math.cos on complex, imaginary part

Percentage Accurate: 65.9% → 99.5%

Time: 10.7s

Alternatives: 12

Speedup: 2.8×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 65.9% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-6)))
     (* (* 0.5 (sin re)) t_0)
     (* (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 <= -((double) INFINITY)) || !(t_0 <= 2e-6)) {
		tmp = (0.5 * sin(re)) * t_0;
	} else {
		tmp = sin(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 <= 2e-6)) {
		tmp = (0.5 * Math.sin(re)) * t_0;
	} 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 <= -math.inf) or not (t_0 <= 2e-6):
		tmp = (0.5 * math.sin(re)) * t_0
	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 <= Float64(-Inf)) || !(t_0 <= 2e-6))
		tmp = Float64(Float64(0.5 * sin(re)) * t_0);
	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 <= -Inf) || ~((t_0 <= 2e-6)))
		tmp = (0.5 * sin(re)) * t_0;
	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, (-Infinity)], N[Not[LessEqual[t$95$0, 2e-6]], $MachinePrecision]], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $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)) < -inf.0 or 1.99999999999999991e-6 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.9%

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

   if -inf.0 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 1.99999999999999991e-6

   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%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 2: 96.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     t_0
     (*
      (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 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0;
	} 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;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} 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 = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_0
	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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_0;
	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 = (0.5 * sin(re)) * (exp(-im) - exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0;
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$0, 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 (*.f64 1/2 (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

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

   if -inf.0 < (*.f64 (*.f64 1/2 (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 53.9%

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

   2. Taylor expanded in im around 0 98.3%

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

      1. associate-+r+ 98.3%

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

      2. +-commutative 98.3%

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

      3. +-commutative 98.3%

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

      4. mul-1-neg 98.3%

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

      5. *-commutative 98.3%

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

      6. distribute-lft-neg-in 98.3%

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

      7. *-commutative 98.3%

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

      8. associate-*r* 98.3%

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

      9. distribute-rgt-out 98.3%

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

      10. *-commutative 98.3%

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

      11. associate-*r* 98.3%

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

      12. *-commutative 98.3%

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

      13. associate-*r* 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 98.8%

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

Alternative 3: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq 0.01:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (<= t_0 0.01)
     (*
      (sin re)
      (+
       (* (pow im 5.0) -0.008333333333333333)
       (- (* (pow im 3.0) -0.16666666666666666) im)))
     (* (* 0.5 (sin re)) t_0))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = sin(re) * ((pow(im, 5.0) * -0.008333333333333333) + ((pow(im, 3.0) * -0.16666666666666666) - im));
	} else {
		tmp = (0.5 * sin(re)) * 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 = exp(-im) - exp(im)
    if (t_0 <= 0.01d0) then
        tmp = sin(re) * (((im ** 5.0d0) * (-0.008333333333333333d0)) + (((im ** 3.0d0) * (-0.16666666666666666d0)) - im))
    else
        tmp = (0.5d0 * sin(re)) * t_0
    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 <= 0.01) {
		tmp = Math.sin(re) * ((Math.pow(im, 5.0) * -0.008333333333333333) + ((Math.pow(im, 3.0) * -0.16666666666666666) - im));
	} else {
		tmp = (0.5 * Math.sin(re)) * t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if t_0 <= 0.01:
		tmp = math.sin(re) * ((math.pow(im, 5.0) * -0.008333333333333333) + ((math.pow(im, 3.0) * -0.16666666666666666) - im))
	else:
		tmp = (0.5 * math.sin(re)) * t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 5.0) * -0.008333333333333333) + Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = sin(re) * (((im ^ 5.0) * -0.008333333333333333) + (((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = (0.5 * sin(re)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 0.0100000000000000002

   1. Initial program 53.4%

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

   2. Taylor expanded in im around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. associate-+r+ 97.8%

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

      3. +-commutative 97.8%

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

      4. mul-1-neg 97.8%

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

      5. *-commutative 97.8%

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

      6. distribute-lft-neg-in 97.8%

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

      7. *-commutative 97.8%

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

      8. associate-*r* 97.8%

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

      9. distribute-rgt-out 97.8%

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

      10. associate-*r* 97.8%

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

      11. *-commutative 97.8%

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

      12. associate-*l* 97.8%

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

      13. distribute-lft-out 97.8%

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

   4. Simplified 97.8%

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

   if 0.0100000000000000002 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))

   1. Initial program 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq 0.01:\\ \;\;\;\;\sin re \cdot \left({im}^{5} \cdot -0.008333333333333333 + \left({im}^{3} \cdot -0.16666666666666666 - im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Alternative 4: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+36} \lor \neg \left(im \leq 29000000000\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \left(-im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.3e+36) (not (<= im 29000000000.0)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (log1p (expm1 (* (sin re) (- im))))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.3e+36) || !(im <= 29000000000.0)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} else {
		tmp = log1p(expm1((sin(re) * -im)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.3e+36) || !(im <= 29000000000.0)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = Math.log1p(Math.expm1((Math.sin(re) * -im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.3e+36) or not (im <= 29000000000.0):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = math.log1p(math.expm1((math.sin(re) * -im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.3e+36) || !(im <= 29000000000.0))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = log1p(expm1(Float64(sin(re) * Float64(-im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.3e+36], N[Not[LessEqual[im, 29000000000.0]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.3000000000000001e36 or 2.9e10 < 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 97.6%

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

      1. associate-+r+ 97.6%

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

      2. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

      4. mul-1-neg 97.6%

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

      5. *-commutative 97.6%

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

      6. distribute-lft-neg-in 97.6%

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

      7. *-commutative 97.6%

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

      8. associate-*r* 97.6%

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

      9. distribute-rgt-out 97.6%

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

      10. *-commutative 97.6%

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

      11. associate-*r* 97.6%

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

      12. *-commutative 97.6%

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

      13. associate-*r* 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around inf 97.6%

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

      1. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   if -1.3000000000000001e36 < im < 2.9e10

   1. Initial program 37.0%

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

   2. Taylor expanded in im around 0 91.8%

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

      1. mul-1-neg 91.8%

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

      2. *-commutative 91.8%

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

      3. distribute-rgt-neg-in 91.8%

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

   4. Simplified 91.8%

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

      1. log1p-expm1-u 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{+36} \lor \neg \left(im \leq 29000000000\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin re \cdot \left(-im\right)\right)\right)\\ \end{array} \]

Alternative 5: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.0027:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{-im} - e^{im}\right)\right)\\ \mathbf{elif}\;im \leq 5.6:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -1.1e+44)
     t_0
     (if (<= im -0.0027)
       (* 0.5 (* re (- (exp (- im)) (exp im))))
       (if (<= im 5.6)
         (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im))
         t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.0027) {
		tmp = 0.5 * (re * (exp(-im) - exp(im)));
	} else if (im <= 5.6) {
		tmp = sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - 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 = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    if (im <= (-1.1d+44)) then
        tmp = t_0
    else if (im <= (-0.0027d0)) then
        tmp = 0.5d0 * (re * (exp(-im) - exp(im)))
    else if (im <= 5.6d0) then
        tmp = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.0027) {
		tmp = 0.5 * (re * (Math.exp(-im) - Math.exp(im)));
	} else if (im <= 5.6) {
		tmp = Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_0
	elif im <= -0.0027:
		tmp = 0.5 * (re * (math.exp(-im) - math.exp(im)))
	elif im <= 5.6:
		tmp = math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.0027)
		tmp = Float64(0.5 * Float64(re * Float64(exp(Float64(-im)) - exp(im))));
	elseif (im <= 5.6)
		tmp = Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.0027)
		tmp = 0.5 * (re * (exp(-im) - exp(im)));
	elseif (im <= 5.6)
		tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$0, If[LessEqual[im, -0.0027], N[(0.5 * N[(re * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.6], N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 or 5.5999999999999996 < 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 97.6%

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

      1. associate-+r+ 97.6%

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

      2. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

      4. mul-1-neg 97.6%

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

      5. *-commutative 97.6%

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

      6. distribute-lft-neg-in 97.6%

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

      7. *-commutative 97.6%

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

      8. associate-*r* 97.6%

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

      9. distribute-rgt-out 97.6%

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

      10. *-commutative 97.6%

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

      11. associate-*r* 97.6%

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

      12. *-commutative 97.6%

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

      13. associate-*r* 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around inf 97.6%

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

      1. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   if -1.09999999999999998e44 < im < -0.0027000000000000001

   1. Initial program 98.9%

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

   2. Taylor expanded in re around 0 65.6%

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

   if -0.0027000000000000001 < im < 5.5999999999999996

   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%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + -1 \cdot \left(\sin re \cdot im\right)} \]
   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 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.0027:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{-im} - e^{im}\right)\right)\\ \mathbf{elif}\;im \leq 5.6:\\ \;\;\;\;\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 6: 95.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -0.0025:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{-im} - e^{im}\right)\right)\\ \mathbf{elif}\;im \leq 4.2:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -1.1e+44)
     t_0
     (if (<= im -0.0025)
       (* 0.5 (* re (- (exp (- im)) (exp im))))
       (if (<= im 4.2) (* (sin re) (- im)) t_0)))))

C

double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.0025) {
		tmp = 0.5 * (re * (exp(-im) - exp(im)));
	} else if (im <= 4.2) {
		tmp = sin(re) * -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 = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    if (im <= (-1.1d+44)) then
        tmp = t_0
    else if (im <= (-0.0025d0)) then
        tmp = 0.5d0 * (re * (exp(-im) - exp(im)))
    else if (im <= 4.2d0) then
        tmp = sin(re) * -im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	double tmp;
	if (im <= -1.1e+44) {
		tmp = t_0;
	} else if (im <= -0.0025) {
		tmp = 0.5 * (re * (Math.exp(-im) - Math.exp(im)));
	} else if (im <= 4.2) {
		tmp = Math.sin(re) * -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	tmp = 0
	if im <= -1.1e+44:
		tmp = t_0
	elif im <= -0.0025:
		tmp = 0.5 * (re * (math.exp(-im) - math.exp(im)))
	elif im <= 4.2:
		tmp = math.sin(re) * -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)))
	tmp = 0.0
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.0025)
		tmp = Float64(0.5 * Float64(re * Float64(exp(Float64(-im)) - exp(im))));
	elseif (im <= 4.2)
		tmp = Float64(sin(re) * Float64(-im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	tmp = 0.0;
	if (im <= -1.1e+44)
		tmp = t_0;
	elseif (im <= -0.0025)
		tmp = 0.5 * (re * (exp(-im) - exp(im)));
	elseif (im <= 4.2)
		tmp = sin(re) * -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -1.1e+44], t$95$0, If[LessEqual[im, -0.0025], N[(0.5 * N[(re * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.2], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 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 97.6%

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

      1. associate-+r+ 97.6%

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

      2. +-commutative 97.6%

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

      3. +-commutative 97.6%

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

      4. mul-1-neg 97.6%

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

      5. *-commutative 97.6%

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

      6. distribute-lft-neg-in 97.6%

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

      7. *-commutative 97.6%

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

      8. associate-*r* 97.6%

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

      9. distribute-rgt-out 97.6%

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

      10. *-commutative 97.6%

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

      11. associate-*r* 97.6%

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

      12. *-commutative 97.6%

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

      13. associate-*r* 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in im around inf 97.6%

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

      1. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   if -1.09999999999999998e44 < im < -0.00250000000000000005

   1. Initial program 98.9%

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

   2. Taylor expanded in re around 0 65.6%

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

   if -0.00250000000000000005 < 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 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.0025:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(e^{-im} - e^{im}\right)\right)\\ \mathbf{elif}\;im \leq 4.2:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 7: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -4.1 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -4.1) (not (<= im 4.2)))
   (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))
   (* (sin re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -4.1) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (sin(re) * pow(im, 7.0));
	} else {
		tmp = sin(re) * -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 <= (-4.1d0)) .or. (.not. (im <= 4.2d0))) then
        tmp = (-0.0001984126984126984d0) * (sin(re) * (im ** 7.0d0))
    else
        tmp = sin(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -4.1) || !(im <= 4.2)) {
		tmp = -0.0001984126984126984 * (Math.sin(re) * Math.pow(im, 7.0));
	} else {
		tmp = Math.sin(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -4.1) or not (im <= 4.2):
		tmp = -0.0001984126984126984 * (math.sin(re) * math.pow(im, 7.0))
	else:
		tmp = math.sin(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -4.1) || !(im <= 4.2))
		tmp = Float64(-0.0001984126984126984 * Float64(sin(re) * (im ^ 7.0)));
	else
		tmp = Float64(sin(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -4.1) || ~((im <= 4.2)))
		tmp = -0.0001984126984126984 * (sin(re) * (im ^ 7.0));
	else
		tmp = sin(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -4.1], N[Not[LessEqual[im, 4.2]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -4.0999999999999996 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 90.3%

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

      1. associate-+r+ 90.3%

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

      2. +-commutative 90.3%

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

      3. +-commutative 90.3%

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

      4. mul-1-neg 90.3%

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

      5. *-commutative 90.3%

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

      6. distribute-lft-neg-in 90.3%

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

      7. *-commutative 90.3%

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

      8. associate-*r* 90.3%

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

      9. distribute-rgt-out 90.3%

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

      10. *-commutative 90.3%

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

      11. associate-*r* 90.3%

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

      12. *-commutative 90.3%

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

      13. associate-*r* 90.3%

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

   4. Simplified 90.3%

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

   5. Taylor expanded in im around inf 90.3%

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

      1. *-commutative 90.3%

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

   7. Simplified 90.3%

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

   if -4.0999999999999996 < im < 4.20000000000000018

   1. Initial program 32.0%

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

   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 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.1 \lor \neg \left(im \leq 4.2\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \end{array} \]

Alternative 8: 82.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+21} \lor \neg \left(im \leq 1.85 \cdot 10^{+56}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.2e+21) (not (<= im 1.85e+56)))
   (* -0.0001984126984126984 (* re (pow im 7.0)))
   (* (sin re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.2e+21) || !(im <= 1.85e+56)) {
		tmp = -0.0001984126984126984 * (re * pow(im, 7.0));
	} else {
		tmp = sin(re) * -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 <= (-1.2d+21)) .or. (.not. (im <= 1.85d+56))) then
        tmp = (-0.0001984126984126984d0) * (re * (im ** 7.0d0))
    else
        tmp = sin(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.2e+21) || !(im <= 1.85e+56)) {
		tmp = -0.0001984126984126984 * (re * Math.pow(im, 7.0));
	} else {
		tmp = Math.sin(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.2e+21) or not (im <= 1.85e+56):
		tmp = -0.0001984126984126984 * (re * math.pow(im, 7.0))
	else:
		tmp = math.sin(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.2e+21) || !(im <= 1.85e+56))
		tmp = Float64(-0.0001984126984126984 * Float64(re * (im ^ 7.0)));
	else
		tmp = Float64(sin(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.2e+21) || ~((im <= 1.85e+56)))
		tmp = -0.0001984126984126984 * (re * (im ^ 7.0));
	else
		tmp = sin(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.2e+21], N[Not[LessEqual[im, 1.85e+56]], $MachinePrecision]], N[(-0.0001984126984126984 * N[(re * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.2e21 or 1.84999999999999998e56 < 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 96.8%

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

      1. associate-+r+ 96.8%

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

      2. +-commutative 96.8%

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

      3. +-commutative 96.8%

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

      4. mul-1-neg 96.8%

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

      5. *-commutative 96.8%

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

      6. distribute-lft-neg-in 96.8%

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

      7. *-commutative 96.8%

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

      8. associate-*r* 96.8%

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

      9. distribute-rgt-out 96.8%

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

      10. *-commutative 96.8%

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

      11. associate-*r* 96.8%

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

      12. *-commutative 96.8%

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

      13. associate-*r* 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in re around 0 69.2%

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

   6. Taylor expanded in im around inf 69.2%

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

   if -1.2e21 < im < 1.84999999999999998e56

   1. Initial program 37.0%

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

   2. Taylor expanded in im around 0 91.8%

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

      1. mul-1-neg 91.8%

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

      2. *-commutative 91.8%

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

      3. distribute-rgt-neg-in 91.8%

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

   4. Simplified 91.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.2 \cdot 10^{+21} \lor \neg \left(im \leq 1.85 \cdot 10^{+56}\right):\\ \;\;\;\;-0.0001984126984126984 \cdot \left(re \cdot {im}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \end{array} \]

Alternative 9: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+27} \lor \neg \left(im \leq 8 \cdot 10^{+68}\right):\\ \;\;\;\;re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -5.5e+27) (not (<= im 8e+68)))
   (* re (- im))
   (* (sin re) (- im))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -5.5e+27) || !(im <= 8e+68)) {
		tmp = re * -im;
	} else {
		tmp = sin(re) * -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 <= (-5.5d+27)) .or. (.not. (im <= 8d+68))) then
        tmp = re * -im
    else
        tmp = sin(re) * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -5.5e+27) || !(im <= 8e+68)) {
		tmp = re * -im;
	} else {
		tmp = Math.sin(re) * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -5.5e+27) or not (im <= 8e+68):
		tmp = re * -im
	else:
		tmp = math.sin(re) * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -5.5e+27) || !(im <= 8e+68))
		tmp = Float64(re * Float64(-im));
	else
		tmp = Float64(sin(re) * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -5.5e+27) || ~((im <= 8e+68)))
		tmp = re * -im;
	else
		tmp = sin(re) * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -5.5e+27], N[Not[LessEqual[im, 8e+68]], $MachinePrecision]], N[(re * (-im)), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -5.49999999999999966e27 or 7.99999999999999962e68 < 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 4.9%

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

      1. mul-1-neg 4.9%

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

      2. *-commutative 4.9%

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

      3. distribute-rgt-neg-in 4.9%

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

   4. Simplified 4.9%

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

   5. Taylor expanded in re around 0 16.6%

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

      1. mul-1-neg 16.6%

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

      2. *-commutative 16.6%

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

      3. distribute-rgt-neg-in 16.6%

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

   7. Simplified 16.6%

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

   if -5.49999999999999966e27 < im < 7.99999999999999962e68

   1. Initial program 39.2%

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

   2. Taylor expanded in im around 0 88.6%

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

      1. mul-1-neg 88.6%

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

      2. *-commutative 88.6%

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

      3. distribute-rgt-neg-in 88.6%

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

   4. Simplified 88.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.5 \cdot 10^{+27} \lor \neg \left(im \leq 8 \cdot 10^{+68}\right):\\ \;\;\;\;re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \end{array} \]

Alternative 10: 55.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.45 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{re \cdot \left(re \cdot 9.5367431640625 \cdot 10^{-7}\right)}\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -3.45e+37)
   (sqrt (* re (* re 9.5367431640625e-7)))
   (if (<= im 7.6e+69) (* (sin re) (- im)) (* re (- im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -3.45e+37) {
		tmp = sqrt((re * (re * 9.5367431640625e-7)));
	} else if (im <= 7.6e+69) {
		tmp = sin(re) * -im;
	} else {
		tmp = re * -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 <= (-3.45d+37)) then
        tmp = sqrt((re * (re * 9.5367431640625d-7)))
    else if (im <= 7.6d+69) then
        tmp = sin(re) * -im
    else
        tmp = re * -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -3.45e+37) {
		tmp = Math.sqrt((re * (re * 9.5367431640625e-7)));
	} else if (im <= 7.6e+69) {
		tmp = Math.sin(re) * -im;
	} else {
		tmp = re * -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -3.45e+37:
		tmp = math.sqrt((re * (re * 9.5367431640625e-7)))
	elif im <= 7.6e+69:
		tmp = math.sin(re) * -im
	else:
		tmp = re * -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -3.45e+37)
		tmp = sqrt(Float64(re * Float64(re * 9.5367431640625e-7)));
	elseif (im <= 7.6e+69)
		tmp = Float64(sin(re) * Float64(-im));
	else
		tmp = Float64(re * Float64(-im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -3.45e+37)
		tmp = sqrt((re * (re * 9.5367431640625e-7)));
	elseif (im <= 7.6e+69)
		tmp = sin(re) * -im;
	else
		tmp = re * -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -3.45e+37], N[Sqrt[N[(re * N[(re * 9.5367431640625e-7), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[im, 7.6e+69], N[(N[Sin[re], $MachinePrecision] * (-im)), $MachinePrecision], N[(re * (-im)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -3.4499999999999998e37

   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 73.8%

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

   4. Applied egg-rr 2.4%

      \[\leadsto 0.5 \cdot \left(\color{blue}{0.001953125} \cdot re\right) \]
   5. Step-by-step derivation

      1. log1p-expm1-u 24.2%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \left(0.001953125 \cdot re\right)\right)\right)} \]

      2. associate-*r* 24.2%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(0.5 \cdot 0.001953125\right) \cdot re}\right)\right) \]

      3. metadata-eval 24.2%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{0.0009765625} \cdot re\right)\right) \]

   6. Applied egg-rr 24.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.0009765625 \cdot re\right)\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 2.4%

         \[\leadsto \color{blue}{0.0009765625 \cdot re} \]

      2. add-sqr-sqrt 1.3%

         \[\leadsto \color{blue}{\sqrt{0.0009765625 \cdot re} \cdot \sqrt{0.0009765625 \cdot re}} \]

      3. sqrt-unprod 14.4%

         \[\leadsto \color{blue}{\sqrt{\left(0.0009765625 \cdot re\right) \cdot \left(0.0009765625 \cdot re\right)}} \]

      4. swap-sqr 14.4%

         \[\leadsto \sqrt{\color{blue}{\left(0.0009765625 \cdot 0.0009765625\right) \cdot \left(re \cdot re\right)}} \]

      5. metadata-eval 14.4%

         \[\leadsto \sqrt{\color{blue}{9.5367431640625 \cdot 10^{-7}} \cdot \left(re \cdot re\right)} \]

   8. Applied egg-rr 14.4%

      \[\leadsto \color{blue}{\sqrt{9.5367431640625 \cdot 10^{-7} \cdot \left(re \cdot re\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 14.4%

         \[\leadsto \sqrt{\color{blue}{\left(9.5367431640625 \cdot 10^{-7} \cdot re\right) \cdot re}} \]

   10. Simplified 14.4%

       \[\leadsto \color{blue}{\sqrt{\left(9.5367431640625 \cdot 10^{-7} \cdot re\right) \cdot re}} \]

   if -3.4499999999999998e37 < im < 7.60000000000000055e69

   1. Initial program 40.5%

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

   2. Taylor expanded in im around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. *-commutative 86.8%

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

      3. distribute-rgt-neg-in 86.8%

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

   4. Simplified 86.8%

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

   if 7.60000000000000055e69 < 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 5.5%

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

      1. mul-1-neg 5.5%

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

      2. *-commutative 5.5%

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

      3. distribute-rgt-neg-in 5.5%

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

   4. Simplified 5.5%

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

   5. Taylor expanded in re around 0 23.8%

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

      1. mul-1-neg 23.8%

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

      2. *-commutative 23.8%

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

      3. distribute-rgt-neg-in 23.8%

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

   7. Simplified 23.8%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.45 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{re \cdot \left(re \cdot 9.5367431640625 \cdot 10^{-7}\right)}\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+69}:\\ \;\;\;\;\sin re \cdot \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(-im\right)\\ \end{array} \]

Alternative 11: 32.9% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

2. Taylor expanded in im around 0 51.0%

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

   1. mul-1-neg 51.0%

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

   2. *-commutative 51.0%

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

   3. distribute-rgt-neg-in 51.0%

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

4. Simplified 51.0%

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

5. Taylor expanded in re around 0 34.4%

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

   1. mul-1-neg 34.4%

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

   2. *-commutative 34.4%

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

   3. distribute-rgt-neg-in 34.4%

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

7. Simplified 34.4%

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

8. Final simplification 34.4%

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

Alternative 12: 3.3% accurate, 102.7× speedup

Math

\[\begin{array}{l} \\ re \cdot 0.0009765625 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re 0.0009765625))

C

double code(double re, double im) {
	return re * 0.0009765625;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * 0.0009765625d0
end function

Java

public static double code(double re, double im) {
	return re * 0.0009765625;
}

Python

def code(re, im):
	return re * 0.0009765625

Julia

function code(re, im)
	return Float64(re * 0.0009765625)
end

MATLAB

function tmp = code(re, im)
	tmp = re * 0.0009765625;
end

Wolfram

code[re_, im_] := N[(re * 0.0009765625), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

2. Taylor expanded in re around 0 51.2%

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

4. Applied egg-rr 3.5%

   \[\leadsto 0.5 \cdot \left(\color{blue}{0.001953125} \cdot re\right) \]

5. Taylor expanded in re around 0 3.5%

   \[\leadsto \color{blue}{0.0009765625 \cdot re} \]
6. Step-by-step derivation

   1. *-commutative 3.5%

      \[\leadsto \color{blue}{re \cdot 0.0009765625} \]

7. Simplified 3.5%

   \[\leadsto \color{blue}{re \cdot 0.0009765625} \]

8. Final simplification 3.5%

   \[\leadsto re \cdot 0.0009765625 \]

Developer target: 99.8% 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 2023192 
(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))))