math.sin on complex, imaginary part

Percentage Accurate: 54.9% → 99.7%

Time: 13.1s

Alternatives: 18

Speedup: 2.8×

• Unsound rule application detected in e-graph. Results may not be sound.

• 50.0% 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: 54.9% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.05\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.05)))
     (* (* 0.5 (cos re)) t_0)
     (*
      (cos re)
      (+
       (- (* (pow im 3.0) -0.16666666666666666) im)
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (* (pow im 5.0) -0.008333333333333333)))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.05)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * (((pow(im, 3.0) * -0.16666666666666666) - im) + ((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 5.0) * -0.008333333333333333)));
	}
	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) || !(t_0 <= 0.05)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * (((Math.pow(im, 3.0) * -0.16666666666666666) - im) + ((Math.pow(im, 7.0) * -0.0001984126984126984) + (Math.pow(im, 5.0) * -0.008333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.05):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * (((math.pow(im, 3.0) * -0.16666666666666666) - im) + ((math.pow(im, 7.0) * -0.0001984126984126984) + (math.pow(im, 5.0) * -0.008333333333333333)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.05))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) + Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 5.0) * -0.008333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.05)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * ((((im ^ 3.0) * -0.16666666666666666) - im) + (((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 5.0) * -0.008333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.05]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] + N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 0.050000000000000003 < (-.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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.050000000000000003

   1. Initial program 8.2%

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

      1. sub0-neg 8.2%

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

   3. Simplified 8.2%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)} \]

      2. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\cos re \cdot im\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      3. mul-1-neg 99.8%

         \[\leadsto \left(\color{blue}{\left(-\cos re \cdot im\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      4. *-commutative 99.8%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \cos re}\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      5. distribute-lft-neg-in 99.8%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \cos re} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      6. *-commutative 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + -0.16666666666666666 \cdot \color{blue}{\left({im}^{3} \cdot \cos re\right)}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      7. associate-*r* 99.8%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      8. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      9. *-commutative 99.8%

         \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      11. *-commutative 99.8%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984}\right) \]

      12. associate-*l* 99.8%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\cos re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)}\right) \]

      13. distribute-lft-out 99.8%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.05\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := e^{-im} - e^{im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.05\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (- (exp (- im)) (exp im))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.05)))
     (* t_0 t_1)
     (*
      t_0
      (+
       (* im -2.0)
       (+
        (* (pow im 5.0) -0.016666666666666666)
        (* (pow im 3.0) -0.3333333333333333)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = exp(-im) - exp(im);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.05)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * ((im * -2.0) + ((pow(im, 5.0) * -0.016666666666666666) + (pow(im, 3.0) * -0.3333333333333333)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.05)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * ((im * -2.0) + ((Math.pow(im, 5.0) * -0.016666666666666666) + (Math.pow(im, 3.0) * -0.3333333333333333)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.05):
		tmp = t_0 * t_1
	else:
		tmp = t_0 * ((im * -2.0) + ((math.pow(im, 5.0) * -0.016666666666666666) + (math.pow(im, 3.0) * -0.3333333333333333)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 0.05))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_0 * Float64(Float64(im * -2.0) + Float64(Float64((im ^ 5.0) * -0.016666666666666666) + Float64((im ^ 3.0) * -0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.05)))
		tmp = t_0 * t_1;
	else
		tmp = t_0 * ((im * -2.0) + (((im ^ 5.0) * -0.016666666666666666) + ((im ^ 3.0) * -0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.05]], $MachinePrecision]], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$0 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.016666666666666666), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 0.050000000000000003 < (-.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. sub0-neg 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 0.050000000000000003

   1. Initial program 8.2%

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

      1. sub0-neg 8.2%

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

   3. Simplified 8.2%

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

   4. Taylor expanded in im around 0 99.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.05\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left({im}^{5} \cdot -0.016666666666666666 + {im}^{3} \cdot -0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-6)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-6)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	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) || !(t_0 <= 5e-6)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e-6):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e-6))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e-6)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e-6]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $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 (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 5.00000000000000041e-6 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 5.00000000000000041e-6

   1. Initial program 7.5%

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

      1. sub0-neg 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 5 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ t_1 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0132:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;im \leq 0.15:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0 \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im)))
        (t_1 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -2.1e+55)
     t_1
     (if (<= im -0.0132)
       (* 0.5 t_0)
       (if (<= im 0.15)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) (* t_0 (+ 0.5 (* re (* re -0.25)))) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double t_1 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.0132) {
		tmp = 0.5 * t_0;
	} else if (im <= 0.15) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} 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 = exp(-im) - exp(im)
    t_1 = (im ** 7.0d0) * (cos(re) * (-0.0001984126984126984d0))
    if (im <= (-2.1d+55)) then
        tmp = t_1
    else if (im <= (-0.0132d0)) then
        tmp = 0.5d0 * t_0
    else if (im <= 0.15d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) then
        tmp = t_0 * (0.5d0 + (re * (re * (-0.25d0))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double t_1 = Math.pow(im, 7.0) * (Math.cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.0132) {
		tmp = 0.5 * t_0;
	} else if (im <= 0.15) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	t_1 = math.pow(im, 7.0) * (math.cos(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -2.1e+55:
		tmp = t_1
	elif im <= -0.0132:
		tmp = 0.5 * t_0
	elif im <= 0.15:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = t_0 * (0.5 + (re * (re * -0.25)))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	t_1 = Float64((im ^ 7.0) * Float64(cos(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.0132)
		tmp = Float64(0.5 * t_0);
	elseif (im <= 0.15)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = Float64(t_0 * Float64(0.5 + Float64(re * Float64(re * -0.25))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	t_1 = (im ^ 7.0) * (cos(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.0132)
		tmp = 0.5 * t_0;
	elseif (im <= 0.15)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = t_0 * (0.5 + (re * (re * -0.25)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+55], t$95$1, If[LessEqual[im, -0.0132], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[im, 0.15], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], N[(t$95$0 * N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.1000000000000001e55 or 1.09999999999999998e44 < 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. sub0-neg 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 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\cos re \cdot im\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      3. mul-1-neg 100.0%

         \[\leadsto \left(\color{blue}{\left(-\cos re \cdot im\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \cos re}\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      5. distribute-lft-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \cos re} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      6. *-commutative 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + -0.16666666666666666 \cdot \color{blue}{\left({im}^{3} \cdot \cos re\right)}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      7. associate-*r* 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      8. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      9. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984}\right) \]

      12. associate-*l* 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\cos re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)}\right) \]

      13. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \cos re\right) \cdot {im}^{7}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \cos re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \cos re\right)} \]

   if -2.1000000000000001e55 < im < -0.0132

   1. Initial program 99.7%

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

      1. sub0-neg 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in re around 0 80.6%

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

   if -0.0132 < im < 0.149999999999999994

   1. Initial program 7.5%

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

      1. sub0-neg 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

   if 0.149999999999999994 < im < 1.09999999999999998e44

   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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 88.9%

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

      4. +-commutative 88.9%

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

      5. *-commutative 88.9%

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

      6. unpow2 88.9%

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

      7. associate-*l* 88.9%

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

   6. Simplified 88.9%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -0.0132:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.15:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0048:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -2.1e+55)
     t_1
     (if (<= im -0.0048)
       t_0
       (if (<= im 5.5e-5) (* im (- (cos re))) (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.0048) {
		tmp = t_0;
	} else if (im <= 5.5e-5) {
		tmp = im * -cos(re);
	} else if (im <= 1.1e+44) {
		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 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (im ** 7.0d0) * (cos(re) * (-0.0001984126984126984d0))
    if (im <= (-2.1d+55)) then
        tmp = t_1
    else if (im <= (-0.0048d0)) then
        tmp = t_0
    else if (im <= 5.5d-5) then
        tmp = im * -cos(re)
    else if (im <= 1.1d+44) 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 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = Math.pow(im, 7.0) * (Math.cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.0048) {
		tmp = t_0;
	} else if (im <= 5.5e-5) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.pow(im, 7.0) * (math.cos(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -2.1e+55:
		tmp = t_1
	elif im <= -0.0048:
		tmp = t_0
	elif im <= 5.5e-5:
		tmp = im * -math.cos(re)
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64((im ^ 7.0) * Float64(cos(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.0048)
		tmp = t_0;
	elseif (im <= 5.5e-5)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (im ^ 7.0) * (cos(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.0048)
		tmp = t_0;
	elseif (im <= 5.5e-5)
		tmp = im * -cos(re);
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+55], t$95$1, If[LessEqual[im, -0.0048], t$95$0, If[LessEqual[im, 5.5e-5], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.1000000000000001e55 or 1.09999999999999998e44 < 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. sub0-neg 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 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\cos re \cdot im\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      3. mul-1-neg 100.0%

         \[\leadsto \left(\color{blue}{\left(-\cos re \cdot im\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \cos re}\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      5. distribute-lft-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \cos re} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      6. *-commutative 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + -0.16666666666666666 \cdot \color{blue}{\left({im}^{3} \cdot \cos re\right)}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      7. associate-*r* 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      8. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      9. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984}\right) \]

      12. associate-*l* 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\cos re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)}\right) \]

      13. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \cos re\right) \cdot {im}^{7}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \cos re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \cos re\right)} \]

   if -2.1000000000000001e55 < im < -0.00479999999999999958 or 5.5000000000000002e-5 < im < 1.09999999999999998e44

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 76.4%

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

   if -0.00479999999999999958 < im < 5.5000000000000002e-5

   1. Initial program 7.5%

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

      1. sub0-neg 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in im around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -0.0048:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.21:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (pow im 7.0) (* (cos re) -0.0001984126984126984))))
   (if (<= im -2.1e+55)
     t_1
     (if (<= im -0.014)
       t_0
       (if (<= im 0.21)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 1.1e+44) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = pow(im, 7.0) * (cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.014) {
		tmp = t_0;
	} else if (im <= 0.21) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		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 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (im ** 7.0d0) * (cos(re) * (-0.0001984126984126984d0))
    if (im <= (-2.1d+55)) then
        tmp = t_1
    else if (im <= (-0.014d0)) then
        tmp = t_0
    else if (im <= 0.21d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 1.1d+44) 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 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = Math.pow(im, 7.0) * (Math.cos(re) * -0.0001984126984126984);
	double tmp;
	if (im <= -2.1e+55) {
		tmp = t_1;
	} else if (im <= -0.014) {
		tmp = t_0;
	} else if (im <= 0.21) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 1.1e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.pow(im, 7.0) * (math.cos(re) * -0.0001984126984126984)
	tmp = 0
	if im <= -2.1e+55:
		tmp = t_1
	elif im <= -0.014:
		tmp = t_0
	elif im <= 0.21:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 1.1e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64((im ^ 7.0) * Float64(cos(re) * -0.0001984126984126984))
	tmp = 0.0
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.014)
		tmp = t_0;
	elseif (im <= 0.21)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (im ^ 7.0) * (cos(re) * -0.0001984126984126984);
	tmp = 0.0;
	if (im <= -2.1e+55)
		tmp = t_1;
	elseif (im <= -0.014)
		tmp = t_0;
	elseif (im <= 0.21)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 1.1e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.1e+55], t$95$1, If[LessEqual[im, -0.014], t$95$0, If[LessEqual[im, 0.21], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.1e+44], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.1000000000000001e55 or 1.09999999999999998e44 < 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. sub0-neg 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 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \left(-1 \cdot \left(\cos re \cdot im\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right)} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \left(\cos re \cdot im\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      3. mul-1-neg 100.0%

         \[\leadsto \left(\color{blue}{\left(-\cos re \cdot im\right)} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(\left(-\color{blue}{im \cdot \cos re}\right) + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      5. distribute-lft-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(-im\right) \cdot \cos re} + -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right)\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      6. *-commutative 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + -0.16666666666666666 \cdot \color{blue}{\left({im}^{3} \cdot \cos re\right)}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      7. associate-*r* 100.0%

         \[\leadsto \left(\left(-im\right) \cdot \cos re + \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \cos re}\right) + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      8. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right)} + \left(-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right) + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      9. *-commutative 100.0%

         \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      10. associate-*l* 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)} + -0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)\right) \]

      11. *-commutative 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\left(\cos re \cdot {im}^{7}\right) \cdot -0.0001984126984126984}\right) \]

      12. associate-*l* 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \left(\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right) + \color{blue}{\cos re \cdot \left({im}^{7} \cdot -0.0001984126984126984\right)}\right) \]

      13. distribute-lft-out 100.0%

          \[\leadsto \cos re \cdot \left(\left(-im\right) + -0.16666666666666666 \cdot {im}^{3}\right) + \color{blue}{\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333 + {im}^{7} \cdot -0.0001984126984126984\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left(\left({im}^{3} \cdot -0.16666666666666666 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{5} \cdot -0.008333333333333333\right)\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(\cos re \cdot {im}^{7}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot \cos re\right) \cdot {im}^{7}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \cos re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{7} \cdot \left(-0.0001984126984126984 \cdot \cos re\right)} \]

   if -2.1000000000000001e55 < im < -0.0140000000000000003 or 0.209999999999999992 < im < 1.09999999999999998e44

   1. Initial program 99.8%

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

      1. sub0-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in re around 0 76.4%

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

   if -0.0140000000000000003 < im < 0.209999999999999992

   1. Initial program 7.5%

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

      1. sub0-neg 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \mathbf{elif}\;im \leq -0.014:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.21:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{7} \cdot \left(\cos re \cdot -0.0001984126984126984\right)\\ \end{array} \]

Alternative 7: 86.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;im \leq -0.00068:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0004:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im)))))
   (if (<= im -0.00068)
     t_0
     (if (<= im 0.0004)
       (* im (- (cos re)))
       (if (<= im 2.55e+92)
         t_0
         (*
          (- (* (pow im 3.0) -0.16666666666666666) im)
          (+ (* -0.5 (* re re)) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double tmp;
	if (im <= -0.00068) {
		tmp = t_0;
	} else if (im <= 0.0004) {
		tmp = im * -cos(re);
	} else if (im <= 2.55e+92) {
		tmp = t_0;
	} else {
		tmp = ((pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	}
	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 = 0.5d0 * (exp(-im) - exp(im))
    if (im <= (-0.00068d0)) then
        tmp = t_0
    else if (im <= 0.0004d0) then
        tmp = im * -cos(re)
    else if (im <= 2.55d+92) then
        tmp = t_0
    else
        tmp = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (((-0.5d0) * (re * re)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (im <= -0.00068) {
		tmp = t_0;
	} else if (im <= 0.0004) {
		tmp = im * -Math.cos(re);
	} else if (im <= 2.55e+92) {
		tmp = t_0;
	} else {
		tmp = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	tmp = 0
	if im <= -0.00068:
		tmp = t_0
	elif im <= 0.0004:
		tmp = im * -math.cos(re)
	elif im <= 2.55e+92:
		tmp = t_0
	else:
		tmp = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (im <= -0.00068)
		tmp = t_0;
	elseif (im <= 0.0004)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 2.55e+92)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	tmp = 0.0;
	if (im <= -0.00068)
		tmp = t_0;
	elseif (im <= 0.0004)
		tmp = im * -cos(re);
	elseif (im <= 2.55e+92)
		tmp = t_0;
	else
		tmp = (((im ^ 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -0.00068], t$95$0, If[LessEqual[im, 0.0004], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 2.55e+92], t$95$0, N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.8e-4 or 4.00000000000000019e-4 < im < 2.5500000000000001e92

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in re around 0 71.7%

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

   if -6.8e-4 < im < 4.00000000000000019e-4

   1. Initial program 7.5%

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

      1. sub0-neg 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in im around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-lft-neg-in 99.7%

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

   6. Simplified 99.7%

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

   if 2.5500000000000001e92 < 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. sub0-neg 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 92.6%

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

      1. mul-1-neg 92.6%

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

      2. unsub-neg 92.6%

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

      3. *-commutative 92.6%

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

      4. associate-*l* 92.6%

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

      5. distribute-lft-out-- 92.6%

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

   6. Simplified 92.6%

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

   7. Taylor expanded in re around 0 5.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. *-commutative 5.4%

         \[\leadsto \left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \color{blue}{{im}^{3} \cdot -0.16666666666666666}\right) - im \]

      2. associate-+r- 5.4%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      3. associate-*r* 5.4%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      4. *-commutative 5.4%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      5. *-lft-identity 5.4%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{1 \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      6. distribute-rgt-out 81.1%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right)} \]

      7. *-commutative 81.1%

         \[\leadsto \left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right) \]

      8. unpow2 81.1%

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \]

   9. Simplified 81.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.00068:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.0004:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \]

Alternative 8: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{if}\;im \leq -7.8 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -58000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(re \cdot re\right) \cdot 0.75\right)\right)\\ \mathbf{elif}\;im \leq 2900000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (- (* (pow im 3.0) -0.16666666666666666) im)
          (+ (* -0.5 (* re re)) 1.0))))
   (if (<= im -7.8e+67)
     t_0
     (if (<= im -58000.0)
       (log1p (expm1 (* (* re re) 0.75)))
       (if (<= im 2900000.0) (* im (- (cos re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = ((pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -7.8e+67) {
		tmp = t_0;
	} else if (im <= -58000.0) {
		tmp = log1p(expm1(((re * re) * 0.75)));
	} else if (im <= 2900000.0) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	double tmp;
	if (im <= -7.8e+67) {
		tmp = t_0;
	} else if (im <= -58000.0) {
		tmp = Math.log1p(Math.expm1(((re * re) * 0.75)));
	} else if (im <= 2900000.0) {
		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) * ((-0.5 * (re * re)) + 1.0)
	tmp = 0
	if im <= -7.8e+67:
		tmp = t_0
	elif im <= -58000.0:
		tmp = math.log1p(math.expm1(((re * re) * 0.75)))
	elif im <= 2900000.0:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0))
	tmp = 0.0
	if (im <= -7.8e+67)
		tmp = t_0;
	elseif (im <= -58000.0)
		tmp = log1p(expm1(Float64(Float64(re * re) * 0.75)));
	elseif (im <= 2900000.0)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -7.8e+67], t$95$0, If[LessEqual[im, -58000.0], N[Log[1 + N[(Exp[N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 2900000.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -7.80000000000000013e67 or 2.9e6 < 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. sub0-neg 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 77.0%

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

      1. mul-1-neg 77.0%

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

      2. unsub-neg 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-*l* 77.0%

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

      5. distribute-lft-out-- 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in re around 0 8.6%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto \left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \color{blue}{{im}^{3} \cdot -0.16666666666666666}\right) - im \]

      2. associate-+r- 8.6%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      3. associate-*r* 8.6%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      4. *-commutative 8.6%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      5. *-lft-identity 8.6%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{1 \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      6. distribute-rgt-out 68.4%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right)} \]

      7. *-commutative 68.4%

         \[\leadsto \left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right) \]

      8. unpow2 68.4%

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \]

   9. Simplified 68.4%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)} \]

   if -7.80000000000000013e67 < im < -58000

   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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 54.5%

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

      4. +-commutative 54.5%

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

      5. *-commutative 54.5%

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

      6. unpow2 54.5%

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

      7. associate-*l* 54.5%

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

   6. Simplified 54.5%

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

   8. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 33.1%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 33.1%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 33.1%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

   11. Simplified 33.1%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot 0.75} \]
   12. Step-by-step derivation

       1. log1p-expm1-u 46.0%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(re \cdot re\right) \cdot 0.75\right)\right)} \]

   13. Applied egg-rr 46.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(re \cdot re\right) \cdot 0.75\right)\right)} \]

   if -58000 < im < 2.9e6

   1. Initial program 11.0%

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

      1. sub0-neg 11.0%

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

   3. Simplified 11.0%

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

   4. Taylor expanded in im around 0 96.3%

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

      1. mul-1-neg 96.3%

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

      2. *-commutative 96.3%

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

      3. distribute-lft-neg-in 96.3%

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

   6. Simplified 96.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.8 \cdot 10^{+67}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{elif}\;im \leq -58000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\left(re \cdot re\right) \cdot 0.75\right)\right)\\ \mathbf{elif}\;im \leq 2900000:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \end{array} \]

Alternative 9: 78.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -0.0092 \lor \neg \left(im \leq 2900000\right):\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -0.0092) (not (<= im 2900000.0)))
   (* (- (* (pow im 3.0) -0.16666666666666666) im) (+ (* -0.5 (* re re)) 1.0))
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -0.0092) || !(im <= 2900000.0)) {
		tmp = ((pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	} 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 <= (-0.0092d0)) .or. (.not. (im <= 2900000.0d0))) then
        tmp = (((im ** 3.0d0) * (-0.16666666666666666d0)) - im) * (((-0.5d0) * (re * re)) + 1.0d0)
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -0.0092) || !(im <= 2900000.0)) {
		tmp = ((Math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -0.0092) or not (im <= 2900000.0):
		tmp = ((math.pow(im, 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0)
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -0.0092) || !(im <= 2900000.0))
		tmp = Float64(Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im) * Float64(Float64(-0.5 * Float64(re * re)) + 1.0));
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -0.0092) || ~((im <= 2900000.0)))
		tmp = (((im ^ 3.0) * -0.16666666666666666) - im) * ((-0.5 * (re * re)) + 1.0);
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -0.0092], N[Not[LessEqual[im, 2900000.0]], $MachinePrecision]], N[(N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision] * N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -0.0091999999999999998 or 2.9e6 < im

   1. Initial program 99.9%

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

      1. sub0-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in im around 0 63.1%

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

      1. mul-1-neg 63.1%

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

      2. unsub-neg 63.1%

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

      3. *-commutative 63.1%

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

      4. associate-*l* 63.1%

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

      5. distribute-lft-out-- 63.1%

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

   6. Simplified 63.1%

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

   7. Taylor expanded in re around 0 10.9%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. *-commutative 10.9%

         \[\leadsto \left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \color{blue}{{im}^{3} \cdot -0.16666666666666666}\right) - im \]

      2. associate-+r- 10.9%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      3. associate-*r* 10.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      4. *-commutative 10.9%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) + \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]

      5. *-lft-identity 10.9%

         \[\leadsto \left(-0.5 \cdot {re}^{2}\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) + \color{blue}{1 \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

      6. distribute-rgt-out 58.9%

         \[\leadsto \color{blue}{\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right)} \]

      7. *-commutative 58.9%

         \[\leadsto \left(\color{blue}{-0.16666666666666666 \cdot {im}^{3}} - im\right) \cdot \left(-0.5 \cdot {re}^{2} + 1\right) \]

      8. unpow2 58.9%

         \[\leadsto \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \color{blue}{\left(re \cdot re\right)} + 1\right) \]

   9. Simplified 58.9%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3} - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)} \]

   if -0.0091999999999999998 < im < 2.9e6

   1. Initial program 8.9%

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

      1. sub0-neg 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in im around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -0.0092 \lor \neg \left(im \leq 2900000\right):\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(-0.5 \cdot \left(re \cdot re\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 10: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -2.7 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -58000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 4.05 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -2.7e+68)
     t_0
     (if (<= im -58000.0)
       (* re (* re 0.75))
       (if (<= im 4.05e+46) (* im (- (cos re))) (- t_0 im))))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -2.7e+68) {
		tmp = t_0;
	} else if (im <= -58000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 4.05e+46) {
		tmp = im * -cos(re);
	} else {
		tmp = t_0 - im;
	}
	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)
    if (im <= (-2.7d+68)) then
        tmp = t_0
    else if (im <= (-58000.0d0)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 4.05d+46) then
        tmp = im * -cos(re)
    else
        tmp = t_0 - im
    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;
	double tmp;
	if (im <= -2.7e+68) {
		tmp = t_0;
	} else if (im <= -58000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 4.05e+46) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = t_0 - im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -2.7e+68:
		tmp = t_0
	elif im <= -58000.0:
		tmp = re * (re * 0.75)
	elif im <= 4.05e+46:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0 - im
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -2.7e+68)
		tmp = t_0;
	elseif (im <= -58000.0)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 4.05e+46)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(t_0 - im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -2.7e+68)
		tmp = t_0;
	elseif (im <= -58000.0)
		tmp = re * (re * 0.75);
	elseif (im <= 4.05e+46)
		tmp = im * -cos(re);
	else
		tmp = t_0 - im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -2.7e+68], t$95$0, If[LessEqual[im, -58000.0], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.05e+46], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.69999999999999991e68

   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. sub0-neg 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 85.4%

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

      1. mul-1-neg 85.4%

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

      2. unsub-neg 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-*l* 85.4%

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

      5. distribute-lft-out-- 85.4%

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

   6. Simplified 85.4%

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

   7. Taylor expanded in re around 0 63.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
   8. Step-by-step derivation

      1. fma-neg 63.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {im}^{3}, -im\right)} \]

   9. Simplified 63.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {im}^{3}, -im\right)} \]

   10. Taylor expanded in im around inf 63.2%

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

   if -2.69999999999999991e68 < im < -58000

   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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 54.5%

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

      4. +-commutative 54.5%

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

      5. *-commutative 54.5%

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

      6. unpow2 54.5%

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

      7. associate-*l* 54.5%

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

   6. Simplified 54.5%

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

   8. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 33.1%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 33.1%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 33.1%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 33.1%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 33.1%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   if -58000 < im < 4.05000000000000024e46

   1. Initial program 16.0%

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

      1. sub0-neg 16.0%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in im around 0 91.0%

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

      1. mul-1-neg 91.0%

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

      2. *-commutative 91.0%

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

      3. distribute-lft-neg-in 91.0%

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

   6. Simplified 91.0%

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

   if 4.05000000000000024e46 < 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. sub0-neg 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 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*l* 80.5%

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

      5. distribute-lft-out-- 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in re around 0 56.9%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.7 \cdot 10^{+68}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -58000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 4.05 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 11: 54.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -84000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -4.5e+67)
     t_0
     (if (<= im -84000.0)
       (* re (* re 0.75))
       (if (<= im 1.7e-10) (- im) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -4.5e+67) {
		tmp = t_0;
	} else if (im <= -84000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 1.7e-10) {
		tmp = -im;
	} 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)
    if (im <= (-4.5d+67)) then
        tmp = t_0
    else if (im <= (-84000.0d0)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 1.7d-10) then
        tmp = -im
    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;
	double tmp;
	if (im <= -4.5e+67) {
		tmp = t_0;
	} else if (im <= -84000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 1.7e-10) {
		tmp = -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.pow(im, 3.0) * -0.16666666666666666
	tmp = 0
	if im <= -4.5e+67:
		tmp = t_0
	elif im <= -84000.0:
		tmp = re * (re * 0.75)
	elif im <= 1.7e-10:
		tmp = -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -4.5e+67)
		tmp = t_0;
	elseif (im <= -84000.0)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 1.7e-10)
		tmp = Float64(-im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -4.5e+67)
		tmp = t_0;
	elseif (im <= -84000.0)
		tmp = re * (re * 0.75);
	elseif (im <= 1.7e-10)
		tmp = -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -4.5e+67], t$95$0, If[LessEqual[im, -84000.0], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.7e-10], (-im), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.4999999999999998e67 or 1.70000000000000007e-10 < im

   1. Initial program 99.6%

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

      1. sub0-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in im around 0 75.8%

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. *-commutative 75.8%

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

      4. associate-*l* 75.8%

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

      5. distribute-lft-out-- 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in re around 0 54.2%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
   8. Step-by-step derivation

      1. fma-neg 54.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {im}^{3}, -im\right)} \]

   9. Simplified 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {im}^{3}, -im\right)} \]

   10. Taylor expanded in im around inf 54.2%

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

   if -4.4999999999999998e67 < im < -84000

   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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 54.5%

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

      4. +-commutative 54.5%

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

      5. *-commutative 54.5%

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

      6. unpow2 54.5%

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

      7. associate-*l* 54.5%

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

   6. Simplified 54.5%

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

   8. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 33.1%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 33.1%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 33.1%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 33.1%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 33.1%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   if -84000 < im < 1.70000000000000007e-10

   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. sub0-neg 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 97.7%

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

      1. mul-1-neg 97.7%

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

      2. *-commutative 97.7%

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

      3. distribute-lft-neg-in 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in re around 0 55.0%

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

      1. neg-mul-1 55.0%

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

   9. Simplified 55.0%

      \[\leadsto \color{blue}{-im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+67}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -84000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 12: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -59000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (pow im 3.0) -0.16666666666666666)))
   (if (<= im -4.5e+67)
     t_0
     (if (<= im -59000.0)
       (* re (* re 0.75))
       (if (<= im 4.2e+46) (* im (- (cos re))) t_0)))))

C

double code(double re, double im) {
	double t_0 = pow(im, 3.0) * -0.16666666666666666;
	double tmp;
	if (im <= -4.5e+67) {
		tmp = t_0;
	} else if (im <= -59000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 4.2e+46) {
		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)
    if (im <= (-4.5d+67)) then
        tmp = t_0
    else if (im <= (-59000.0d0)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 4.2d+46) 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;
	double tmp;
	if (im <= -4.5e+67) {
		tmp = t_0;
	} else if (im <= -59000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 4.2e+46) {
		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
	tmp = 0
	if im <= -4.5e+67:
		tmp = t_0
	elif im <= -59000.0:
		tmp = re * (re * 0.75)
	elif im <= 4.2e+46:
		tmp = im * -math.cos(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64((im ^ 3.0) * -0.16666666666666666)
	tmp = 0.0
	if (im <= -4.5e+67)
		tmp = t_0;
	elseif (im <= -59000.0)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 4.2e+46)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im ^ 3.0) * -0.16666666666666666;
	tmp = 0.0;
	if (im <= -4.5e+67)
		tmp = t_0;
	elseif (im <= -59000.0)
		tmp = re * (re * 0.75);
	elseif (im <= 4.2e+46)
		tmp = im * -cos(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]}, If[LessEqual[im, -4.5e+67], t$95$0, If[LessEqual[im, -59000.0], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2e+46], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.4999999999999998e67 or 4.2e46 < 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. sub0-neg 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 83.2%

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

      1. mul-1-neg 83.2%

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

      2. unsub-neg 83.2%

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

      3. *-commutative 83.2%

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

      4. associate-*l* 83.2%

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

      5. distribute-lft-out-- 83.2%

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

   6. Simplified 83.2%

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

   7. Taylor expanded in re around 0 60.3%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
   8. Step-by-step derivation

      1. fma-neg 60.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {im}^{3}, -im\right)} \]

   9. Simplified 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {im}^{3}, -im\right)} \]

   10. Taylor expanded in im around inf 60.3%

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

   if -4.4999999999999998e67 < im < -59000

   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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 54.5%

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

      4. +-commutative 54.5%

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

      5. *-commutative 54.5%

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

      6. unpow2 54.5%

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

      7. associate-*l* 54.5%

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

   6. Simplified 54.5%

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

   8. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 33.1%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 33.1%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 33.1%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 33.1%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 33.1%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   if -59000 < im < 4.2e46

   1. Initial program 16.0%

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

      1. sub0-neg 16.0%

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

   3. Simplified 16.0%

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

   4. Taylor expanded in im around 0 91.0%

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

      1. mul-1-neg 91.0%

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

      2. *-commutative 91.0%

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

      3. distribute-lft-neg-in 91.0%

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

   6. Simplified 91.0%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+67}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \mathbf{elif}\;im \leq -59000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{+46}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 13: 38.6% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{if}\;im \leq -2.9 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -58000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 2200000:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* im (* 0.5 (* re re))) im)))
   (if (<= im -2.9e+55)
     t_0
     (if (<= im -58000.0)
       (* re (* re 0.75))
       (if (<= im 2200000.0) (- im) t_0)))))

C

double code(double re, double im) {
	double t_0 = (im * (0.5 * (re * re))) - im;
	double tmp;
	if (im <= -2.9e+55) {
		tmp = t_0;
	} else if (im <= -58000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 2200000.0) {
		tmp = -im;
	} 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 * (0.5d0 * (re * re))) - im
    if (im <= (-2.9d+55)) then
        tmp = t_0
    else if (im <= (-58000.0d0)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 2200000.0d0) then
        tmp = -im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (im * (0.5 * (re * re))) - im;
	double tmp;
	if (im <= -2.9e+55) {
		tmp = t_0;
	} else if (im <= -58000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 2200000.0) {
		tmp = -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (im * (0.5 * (re * re))) - im
	tmp = 0
	if im <= -2.9e+55:
		tmp = t_0
	elif im <= -58000.0:
		tmp = re * (re * 0.75)
	elif im <= 2200000.0:
		tmp = -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(im * Float64(0.5 * Float64(re * re))) - im)
	tmp = 0.0
	if (im <= -2.9e+55)
		tmp = t_0;
	elseif (im <= -58000.0)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 2200000.0)
		tmp = Float64(-im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (im * (0.5 * (re * re))) - im;
	tmp = 0.0;
	if (im <= -2.9e+55)
		tmp = t_0;
	elseif (im <= -58000.0)
		tmp = re * (re * 0.75);
	elseif (im <= 2200000.0)
		tmp = -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -2.9e+55], t$95$0, If[LessEqual[im, -58000.0], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2200000.0], (-im), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.8999999999999999e55 or 2.2e6 < 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. sub0-neg 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 5.6%

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

      1. mul-1-neg 5.6%

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

      2. *-commutative 5.6%

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

      3. distribute-lft-neg-in 5.6%

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

   6. Simplified 5.6%

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

      1. add-sqr-sqrt 2.9%

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

      2. pow2 2.9%

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

      3. *-commutative 2.9%

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

   8. Applied egg-rr 2.9%

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

   9. Taylor expanded in re around 0 0.0%

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

       1. fma-def 0.0%

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

       2. unpow2 0.0%

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

       3. rem-square-sqrt 0.0%

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

       4. associate-*r* 0.0%

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

       5. unpow2 0.0%

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

       6. rem-square-sqrt 27.7%

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

       7. metadata-eval 27.7%

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

       8. fma-def 27.7%

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

       9. mul-1-neg 27.7%

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

       10. +-commutative 27.7%

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

       11. unsub-neg 27.7%

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

       12. *-commutative 27.7%

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

       13. *-commutative 27.7%

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

       14. associate-*l* 27.7%

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

       15. unpow2 27.7%

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

   11. Simplified 27.7%

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

   if -2.8999999999999999e55 < im < -58000

   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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 55.6%

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

      4. +-commutative 55.6%

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

      5. *-commutative 55.6%

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

      6. unpow2 55.6%

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

      7. associate-*l* 55.6%

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

   6. Simplified 55.6%

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

   8. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 34.6%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 34.6%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 34.6%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 34.6%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 34.6%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   if -58000 < im < 2.2e6

   1. Initial program 11.0%

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

      1. sub0-neg 11.0%

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

   3. Simplified 11.0%

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

   4. Taylor expanded in im around 0 96.3%

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

      1. mul-1-neg 96.3%

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

      2. *-commutative 96.3%

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

      3. distribute-lft-neg-in 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in re around 0 53.8%

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

      1. neg-mul-1 53.8%

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

   9. Simplified 53.8%

      \[\leadsto \color{blue}{-im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.9 \cdot 10^{+55}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq -58000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 2200000:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot \left(re \cdot re\right)\right) - im\\ \end{array} \]

Alternative 14: 35.0% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -80000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 2400000:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -80000.0)
   (* re (* re 0.75))
   (if (<= im 2400000.0) (- im) (* (+ 0.5 (* re (* re -0.25))) -3.0))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -80000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 2400000.0) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-80000.0d0)) then
        tmp = re * (re * 0.75d0)
    else if (im <= 2400000.0d0) then
        tmp = -im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -80000.0) {
		tmp = re * (re * 0.75);
	} else if (im <= 2400000.0) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -80000.0:
		tmp = re * (re * 0.75)
	elif im <= 2400000.0:
		tmp = -im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * -3.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -80000.0)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (im <= 2400000.0)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -80000.0)
		tmp = re * (re * 0.75);
	elseif (im <= 2400000.0)
		tmp = -im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -80000.0], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2400000.0], (-im), N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -8e4

   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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 72.6%

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

      4. +-commutative 72.6%

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

      5. *-commutative 72.6%

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

      6. unpow2 72.6%

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

      7. associate-*l* 72.6%

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

   6. Simplified 72.6%

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

   8. Applied egg-rr 17.4%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 18.0%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 18.0%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 18.0%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 18.0%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 18.0%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   if -8e4 < im < 2.4e6

   1. Initial program 11.0%

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

      1. sub0-neg 11.0%

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

   3. Simplified 11.0%

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

   4. Taylor expanded in im around 0 96.3%

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

      1. mul-1-neg 96.3%

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

      2. *-commutative 96.3%

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

      3. distribute-lft-neg-in 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in re around 0 53.8%

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

      1. neg-mul-1 53.8%

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

   9. Simplified 53.8%

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

   if 2.4e6 < 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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 82.4%

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

      4. +-commutative 82.4%

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

      5. *-commutative 82.4%

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

      6. unpow2 82.4%

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

      7. associate-*l* 82.4%

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

   6. Simplified 82.4%

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

   8. Applied egg-rr 18.0%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -80000:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;im \leq 2400000:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot -3\\ \end{array} \]

Alternative 15: 34.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;re \leq 8.6 \cdot 10^{+154}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e+152)
   (* re (* re 0.75))
   (if (<= re 8.6e+154) (- im) (* (+ 0.5 (* re (* re -0.25))) 27.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+152) {
		tmp = re * (re * 0.75);
	} else if (re <= 8.6e+154) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d+152)) then
        tmp = re * (re * 0.75d0)
    else if (re <= 8.6d+154) then
        tmp = -im
    else
        tmp = (0.5d0 + (re * (re * (-0.25d0)))) * 27.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+152) {
		tmp = re * (re * 0.75);
	} else if (re <= 8.6e+154) {
		tmp = -im;
	} else {
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e+152:
		tmp = re * (re * 0.75)
	elif re <= 8.6e+154:
		tmp = -im
	else:
		tmp = (0.5 + (re * (re * -0.25))) * 27.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e+152)
		tmp = Float64(re * Float64(re * 0.75));
	elseif (re <= 8.6e+154)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(0.5 + Float64(re * Float64(re * -0.25))) * 27.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e+152)
		tmp = re * (re * 0.75);
	elseif (re <= 8.6e+154)
		tmp = -im;
	else
		tmp = (0.5 + (re * (re * -0.25))) * 27.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e+152], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.6e+154], (-im), N[(N[(0.5 + N[(re * N[(re * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 27.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.5000000000000001e152

   1. Initial program 52.9%

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

      1. sub0-neg 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 24.2%

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

      4. +-commutative 24.2%

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

      5. *-commutative 24.2%

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

      6. unpow2 24.2%

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

      7. associate-*l* 24.2%

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

   6. Simplified 24.2%

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

   8. Applied egg-rr 35.4%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 35.4%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 35.4%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 35.4%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 35.4%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 35.4%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   if -4.5000000000000001e152 < re < 8.5999999999999995e154

   1. Initial program 52.7%

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

      1. sub0-neg 52.7%

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

   3. Simplified 52.7%

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

   4. Taylor expanded in im around 0 53.6%

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

      1. mul-1-neg 53.6%

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

      2. *-commutative 53.6%

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

      3. distribute-lft-neg-in 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in re around 0 37.4%

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

      1. neg-mul-1 37.4%

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

   9. Simplified 37.4%

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

   if 8.5999999999999995e154 < re

   1. Initial program 62.3%

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

      1. sub0-neg 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in re around 0 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*r* 0.1%

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

      3. distribute-rgt-out 30.6%

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

      4. +-commutative 30.6%

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

      5. *-commutative 30.6%

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

      6. unpow2 30.6%

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

      7. associate-*l* 30.6%

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

   6. Simplified 30.6%

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

   8. Applied egg-rr 34.1%

      \[\leadsto \color{blue}{27} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{elif}\;re \leq 8.6 \cdot 10^{+154}:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \cdot 27\\ \end{array} \]

Alternative 16: 34.7% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -58000 \lor \neg \left(im \leq 5400000\right):\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -58000.0) (not (<= im 5400000.0))) (* re (* re 0.75)) (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -58000.0) || !(im <= 5400000.0)) {
		tmp = re * (re * 0.75);
	} else {
		tmp = -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 <= (-58000.0d0)) .or. (.not. (im <= 5400000.0d0))) then
        tmp = re * (re * 0.75d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -58000.0) || !(im <= 5400000.0)) {
		tmp = re * (re * 0.75);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -58000.0) or not (im <= 5400000.0):
		tmp = re * (re * 0.75)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -58000.0) || !(im <= 5400000.0))
		tmp = Float64(re * Float64(re * 0.75));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -58000.0) || ~((im <= 5400000.0)))
		tmp = re * (re * 0.75);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -58000.0], N[Not[LessEqual[im, 5400000.0]], $MachinePrecision]], N[(re * N[(re * 0.75), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if im < -58000 or 5.4e6 < 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. sub0-neg 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 re around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. distribute-rgt-out 76.6%

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

      4. +-commutative 76.6%

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

      5. *-commutative 76.6%

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

      6. unpow2 76.6%

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

      7. associate-*l* 76.6%

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

   6. Simplified 76.6%

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

   8. Applied egg-rr 17.7%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 17.6%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 17.6%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 17.6%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

       3. associate-*l* 17.6%

          \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   11. Simplified 17.6%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot 0.75\right)} \]

   if -58000 < im < 5.4e6

   1. Initial program 11.0%

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

      1. sub0-neg 11.0%

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

   3. Simplified 11.0%

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

   4. Taylor expanded in im around 0 96.3%

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

      1. mul-1-neg 96.3%

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

      2. *-commutative 96.3%

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

      3. distribute-lft-neg-in 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in re around 0 53.8%

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

      1. neg-mul-1 53.8%

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

   9. Simplified 53.8%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -58000 \lor \neg \left(im \leq 5400000\right):\\ \;\;\;\;re \cdot \left(re \cdot 0.75\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \]

Alternative 17: 29.8% 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 54.1%

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

   1. sub0-neg 54.1%

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

3. Simplified 54.1%

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

4. Taylor expanded in im around 0 52.2%

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

   1. mul-1-neg 52.2%

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

   2. *-commutative 52.2%

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

   3. distribute-lft-neg-in 52.2%

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

6. Simplified 52.2%

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

7. Taylor expanded in re around 0 29.6%

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

   1. neg-mul-1 29.6%

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

9. Simplified 29.6%

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

10. Final simplification 29.6%

    \[\leadsto -im \]

Alternative 18: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.5)

C

double code(double re, double im) {
	return -1.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.5d0
end function

Java

public static double code(double re, double im) {
	return -1.5;
}

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

function tmp = code(re, im)
	tmp = -1.5;
end

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 54.1%

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

   1. sub0-neg 54.1%

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

3. Simplified 54.1%

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

4. Taylor expanded in re around 0 3.0%

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

   1. *-commutative 3.0%

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

   2. associate-*r* 3.0%

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

   3. distribute-rgt-out 41.3%

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

   4. +-commutative 41.3%

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

   5. *-commutative 41.3%

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

   6. unpow2 41.3%

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

   7. associate-*l* 41.3%

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

6. Simplified 41.3%

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

8. Applied egg-rr 10.0%

   \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

9. Taylor expanded in re around 0 2.7%

   \[\leadsto \color{blue}{-1.5} \]

10. Final simplification 2.7%

    \[\leadsto -1.5 \]

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 2023171 
(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))))