math.cos on complex, imaginary part

Percentage Accurate: 65.0% → 99.8%

Time: 9.5s

Alternatives: 13

Speedup: 2.8×

• 49.0% 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: 65.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -10 \lor \neg \left(t_0 \leq 0.0002\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im}^{5} \cdot -0.008333333333333333 - im, \sin re, \sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + -0.0001984126984126984 \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -10.0) (not (<= t_0 0.0002)))
     (* t_0 (* 0.5 (sin re)))
     (fma
      (- (* (pow im 5.0) -0.008333333333333333) im)
      (sin re)
      (*
       (sin re)
       (+
        (* -0.16666666666666666 (pow im 3.0))
        (* -0.0001984126984126984 (pow im 7.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -10.0) || !(t_0 <= 0.0002)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = fma(((pow(im, 5.0) * -0.008333333333333333) - im), sin(re), (sin(re) * ((-0.16666666666666666 * pow(im, 3.0)) + (-0.0001984126984126984 * pow(im, 7.0)))));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -10.0) || !(t_0 <= 0.0002))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = fma(Float64(Float64((im ^ 5.0) * -0.008333333333333333) - im), sin(re), Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) + Float64(-0.0001984126984126984 * (im ^ 7.0)))));
	end
	return 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, -10.0], N[Not[LessEqual[t$95$0, 0.0002]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision] - im), $MachinePrecision] * N[Sin[re], $MachinePrecision] + N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0001984126984126984 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -10 or 2.0000000000000001e-4 < (-.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 -10 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.0000000000000001e-4

   1. Initial program 32.5%

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

   2. Taylor expanded in im around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. mul-1-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-lft-neg-in 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*r* 99.9%

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

      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 99.9%

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

      1. distribute-rgt-in 99.8%

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

      2. fma-def 99.9%

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

      3. fma-def 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in re around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10 \lor \neg \left(e^{-im} - e^{im} \leq 0.0002\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({im}^{5} \cdot -0.008333333333333333 - im, \sin re, \sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} + -0.0001984126984126984 \cdot {im}^{7}\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -10 \lor \neg \left(t_0 \leq 0.0002\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left(-0.16666666666666666 \cdot {im}^{3} + -0.0001984126984126984 \cdot {im}^{7}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -10.0) (not (<= t_0 0.0002)))
     (* t_0 (* 0.5 (sin re)))
     (*
      (sin re)
      (+
       (- (* (pow im 5.0) -0.008333333333333333) im)
       (+
        (* -0.16666666666666666 (pow im 3.0))
        (* -0.0001984126984126984 (pow im 7.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -10.0) || !(t_0 <= 0.0002)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * (((pow(im, 5.0) * -0.008333333333333333) - im) + ((-0.16666666666666666 * pow(im, 3.0)) + (-0.0001984126984126984 * pow(im, 7.0))));
	}
	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 <= (-10.0d0)) .or. (.not. (t_0 <= 0.0002d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * ((((im ** 5.0d0) * (-0.008333333333333333d0)) - im) + (((-0.16666666666666666d0) * (im ** 3.0d0)) + ((-0.0001984126984126984d0) * (im ** 7.0d0))))
    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 <= -10.0) || !(t_0 <= 0.0002)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * (((Math.pow(im, 5.0) * -0.008333333333333333) - im) + ((-0.16666666666666666 * Math.pow(im, 3.0)) + (-0.0001984126984126984 * Math.pow(im, 7.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -10.0) or not (t_0 <= 0.0002):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * (((math.pow(im, 5.0) * -0.008333333333333333) - im) + ((-0.16666666666666666 * math.pow(im, 3.0)) + (-0.0001984126984126984 * math.pow(im, 7.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= -10.0) || !(t_0 <= 0.0002))
		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(-0.16666666666666666 * (im ^ 3.0)) + Float64(-0.0001984126984126984 * (im ^ 7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -10.0) || ~((t_0 <= 0.0002)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((((im ^ 5.0) * -0.008333333333333333) - im) + ((-0.16666666666666666 * (im ^ 3.0)) + (-0.0001984126984126984 * (im ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10.0], N[Not[LessEqual[t$95$0, 0.0002]], $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[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0001984126984126984 * N[Power[im, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < -10 or 2.0000000000000001e-4 < (-.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 -10 < (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)) < 2.0000000000000001e-4

   1. Initial program 32.5%

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

   2. Taylor expanded in im around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. mul-1-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-lft-neg-in 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*r* 99.9%

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

      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 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -10 \lor \neg \left(e^{-im} - e^{im} \leq 0.0002\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(\left({im}^{5} \cdot -0.008333333333333333 - im\right) + \left(-0.16666666666666666 \cdot {im}^{3} + -0.0001984126984126984 \cdot {im}^{7}\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -0.05 \lor \neg \left(t_0 \leq 0.0002\right):\\ \;\;\;\;t_0 \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 -0.05) (not (<= t_0 0.0002)))
     (* t_0 (* 0.5 (sin re)))
     (* (sin re) (- (* -0.16666666666666666 (pow im 3.0)) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -0.05) || !(t_0 <= 0.0002)) {
		tmp = t_0 * (0.5 * sin(re));
	} else {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-im) - exp(im)
    if ((t_0 <= (-0.05d0)) .or. (.not. (t_0 <= 0.0002d0))) then
        tmp = t_0 * (0.5d0 * sin(re))
    else
        tmp = sin(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - 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 <= 0.0002)) {
		tmp = t_0 * (0.5 * Math.sin(re));
	} else {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - 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 <= 0.0002):
		tmp = t_0 * (0.5 * math.sin(re))
	else:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - 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 <= 0.0002))
		tmp = Float64(t_0 * Float64(0.5 * sin(re)));
	else
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - 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 <= 0.0002)))
		tmp = t_0 * (0.5 * sin(re));
	else
		tmp = sin(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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, 0.0002]], $MachinePrecision]], N[(t$95$0 * N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $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 2.0000000000000001e-4 < (-.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)) < 2.0000000000000001e-4

   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 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-lft-out-- 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -0.05 \lor \neg \left(e^{-im} - e^{im} \leq 0.0002\right):\\ \;\;\;\;\left(e^{-im} - e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 -2e-5)
       (* 0.5 (* (- (exp (- im)) (exp im)) re))
       (if (<= im 8.5e+42) (log1p (expm1 (* im (- (sin re))))) 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 <= -2e-5) {
		tmp = 0.5 * ((exp(-im) - exp(im)) * re);
	} else if (im <= 8.5e+42) {
		tmp = log1p(expm1((im * -sin(re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 <= -2e-5) {
		tmp = 0.5 * ((Math.exp(-im) - Math.exp(im)) * re);
	} else if (im <= 8.5e+42) {
		tmp = Math.log1p(Math.expm1((im * -Math.sin(re))));
	} 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 <= -2e-5:
		tmp = 0.5 * ((math.exp(-im) - math.exp(im)) * re)
	elif im <= 8.5e+42:
		tmp = math.log1p(math.expm1((im * -math.sin(re))))
	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 <= -2e-5)
		tmp = Float64(0.5 * Float64(Float64(exp(Float64(-im)) - exp(im)) * re));
	elseif (im <= 8.5e+42)
		tmp = log1p(expm1(Float64(im * Float64(-sin(re)))));
	else
		tmp = t_0;
	end
	return 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, -2e-5], N[(0.5 * N[(N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.5e+42], N[Log[1 + N[(Exp[N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -1.09999999999999998e44 or 8.5000000000000003e42 < 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 98.9%

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

      1. associate-+r+ 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. mul-1-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. distribute-lft-neg-in 98.9%

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

      7. *-commutative 98.9%

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

      8. associate-*r* 98.9%

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

      9. distribute-rgt-out 98.9%

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

      10. *-commutative 98.9%

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

      11. associate-*r* 98.9%

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

      12. *-commutative 98.9%

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

      13. associate-*r* 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in im around inf 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if -1.09999999999999998e44 < im < -2.00000000000000016e-5

   1. Initial program 97.6%

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

   2. Taylor expanded in re around 0 91.4%

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

   if -2.00000000000000016e-5 < im < 8.5000000000000003e42

   1. Initial program 36.3%

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

   2. Taylor expanded in im around 0 93.0%

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

      1. mul-1-neg 93.0%

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

      2. *-commutative 93.0%

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

      3. distribute-rgt-neg-in 93.0%

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

   4. Simplified 93.0%

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

      1. log1p-expm1-u 96.8%

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

      2. distribute-rgt-neg-out 96.8%

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

   6. Applied egg-rr 96.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 8.5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-\sin re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (- (exp (- im)) (exp im)) re)))
        (t_1 (* -0.0001984126984126984 (* (sin re) (pow im 7.0)))))
   (if (<= im -1.1e+44)
     t_1
     (if (<= im -2e-5)
       t_0
       (if (<= im 2.5) (* 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 <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -2e-5) {
		tmp = t_0;
	} else if (im <= 2.5) {
		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 <= (-1.1d+44)) then
        tmp = t_1
    else if (im <= (-2d-5)) then
        tmp = t_0
    else if (im <= 2.5d0) 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 <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -2e-5) {
		tmp = t_0;
	} else if (im <= 2.5) {
		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 <= -1.1e+44:
		tmp = t_1
	elif im <= -2e-5:
		tmp = t_0
	elif im <= 2.5:
		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 <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -2e-5)
		tmp = t_0;
	elseif (im <= 2.5)
		tmp = Float64(im * Float64(-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 <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -2e-5)
		tmp = t_0;
	elseif (im <= 2.5)
		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, -1.1e+44], t$95$1, If[LessEqual[im, -2e-5], t$95$0, If[LessEqual[im, 2.5], 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 < -1.09999999999999998e44 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 -1.09999999999999998e44 < im < -2.00000000000000016e-5 or 2.5 < im < 1.09999999999999998e44

   1. Initial program 98.5%

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

   2. Taylor expanded in re around 0 83.1%

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

   if -2.00000000000000016e-5 < im < 2.5

   1. Initial program 32.3%

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

   2. Taylor expanded in im around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

   4. Simplified 98.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 6: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ t_1 := -0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.088:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 -1.1e+44)
     t_1
     (if (<= im -0.088)
       t_0
       (if (<= im 2.5)
         (* (sin re) (- (* -0.16666666666666666 (pow im 3.0)) 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 <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.088) {
		tmp = t_0;
	} else if (im <= 2.5) {
		tmp = sin(re) * ((-0.16666666666666666 * pow(im, 3.0)) - 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 <= (-1.1d+44)) then
        tmp = t_1
    else if (im <= (-0.088d0)) then
        tmp = t_0
    else if (im <= 2.5d0) then
        tmp = sin(re) * (((-0.16666666666666666d0) * (im ** 3.0d0)) - 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 <= -1.1e+44) {
		tmp = t_1;
	} else if (im <= -0.088) {
		tmp = t_0;
	} else if (im <= 2.5) {
		tmp = Math.sin(re) * ((-0.16666666666666666 * Math.pow(im, 3.0)) - 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 <= -1.1e+44:
		tmp = t_1
	elif im <= -0.088:
		tmp = t_0
	elif im <= 2.5:
		tmp = math.sin(re) * ((-0.16666666666666666 * math.pow(im, 3.0)) - 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 <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.088)
		tmp = t_0;
	elseif (im <= 2.5)
		tmp = Float64(sin(re) * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - 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 <= -1.1e+44)
		tmp = t_1;
	elseif (im <= -0.088)
		tmp = t_0;
	elseif (im <= 2.5)
		tmp = sin(re) * ((-0.16666666666666666 * (im ^ 3.0)) - 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, -1.1e+44], t$95$1, If[LessEqual[im, -0.088], t$95$0, If[LessEqual[im, 2.5], N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $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 < -1.09999999999999998e44 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 -1.09999999999999998e44 < im < -0.087999999999999995 or 2.5 < im < 1.09999999999999998e44

   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 83.3%

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

   if -0.087999999999999995 < im < 2.5

   1. Initial program 32.9%

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

   2. Taylor expanded in im around 0 99.0%

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

      1. mul-1-neg 99.0%

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

      2. unsub-neg 99.0%

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

      3. *-commutative 99.0%

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

      4. associate-*l* 99.0%

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

      5. distribute-lft-out-- 99.0%

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

   4. Simplified 99.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.1 \cdot 10^{+44}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \mathbf{elif}\;im \leq -0.088:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{elif}\;im \leq 2.5:\\ \;\;\;\;\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;-0.0001984126984126984 \cdot \left(\sin re \cdot {im}^{7}\right)\\ \end{array} \]

Alternative 7: 83.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.4 \lor \neg \left(im \leq 2.4\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.4) (not (<= im 2.4)))
   (* -0.16666666666666666 (* (sin re) (pow im 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.4) || !(im <= 2.4)) {
		tmp = -0.16666666666666666 * (sin(re) * pow(im, 3.0));
	} 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 <= (-2.4d0)) .or. (.not. (im <= 2.4d0))) then
        tmp = (-0.16666666666666666d0) * (sin(re) * (im ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.4) || !(im <= 2.4)) {
		tmp = -0.16666666666666666 * (Math.sin(re) * Math.pow(im, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.4) or not (im <= 2.4):
		tmp = -0.16666666666666666 * (math.sin(re) * math.pow(im, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.4) || !(im <= 2.4))
		tmp = Float64(-0.16666666666666666 * Float64(sin(re) * (im ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.4) || ~((im <= 2.4)))
		tmp = -0.16666666666666666 * (sin(re) * (im ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.4], N[Not[LessEqual[im, 2.4]], $MachinePrecision]], N[(-0.16666666666666666 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.39999999999999991 or 2.39999999999999991 < 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 61.3%

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

      1. mul-1-neg 61.3%

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

      2. unsub-neg 61.3%

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

      3. *-commutative 61.3%

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

      4. associate-*l* 61.3%

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

      5. distribute-lft-out-- 61.3%

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

   4. Simplified 61.3%

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

   5. Taylor expanded in im around inf 61.2%

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

      1. *-commutative 61.2%

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

   7. Simplified 61.2%

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

   if -2.39999999999999991 < im < 2.39999999999999991

   1. Initial program 32.5%

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

   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 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.4 \lor \neg \left(im \leq 2.4\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 8: 92.7% accurate, 1.5× speedup

Math

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

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(im * Float64(-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 79.6%

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

      1. associate-+r+ 79.6%

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

      2. +-commutative 79.6%

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

      3. +-commutative 79.6%

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

      4. mul-1-neg 79.6%

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

      5. *-commutative 79.6%

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

      6. distribute-lft-neg-in 79.6%

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

      7. *-commutative 79.6%

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

      8. associate-*r* 79.6%

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

      9. distribute-rgt-out 79.6%

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

      10. *-commutative 79.6%

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

      11. associate-*r* 79.6%

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

      12. *-commutative 79.6%

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

      13. associate-*r* 79.6%

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

   4. Simplified 79.6%

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

   5. Taylor expanded in im around inf 79.6%

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

      1. *-commutative 79.6%

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

   7. Simplified 79.6%

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

   if -4.20000000000000018 < im < 4.20000000000000018

   1. Initial program 32.9%

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

   2. Taylor expanded in im around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

   4. Simplified 98.3%

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

4. Final simplification 90.0%

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

Alternative 9: 77.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.9 \cdot 10^{-6} \lor \neg \left(im \leq 7.2 \cdot 10^{+19}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -2.9e-6) (not (<= im 7.2e+19)))
   (* re (- (* -0.16666666666666666 (pow im 3.0)) im))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -2.9e-6) || !(im <= 7.2e+19)) {
		tmp = re * ((-0.16666666666666666 * pow(im, 3.0)) - im);
	} 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 <= (-2.9d-6)) .or. (.not. (im <= 7.2d+19))) then
        tmp = re * (((-0.16666666666666666d0) * (im ** 3.0d0)) - im)
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -2.9e-6) || !(im <= 7.2e+19)) {
		tmp = re * ((-0.16666666666666666 * Math.pow(im, 3.0)) - im);
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -2.9e-6) or not (im <= 7.2e+19):
		tmp = re * ((-0.16666666666666666 * math.pow(im, 3.0)) - im)
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -2.9e-6) || !(im <= 7.2e+19))
		tmp = Float64(re * Float64(Float64(-0.16666666666666666 * (im ^ 3.0)) - im));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -2.9e-6) || ~((im <= 7.2e+19)))
		tmp = re * ((-0.16666666666666666 * (im ^ 3.0)) - im);
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -2.9e-6], N[Not[LessEqual[im, 7.2e+19]], $MachinePrecision]], N[(re * N[(N[(-0.16666666666666666 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.9000000000000002e-6 or 7.2e19 < im

   1. Initial program 99.4%

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

   2. Taylor expanded in re around 0 77.8%

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

   3. Taylor expanded in im around 0 50.8%

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

      1. mul-1-neg 50.8%

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

      2. distribute-rgt-neg-in 50.8%

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

      3. *-commutative 50.8%

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

      4. associate-*l* 50.8%

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

      5. distribute-lft-in 50.8%

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

      6. +-commutative 50.8%

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

      7. sub-neg 50.8%

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

      8. *-commutative 50.8%

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

   5. Simplified 50.8%

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

   if -2.9000000000000002e-6 < im < 7.2e19

   1. Initial program 34.3%

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

   2. Taylor expanded in im around 0 95.5%

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

      1. mul-1-neg 95.5%

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

      2. *-commutative 95.5%

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

      3. distribute-rgt-neg-in 95.5%

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

   4. Simplified 95.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.9 \cdot 10^{-6} \lor \neg \left(im \leq 7.2 \cdot 10^{+19}\right):\\ \;\;\;\;re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 10: 77.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -280000000000 \lor \neg \left(im \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -280000000000.0) (not (<= im 2.2e+15)))
   (* -0.16666666666666666 (* re (pow im 3.0)))
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -280000000000.0) || !(im <= 2.2e+15)) {
		tmp = -0.16666666666666666 * (re * pow(im, 3.0));
	} 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 <= (-280000000000.0d0)) .or. (.not. (im <= 2.2d+15))) then
        tmp = (-0.16666666666666666d0) * (re * (im ** 3.0d0))
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -280000000000.0) || !(im <= 2.2e+15)) {
		tmp = -0.16666666666666666 * (re * Math.pow(im, 3.0));
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -280000000000.0) or not (im <= 2.2e+15):
		tmp = -0.16666666666666666 * (re * math.pow(im, 3.0))
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -280000000000.0) || !(im <= 2.2e+15))
		tmp = Float64(-0.16666666666666666 * Float64(re * (im ^ 3.0)));
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -280000000000.0) || ~((im <= 2.2e+15)))
		tmp = -0.16666666666666666 * (re * (im ^ 3.0));
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -280000000000.0], N[Not[LessEqual[im, 2.2e+15]], $MachinePrecision]], N[(-0.16666666666666666 * N[(re * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -2.8e11 or 2.2e15 < 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 78.3%

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

   3. Taylor expanded in im around 0 50.3%

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

   4. Taylor expanded in im around inf 50.3%

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

   if -2.8e11 < im < 2.2e15

   1. Initial program 36.1%

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

   2. Taylor expanded in im around 0 93.8%

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

      1. mul-1-neg 93.8%

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

      2. *-commutative 93.8%

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

      3. distribute-rgt-neg-in 93.8%

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

   4. Simplified 93.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -280000000000 \lor \neg \left(im \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;-0.16666666666666666 \cdot \left(re \cdot {im}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 11: 57.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+18} \lor \neg \left(im \leq 3.8 \cdot 10^{+121}\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.5e+18) (not (<= im 3.8e+121)))
   (* (- im) re)
   (* im (- (sin re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.5e+18) || !(im <= 3.8e+121)) {
		tmp = -im * re;
	} 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 <= (-3.5d+18)) .or. (.not. (im <= 3.8d+121))) then
        tmp = -im * re
    else
        tmp = im * -sin(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -3.5e+18) || !(im <= 3.8e+121)) {
		tmp = -im * re;
	} else {
		tmp = im * -Math.sin(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -3.5e+18) or not (im <= 3.8e+121):
		tmp = -im * re
	else:
		tmp = im * -math.sin(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.5e+18) || !(im <= 3.8e+121))
		tmp = Float64(Float64(-im) * re);
	else
		tmp = Float64(im * Float64(-sin(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -3.5e+18) || ~((im <= 3.8e+121)))
		tmp = -im * re;
	else
		tmp = im * -sin(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.5e+18], N[Not[LessEqual[im, 3.8e+121]], $MachinePrecision]], N[((-im) * re), $MachinePrecision], N[(im * (-N[Sin[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.5e18 or 3.8e121 < 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 4.4%

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

      1. mul-1-neg 4.4%

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

      2. *-commutative 4.4%

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

      3. distribute-rgt-neg-in 4.4%

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

   4. Simplified 4.4%

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

   5. Taylor expanded in re around 0 16.5%

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

      1. mul-1-neg 16.5%

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

      2. *-commutative 16.5%

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

      3. distribute-rgt-neg-in 16.5%

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

   7. Simplified 16.5%

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

   if -3.5e18 < im < 3.8e121

   1. Initial program 43.6%

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

   2. Taylor expanded in im around 0 83.1%

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

      1. mul-1-neg 83.1%

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

      2. *-commutative 83.1%

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

      3. distribute-rgt-neg-in 83.1%

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

   4. Simplified 83.1%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.5 \cdot 10^{+18} \lor \neg \left(im \leq 3.8 \cdot 10^{+121}\right):\\ \;\;\;\;\left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\sin re\right)\\ \end{array} \]

Alternative 12: 33.1% accurate, 77.0× speedup

Math

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

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(Float64(-im) * re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in im around 0 56.7%

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

   1. mul-1-neg 56.7%

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

   2. *-commutative 56.7%

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

   3. distribute-rgt-neg-in 56.7%

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

4. Simplified 56.7%

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

5. Taylor expanded in re around 0 38.5%

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

   1. mul-1-neg 38.5%

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

   2. *-commutative 38.5%

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

   3. distribute-rgt-neg-in 38.5%

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

7. Simplified 38.5%

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

8. Final simplification 38.5%

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

Alternative 13: 3.2% accurate, 102.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re * -1.5

Julia

function code(re, im)
	return Float64(re * -1.5)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re * -1.5), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in re around 0 51.6%

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

4. Applied egg-rr 3.3%

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

5. Taylor expanded in re around 0 3.3%

   \[\leadsto \color{blue}{-1.5 \cdot re} \]
6. Step-by-step derivation

   1. *-commutative 3.3%

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

7. Simplified 3.3%

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

8. Final simplification 3.3%

   \[\leadsto re \cdot -1.5 \]

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 2023199 
(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))))