%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 5.4%

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

      2. neg-mul-1 5.4%

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

   6. Simplified 5.4%

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

   7. Taylor expanded in re around 0 28.5%

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

      1. mul-1-neg 28.5%

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

      2. +-commutative 28.5%

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

      3. *-commutative 28.5%

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

      4. associate-*l* 28.5%

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

      5. mul-1-neg 28.5%

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

      6. *-commutative 28.5%

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

      7. distribute-lft-out 28.5%

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

   9. Simplified 28.5%

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

   10. Taylor expanded in re around inf 26.2%

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

       1. associate-*r* 26.2%

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

       2. *-commutative 26.2%

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

       3. *-commutative 26.2%

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

   12. Simplified 26.2%

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

   if -650 < im < 1.6999999999999999e50

   1. Initial program 16.7%

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

      1. neg-sub0 16.7%

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

   3. Simplified 16.7%

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

   4. Taylor expanded in im around 0 90.5%

      \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 90.5%

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

      2. neg-mul-1 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -650 \lor \neg \left(im \leq 1.7 \cdot 10^{+50}\right):\\ \;\;\;\;{re}^{2} \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \cos re\\ \end{array} \]

Alternative 9: 51.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(-im\right) \cdot \cos re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (- im) (cos re)))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return -im * Math.cos(re);
}

Python

def code(re, im):
	return -im * math.cos(re)

Julia

function code(re, im)
	return Float64(Float64(-im) * cos(re))
end

MATLAB

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

Wolfram

code[re_, im_] := N[((-im) * N[Cos[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

   1. neg-sub0 56.4%

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

3. Simplified 56.4%

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

4. Taylor expanded in im around 0 50.0%

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
5. Step-by-step derivation

   1. associate-*r* 50.0%

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

   2. neg-mul-1 50.0%

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

6. Simplified 50.0%

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

7. Final simplification 50.0%

   \[\leadsto \left(-im\right) \cdot \cos re \]

Alternative 10: 29.9% accurate, 154.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

function code(re, im)
	return Float64(-im)
end

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 56.4%

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

   1. neg-sub0 56.4%

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

3. Simplified 56.4%

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

4. Taylor expanded in im around 0 50.0%

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \cos re\right)} \]
5. Step-by-step derivation

   1. associate-*r* 50.0%

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

   2. neg-mul-1 50.0%

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

6. Simplified 50.0%

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

7. Taylor expanded in re around 0 28.6%

   \[\leadsto \color{blue}{-1 \cdot im} \]
8. Step-by-step derivation

   1. mul-1-neg 28.6%

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

9. Simplified 28.6%

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

10. Final simplification 28.6%

    \[\leadsto -im \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024024 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))