math.cos on complex, imaginary part

Percentage Accurate: 65.6% → 99.6%

Time: 8.0s

Alternatives: 11

Speedup: 2.8×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+66} \lor \neg \left(t_0 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -5e+66) (not (<= t_0 2e-5)))
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (- (* (pow im 5.0) -0.008333333333333333) im)
       (+
        (* (pow im 7.0) -0.0001984126984126984)
        (* (pow im 3.0) -0.16666666666666666)))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * (((pow(im, 5.0) * -0.008333333333333333) - im) + ((pow(im, 7.0) * -0.0001984126984126984) + (pow(im, 3.0) * -0.16666666666666666)));
	}
	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 = exp(-im) - exp(im)
    if ((t_0 <= (-5d+66)) .or. (.not. (t_0 <= 2d-5))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * ((((im ** 5.0d0) * (-0.008333333333333333d0)) - im) + (((im ** 7.0d0) * (-0.0001984126984126984d0)) + ((im ** 3.0d0) * (-0.16666666666666666d0))))
    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 tmp;
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * (((Math.pow(im, 5.0) * -0.008333333333333333) - im) + ((Math.pow(im, 7.0) * -0.0001984126984126984) + (Math.pow(im, 3.0) * -0.16666666666666666)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -5e+66) or not (t_0 <= 2e-5):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * (((math.pow(im, 5.0) * -0.008333333333333333) - im) + ((math.pow(im, 7.0) * -0.0001984126984126984) + (math.pow(im, 3.0) * -0.16666666666666666)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -5e+66) || !(t_0 <= 2e-5))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im) + Float64(Float64((im ^ 7.0) * -0.0001984126984126984) + Float64((im ^ 3.0) * -0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -5e+66) || ~((t_0 <= 2e-5)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((((im ^ 5.0) * -0.008333333333333333) - im) + (((im ^ 7.0) * -0.0001984126984126984) + ((im ^ 3.0) * -0.16666666666666666)));
	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, -5e+66], N[Not[LessEqual[t$95$0, 2e-5]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision] + N[(N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -4.99999999999999991e66 or 2.00000000000000016e-5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   if -4.99999999999999991e66 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.00000000000000016e-5

   1. Initial program 31.0%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. associate-+r+ 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

      4. mul-1-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-lft-neg-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. *-commutative 99.8%

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

      11. associate-*r* 99.8%

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

      12. *-commutative 99.8%

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

      13. associate-*r* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -5 \cdot 10^{+66} \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left({im}^{7} \cdot -0.0001984126984126984 + {im}^{3} \cdot -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.05 \lor \neg \left(t_0 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.05) (not (<= t_0 2e-5)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin 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 <= -0.05) || !(t_0 <= 2e-5)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.05d0)) .or. (.not. (t_0 <= 2d-5))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    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 tmp;
	if ((t_0 <= -0.05) || !(t_0 <= 2e-5)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(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 <= -0.05) or not (t_0 <= 2e-5):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(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 <= -0.05) || !(t_0 <= 2e-5))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(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 <= -0.05) || ~((t_0 <= 2e-5)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(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, -0.05], N[Not[LessEqual[t$95$0, 2e-5]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[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 (neg.f64 im)) (exp.f64 im)) < -0.050000000000000003 or 2.00000000000000016e-5 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 100.0%

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

   if -0.050000000000000003 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.00000000000000016e-5

   1. Initial program 30.4%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05 \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0013:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.0015:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -1.05e+44)
     t_1
     (if (<= im -0.0013)
       t_0
       (if (<= im 0.0015) (* (- im) (sin 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)) * re);
	double t_1 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.05e+44) {
		tmp = t_1;
	} else if (im <= -0.0013) {
		tmp = t_0;
	} else if (im <= 0.0015) {
		tmp = -im * sin(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)) * re)
    t_1 = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    if (im <= (-1.05d+44)) then
        tmp = t_1
    else if (im <= (-0.0013d0)) then
        tmp = t_0
    else if (im <= 0.0015d0) then
        tmp = -im * sin(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)) * re);
	double t_1 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.05e+44) {
		tmp = t_1;
	} else if (im <= -0.0013) {
		tmp = t_0;
	} else if (im <= 0.0015) {
		tmp = -im * Math.sin(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)) * re)
	t_1 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.05e+44:
		tmp = t_1
	elif im <= -0.0013:
		tmp = t_0
	elif im <= 0.0015:
		tmp = -im * math.sin(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(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.05e+44)
		tmp = t_1;
	elseif (im <= -0.0013)
		tmp = t_0;
	elseif (im <= 0.0015)
		tmp = Float64(Float64(-im) * sin(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)) * re);
	t_1 = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.05e+44)
		tmp = t_1;
	elseif (im <= -0.0013)
		tmp = t_0;
	elseif (im <= 0.0015)
		tmp = -im * sin(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[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.05e+44], t$95$1, If[LessEqual[im, -0.0013], t$95$0, If[LessEqual[im, 0.0015], N[((-im) * N[Sin[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 < -1.04999999999999993e44 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*r* 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.04999999999999993e44 < im < -0.0012999999999999999 or 0.0015 < im < 1.09999999999999998e44

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 73.5%

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

   if -0.0012999999999999999 < im < 0.0015

   1. Initial program 30.4%

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

   2. Taylor expanded in im around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.0013:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 0.0015:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.095:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.023:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -1.05e+44)
     t_1
     (if (<= im -0.095)
       t_0
       (if (<= im 0.023)
         (* (sin 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)) * re);
	double t_1 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.05e+44) {
		tmp = t_1;
	} else if (im <= -0.095) {
		tmp = t_0;
	} else if (im <= 0.023) {
		tmp = sin(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)) * re)
    t_1 = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    if (im <= (-1.05d+44)) then
        tmp = t_1
    else if (im <= (-0.095d0)) then
        tmp = t_0
    else if (im <= 0.023d0) then
        tmp = sin(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)) * re);
	double t_1 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.05e+44) {
		tmp = t_1;
	} else if (im <= -0.095) {
		tmp = t_0;
	} else if (im <= 0.023) {
		tmp = Math.sin(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)) * re)
	t_1 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.05e+44:
		tmp = t_1
	elif im <= -0.095:
		tmp = t_0
	elif im <= 0.023:
		tmp = math.sin(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(Float64(exp(Float64(-im)) - exp(im)) * re))
	t_1 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.05e+44)
		tmp = t_1;
	elseif (im <= -0.095)
		tmp = t_0;
	elseif (im <= 0.023)
		tmp = Float64(sin(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)) * re);
	t_1 = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.05e+44)
		tmp = t_1;
	elseif (im <= -0.095)
		tmp = t_0;
	elseif (im <= 0.023)
		tmp = sin(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[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.05e+44], t$95$1, If[LessEqual[im, -0.095], t$95$0, If[LessEqual[im, 0.023], N[(N[Sin[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 < -1.04999999999999993e44 or 1.09999999999999998e44 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-lft-neg-in 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*r* 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in im around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -1.04999999999999993e44 < im < -0.095000000000000001 or 0.023 < im < 1.09999999999999998e44

   1. Initial program 99.7%

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

   2. Taylor expanded in re around 0 73.5%

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

   if -0.095000000000000001 < im < 0.023

   1. Initial program 30.4%

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

   2. Taylor expanded in im around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-out-- 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.095:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 0.023:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 5: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.2 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.2) (not (<= im 4.2)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.2) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} else {
		tmp = -im * sin(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 <= (-4.2d0)) .or. (.not. (im <= 4.2d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.2) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.2) or not (im <= 4.2):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.2) || !(im <= 4.2))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.2) || ~((im <= 4.2)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.2], N[Not[LessEqual[im, 4.2]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.20000000000000018 or 4.20000000000000018 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 87.2%

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

      1. associate-+r+ 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. mul-1-neg 87.2%

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

      5. *-commutative 87.2%

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

      6. distribute-lft-neg-in 87.2%

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

      7. *-commutative 87.2%

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

      8. associate-*r* 87.2%

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

      9. distribute-rgt-out 87.2%

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

      10. *-commutative 87.2%

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

      11. associate-*r* 87.2%

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

      12. *-commutative 87.2%

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

      13. associate-*r* 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in im around inf 87.2%

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

      1. *-commutative 87.2%

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

   7. Simplified 87.2%

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

   if -4.20000000000000018 < im < 4.20000000000000018

   1. Initial program 31.0%

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

   2. Taylor expanded in im around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. *-commutative 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.2 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 6: 81.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ t_1 := {im}^{7} \cdot \left(re \cdot \left(-0.0001984126984126984 + 3.306878306878307 \cdot 10^{-5} \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{if}\;im \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -5.3 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+17}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* re (pow im 7.0))))
        (t_1
         (*
          (pow im 7.0)
          (*
           re
           (+ -0.0001984126984126984 (* 3.306878306878307e-5 (* re re)))))))
   (if (<= im -2.3e+95)
     t_1
     (if (<= im -5e+50)
       t_0
       (if (<= im -5.3e+14)
         t_1
         (if (<= im 4.8e+17) (* (- im) (sin re)) t_0))))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (re * pow(im, 7.0));
	double t_1 = pow(im, 7.0) * (re * (-0.0001984126984126984 + (3.306878306878307e-5 * (re * re))));
	double tmp;
	if (im <= -2.3e+95) {
		tmp = t_1;
	} else if (im <= -5e+50) {
		tmp = t_0;
	} else if (im <= -5.3e+14) {
		tmp = t_1;
	} else if (im <= 4.8e+17) {
		tmp = -im * sin(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) :: t_1
    real(8) :: tmp
    t_0 = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    t_1 = (im ** 7.0d0) * (re * ((-0.0001984126984126984d0) + (3.306878306878307d-5 * (re * re))))
    if (im <= (-2.3d+95)) then
        tmp = t_1
    else if (im <= (-5d+50)) then
        tmp = t_0
    else if (im <= (-5.3d+14)) then
        tmp = t_1
    else if (im <= 4.8d+17) then
        tmp = -im * sin(re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	double t_1 = Math.pow(im, 7.0) * (re * (-0.0001984126984126984 + (3.306878306878307e-5 * (re * re))));
	double tmp;
	if (im <= -2.3e+95) {
		tmp = t_1;
	} else if (im <= -5e+50) {
		tmp = t_0;
	} else if (im <= -5.3e+14) {
		tmp = t_1;
	} else if (im <= 4.8e+17) {
		tmp = -im * Math.sin(re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	t_1 = math.pow(im, 7.0) * (re * (-0.0001984126984126984 + (3.306878306878307e-5 * (re * re))))
	tmp = 0
	if im <= -2.3e+95:
		tmp = t_1
	elif im <= -5e+50:
		tmp = t_0
	elif im <= -5.3e+14:
		tmp = t_1
	elif im <= 4.8e+17:
		tmp = -im * math.sin(re)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)))
	t_1 = Float64((im ^ 7.0) * Float64(re * Float64(-0.0001984126984126984 + Float64(3.306878306878307e-5 * Float64(re * re)))))
	tmp = 0.0
	if (im <= -2.3e+95)
		tmp = t_1;
	elseif (im <= -5e+50)
		tmp = t_0;
	elseif (im <= -5.3e+14)
		tmp = t_1;
	elseif (im <= 4.8e+17)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.0001984126984126984 * (re * (im ^ 7.0));
	t_1 = (im ^ 7.0) * (re * (-0.0001984126984126984 + (3.306878306878307e-5 * (re * re))));
	tmp = 0.0;
	if (im <= -2.3e+95)
		tmp = t_1;
	elseif (im <= -5e+50)
		tmp = t_0;
	elseif (im <= -5.3e+14)
		tmp = t_1;
	elseif (im <= 4.8e+17)
		tmp = -im * sin(re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 7.0], $MachinePrecision] * N[(re * N[(-0.0001984126984126984 + N[(3.306878306878307e-5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -2.3e+95], t$95$1, If[LessEqual[im, -5e+50], t$95$0, If[LessEqual[im, -5.3e+14], t$95$1, If[LessEqual[im, 4.8e+17], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -2.29999999999999997e95 or -5e50 < im < -5.3e14

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.1%

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

      1. associate-+r+ 91.1%

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

      2. +-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. mul-1-neg 91.1%

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

      5. *-commutative 91.1%

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

      6. distribute-lft-neg-in 91.1%

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

      7. *-commutative 91.1%

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

      8. associate-*r* 91.1%

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

      9. distribute-rgt-out 91.1%

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

      10. *-commutative 91.1%

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

      11. associate-*r* 91.1%

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

      12. *-commutative 91.1%

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

      13. associate-*r* 91.1%

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

   4. Simplified 91.1%

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

   5. Taylor expanded in im around inf 91.1%

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

      1. *-commutative 91.1%

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

   7. Simplified 91.1%

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

   8. Taylor expanded in re around 0 2.0%

      \[\leadsto \color{blue}{-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right) + 3.306878306878307 \cdot 10^{-5} \cdot \left({re}^{3} \cdot {im}^{7}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 2.0%

         \[\leadsto \color{blue}{\left(-0.0001984126984126984 \cdot re\right) \cdot {im}^{7}} + 3.306878306878307 \cdot 10^{-5} \cdot \left({re}^{3} \cdot {im}^{7}\right) \]

      2. *-commutative 2.0%

         \[\leadsto \color{blue}{\left(re \cdot -0.0001984126984126984\right)} \cdot {im}^{7} + 3.306878306878307 \cdot 10^{-5} \cdot \left({re}^{3} \cdot {im}^{7}\right) \]

      3. associate-*r* 2.0%

         \[\leadsto \left(re \cdot -0.0001984126984126984\right) \cdot {im}^{7} + \color{blue}{\left(3.306878306878307 \cdot 10^{-5} \cdot {re}^{3}\right) \cdot {im}^{7}} \]

      4. distribute-rgt-out 87.2%

         \[\leadsto \color{blue}{{im}^{7} \cdot \left(re \cdot -0.0001984126984126984 + 3.306878306878307 \cdot 10^{-5} \cdot {re}^{3}\right)} \]

      5. *-commutative 87.2%

         \[\leadsto {im}^{7} \cdot \left(\color{blue}{-0.0001984126984126984 \cdot re} + 3.306878306878307 \cdot 10^{-5} \cdot {re}^{3}\right) \]

      6. unpow3 87.2%

         \[\leadsto {im}^{7} \cdot \left(-0.0001984126984126984 \cdot re + 3.306878306878307 \cdot 10^{-5} \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot re\right)}\right) \]

      7. unpow2 87.2%

         \[\leadsto {im}^{7} \cdot \left(-0.0001984126984126984 \cdot re + 3.306878306878307 \cdot 10^{-5} \cdot \left(\color{blue}{{re}^{2}} \cdot re\right)\right) \]

      8. associate-*r* 87.2%

         \[\leadsto {im}^{7} \cdot \left(-0.0001984126984126984 \cdot re + \color{blue}{\left(3.306878306878307 \cdot 10^{-5} \cdot {re}^{2}\right) \cdot re}\right) \]

      9. distribute-rgt-out 87.2%

         \[\leadsto {im}^{7} \cdot \color{blue}{\left(re \cdot \left(-0.0001984126984126984 + 3.306878306878307 \cdot 10^{-5} \cdot {re}^{2}\right)\right)} \]

      10. unpow2 87.2%

          \[\leadsto {im}^{7} \cdot \left(re \cdot \left(-0.0001984126984126984 + 3.306878306878307 \cdot 10^{-5} \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \]

   10. Simplified 87.2%

       \[\leadsto \color{blue}{{im}^{7} \cdot \left(re \cdot \left(-0.0001984126984126984 + 3.306878306878307 \cdot 10^{-5} \cdot \left(re \cdot re\right)\right)\right)} \]

   if -2.29999999999999997e95 < im < -5e50 or 4.8e17 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 91.0%

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

      1. associate-+r+ 91.0%

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

      2. +-commutative 91.0%

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

      3. +-commutative 91.0%

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

      4. mul-1-neg 91.0%

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

      5. *-commutative 91.0%

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

      6. distribute-lft-neg-in 91.0%

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

      7. *-commutative 91.0%

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

      8. associate-*r* 91.0%

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

      9. distribute-rgt-out 91.0%

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

      10. *-commutative 91.0%

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

      11. associate-*r* 91.0%

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

      12. *-commutative 91.0%

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

      13. associate-*r* 91.0%

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

   4. Simplified 91.0%

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

   5. Taylor expanded in re around 0 78.8%

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

   6. Taylor expanded in im around inf 78.8%

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

   if -5.3e14 < im < 4.8e17

   1. Initial program 34.2%

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

   2. Taylor expanded in im around 0 94.3%

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

      1. mul-1-neg 94.3%

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

      2. *-commutative 94.3%

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

      3. distribute-rgt-neg-in 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;{im}^{7} \cdot \left(re \cdot \left(-0.0001984126984126984 + 3.306878306878307 \cdot 10^{-5} \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq -5 \cdot 10^{+50}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -5.3 \cdot 10^{+14}:\\ \;\;\;\;{im}^{7} \cdot \left(re \cdot \left(-0.0001984126984126984 + 3.306878306878307 \cdot 10^{-5} \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{elif}\;im \leq 4.8 \cdot 10^{+17}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 7: 82.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -30000 \lor \neg \left(im \leq 4.5 \cdot 10^{+17}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -30000.0) (not (<= im 4.5e+17)))
   (* -0.0001984126984126984 (* re (pow im 7.0)))
   (* (- im) (sin re))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -30000.0) || !(im <= 4.5e+17)) {
		tmp = -0.0001984126984126984 * (re * pow(im, 7.0));
	} else {
		tmp = -im * sin(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 <= (-30000.0d0)) .or. (.not. (im <= 4.5d+17))) then
        tmp = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    else
        tmp = -im * sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -30000.0) || !(im <= 4.5e+17)) {
		tmp = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	} else {
		tmp = -im * Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -30000.0) or not (im <= 4.5e+17):
		tmp = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	else:
		tmp = -im * math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -30000.0) || !(im <= 4.5e+17))
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)));
	else
		tmp = Float64(Float64(-im) * sin(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -30000.0) || ~((im <= 4.5e+17)))
		tmp = -0.0001984126984126984 * (re * (im ^ 7.0));
	else
		tmp = -im * sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -30000.0], N[Not[LessEqual[im, 4.5e+17]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3e4 or 4.5e17 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0 89.1%

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

      1. associate-+r+ 89.1%

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

      2. +-commutative 89.1%

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

      3. +-commutative 89.1%

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

      4. mul-1-neg 89.1%

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

      5. *-commutative 89.1%

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

      6. distribute-lft-neg-in 89.1%

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

      7. *-commutative 89.1%

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

      8. associate-*r* 89.1%

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

      9. distribute-rgt-out 89.1%

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

      10. *-commutative 89.1%

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

      11. associate-*r* 89.1%

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

      12. *-commutative 89.1%

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

      13. associate-*r* 89.1%

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

   4. Simplified 89.1%

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

   5. Taylor expanded in re around 0 73.0%

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

   6. Taylor expanded in im around inf 73.0%

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

   if -3e4 < im < 4.5e17

   1. Initial program 32.6%

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

   2. Taylor expanded in im around 0 96.5%

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

      1. mul-1-neg 96.5%

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

      2. *-commutative 96.5%

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

      3. distribute-rgt-neg-in 96.5%

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

   4. Simplified 96.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -30000 \lor \neg \left(im \leq 4.5 \cdot 10^{+17}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \end{array} \]

Alternative 8: 56.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.3 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{\left(re \cdot re\right) \cdot 182.25}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.3e+14)
   (sqrt (* (* re re) 182.25))
   (if (<= im 2.5e+32) (* (- im) (sin re)) (* im (- re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.3e+14) {
		tmp = sqrt(((re * re) * 182.25));
	} else if (im <= 2.5e+32) {
		tmp = -im * sin(re);
	} else {
		tmp = im * -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 <= (-5.3d+14)) then
        tmp = sqrt(((re * re) * 182.25d0))
    else if (im <= 2.5d+32) then
        tmp = -im * sin(re)
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.3e+14) {
		tmp = Math.sqrt(((re * re) * 182.25));
	} else if (im <= 2.5e+32) {
		tmp = -im * Math.sin(re);
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.3e+14:
		tmp = math.sqrt(((re * re) * 182.25))
	elif im <= 2.5e+32:
		tmp = -im * math.sin(re)
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.3e+14)
		tmp = sqrt(Float64(Float64(re * re) * 182.25));
	elseif (im <= 2.5e+32)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.3e+14)
		tmp = sqrt(((re * re) * 182.25));
	elseif (im <= 2.5e+32)
		tmp = -im * sin(re);
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.3e+14], N[Sqrt[N[(N[(re * re), $MachinePrecision] * 182.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 2.5e+32], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(im * (-re)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.3e14

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 79.4%

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

   4. Applied egg-rr 2.2%

      \[\leadsto 0.5 \cdot \left(\color{blue}{27} \cdot re\right) \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 1.1%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \left(27 \cdot re\right)} \cdot \sqrt{0.5 \cdot \left(27 \cdot re\right)}} \]

      2. sqrt-unprod 11.7%

         \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \left(27 \cdot re\right)\right) \cdot \left(0.5 \cdot \left(27 \cdot re\right)\right)}} \]

      3. associate-*r* 11.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left(0.5 \cdot 27\right) \cdot re\right)} \cdot \left(0.5 \cdot \left(27 \cdot re\right)\right)} \]

      4. associate-*r* 11.7%

         \[\leadsto \sqrt{\left(\left(0.5 \cdot 27\right) \cdot re\right) \cdot \color{blue}{\left(\left(0.5 \cdot 27\right) \cdot re\right)}} \]

      5. swap-sqr 11.7%

         \[\leadsto \sqrt{\color{blue}{\left(\left(0.5 \cdot 27\right) \cdot \left(0.5 \cdot 27\right)\right) \cdot \left(re \cdot re\right)}} \]

      6. metadata-eval 11.7%

         \[\leadsto \sqrt{\left(\color{blue}{13.5} \cdot \left(0.5 \cdot 27\right)\right) \cdot \left(re \cdot re\right)} \]

      7. metadata-eval 11.7%

         \[\leadsto \sqrt{\left(13.5 \cdot \color{blue}{13.5}\right) \cdot \left(re \cdot re\right)} \]

      8. metadata-eval 11.7%

         \[\leadsto \sqrt{\color{blue}{182.25} \cdot \left(re \cdot re\right)} \]

   6. Applied egg-rr 11.7%

      \[\leadsto \color{blue}{\sqrt{182.25 \cdot \left(re \cdot re\right)}} \]

   if -5.3e14 < im < 2.4999999999999999e32

   1. Initial program 36.2%

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

   2. Taylor expanded in im around 0 91.5%

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

      1. mul-1-neg 91.5%

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

      2. *-commutative 91.5%

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

      3. distribute-rgt-neg-in 91.5%

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

   4. Simplified 91.5%

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

   if 2.4999999999999999e32 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 82.1%

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

   3. Taylor expanded in im around 0 23.9%

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

      1. mul-1-neg 23.9%

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

      2. distribute-rgt-neg-in 23.9%

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

   5. Simplified 23.9%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.3 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{\left(re \cdot re\right) \cdot 182.25}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+32}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 9: 53.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.4 \cdot 10^{+31}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.4e+31) (* (- im) (sin re)) (* im (- re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.4e+31) {
		tmp = -im * sin(re);
	} else {
		tmp = im * -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 <= 2.4d+31) then
        tmp = -im * sin(re)
    else
        tmp = im * -re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.4e+31) {
		tmp = -im * Math.sin(re);
	} else {
		tmp = im * -re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.4e+31:
		tmp = -im * math.sin(re)
	else:
		tmp = im * -re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.4e+31)
		tmp = Float64(Float64(-im) * sin(re));
	else
		tmp = Float64(im * Float64(-re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.4e+31)
		tmp = -im * sin(re);
	else
		tmp = im * -re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.4e+31], N[((-im) * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(im * (-re)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.39999999999999982e31

   1. Initial program 57.9%

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

   2. Taylor expanded in im around 0 61.9%

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

      1. mul-1-neg 61.9%

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

      2. *-commutative 61.9%

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

      3. distribute-rgt-neg-in 61.9%

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

   4. Simplified 61.9%

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

   if 2.39999999999999982e31 < im

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0 82.1%

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

   3. Taylor expanded in im around 0 23.9%

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

      1. mul-1-neg 23.9%

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

      2. distribute-rgt-neg-in 23.9%

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

   5. Simplified 23.9%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.4 \cdot 10^{+31}:\\ \;\;\;\;\left(-im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-re\right)\\ \end{array} \]

Alternative 10: 32.8% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im * -re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.1%

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

2. Taylor expanded in re around 0 56.1%

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

3. Taylor expanded in im around 0 30.3%

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

   1. mul-1-neg 30.3%

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

   2. distribute-rgt-neg-in 30.3%

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

5. Simplified 30.3%

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

6. Final simplification 30.3%

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

Alternative 11: 14.5% accurate, 308.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 67.1%

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

2. Taylor expanded in re around 0 56.1%

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

   1. expm1-log1p-u 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\right)\right)} \]

   2. expm1-udef 33.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\right)} - 1} \]

   3. add-sqr-sqrt 16.0%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} - e^{im}\right) \cdot re\right)\right)} - 1 \]

   4. sqrt-unprod 24.1%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} - e^{im}\right) \cdot re\right)\right)} - 1 \]

   5. sqr-neg 24.1%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{\sqrt{\color{blue}{im \cdot im}}} - e^{im}\right) \cdot re\right)\right)} - 1 \]

   6. sqrt-unprod 8.2%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} - e^{im}\right) \cdot re\right)\right)} - 1 \]

   7. add-sqr-sqrt 14.0%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{\color{blue}{im}} - e^{im}\right) \cdot re\right)\right)} - 1 \]

4. Applied egg-rr 14.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{im} - e^{im}\right) \cdot re\right)\right)} - 1} \]
5. Step-by-step derivation

   1. expm1-def 14.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \left(\left(e^{im} - e^{im}\right) \cdot re\right)\right)\right)} \]

   2. expm1-log1p 14.0%

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

   3. +-inverses 14.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{0} \cdot re\right) \]

   4. mul0-lft 14.4%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

   5. metadata-eval 14.4%

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

6. Simplified 14.4%

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

7. Final simplification 14.4%

   \[\leadsto 0 \]

Developer target: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-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 = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-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.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(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 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[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[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023240 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

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

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))