math.sin on complex, imaginary part

Percentage Accurate: 55.1% → 99.9%

Time: 13.1s

Alternatives: 13

Speedup: 13.4×

• 50.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return cos(re) * (0.0 - sinh(im));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right), \color{blue}{\frac{1}{2}}\right) \]

4. Applied egg-rr 99.9%

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

   1. remove-double-div N/A

      \[\leadsto \left(\cos re \cdot \frac{1}{\frac{1}{-2 \cdot \sinh im}}\right) \cdot \frac{1}{2} \]

   2. un-div-inv N/A

      \[\leadsto \frac{\cos re}{\frac{1}{-2 \cdot \sinh im}} \cdot \frac{1}{2} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\cos re}{\frac{\frac{1}{\sinh im}}{-2}} \cdot \frac{1}{2} \]

   4. associate-/r/ N/A

      \[\leadsto \left(\frac{\cos re}{\frac{1}{\sinh im}} \cdot -2\right) \cdot \frac{1}{2} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot \color{blue}{\left(-2 \cdot \frac{1}{2}\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot -1 \]

   7. *-commutative N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\cos re}{\frac{1}{\sinh im}}} \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\cos re}{\frac{1}{\sinh im}}\right) \]

   9. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{\frac{1}{\sinh im}}\right)\right) \]

   10. associate-/r/ N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{1} \cdot \sinh im\right)\right) \]

   11. /-rgt-identity N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\cos re \cdot \sinh im\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\cos re, \sinh im\right)\right) \]

   13. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \sinh im\right)\right) \]

   14. sinh-lowering-sinh.f64 99.9%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{sinh.f64}\left(im\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \cos re \cdot \left(0 - \sinh im\right) \]
8. Add Preprocessing

Alternative 2: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.983:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \sinh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.983)
   (*
    (* (cos re) 0.5)
    (*
     im
     (+
      -2.0
      (*
       (* im im)
       (+
        -0.3333333333333333
        (*
         (* im im)
         (+ -0.016666666666666666 (* (* im im) -0.0003968253968253968))))))))
   (- 0.0 (sinh im))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.983) {
		tmp = (cos(re) * 0.5) * (im * (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))));
	} else {
		tmp = 0.0 - sinh(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.983d0) then
        tmp = (cos(re) * 0.5d0) * (im * ((-2.0d0) + ((im * im) * ((-0.3333333333333333d0) + ((im * im) * ((-0.016666666666666666d0) + ((im * im) * (-0.0003968253968253968d0))))))))
    else
        tmp = 0.0d0 - sinh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.983) {
		tmp = (Math.cos(re) * 0.5) * (im * (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))));
	} else {
		tmp = 0.0 - Math.sinh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.983:
		tmp = (math.cos(re) * 0.5) * (im * (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))))
	else:
		tmp = 0.0 - math.sinh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.983)
		tmp = Float64(Float64(cos(re) * 0.5) * Float64(im * Float64(-2.0 + Float64(Float64(im * im) * Float64(-0.3333333333333333 + Float64(Float64(im * im) * Float64(-0.016666666666666666 + Float64(Float64(im * im) * -0.0003968253968253968))))))));
	else
		tmp = Float64(0.0 - sinh(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.983)
		tmp = (cos(re) * 0.5) * (im * (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))));
	else
		tmp = 0.0 - sinh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.983], N[(N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(im * N[(-2.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(im * im), $MachinePrecision] * N[(-0.016666666666666666 + N[(N[(im * im), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sinh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.982999999999999985

   1. Initial program 55.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} + \color{blue}{\frac{-1}{2520} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 96.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 96.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)} \]

   if 0.982999999999999985 < (cos.f64 re)

   1. Initial program 56.1%

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

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   4. Applied egg-rr 100.0%

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

      1. remove-double-div N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{\frac{1}{-2 \cdot \sinh im}}\right) \cdot \frac{1}{2} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos re}{\frac{1}{-2 \cdot \sinh im}} \cdot \frac{1}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{\cos re}{\frac{\frac{1}{\sinh im}}{-2}} \cdot \frac{1}{2} \]

      4. associate-/r/ N/A

         \[\leadsto \left(\frac{\cos re}{\frac{1}{\sinh im}} \cdot -2\right) \cdot \frac{1}{2} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot \color{blue}{\left(-2 \cdot \frac{1}{2}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot -1 \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\cos re}{\frac{1}{\sinh im}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\cos re}{\frac{1}{\sinh im}}\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{\frac{1}{\sinh im}}\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{1} \cdot \sinh im\right)\right) \]

      11. /-rgt-identity N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\cos re \cdot \sinh im\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\cos re, \sinh im\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \sinh im\right)\right) \]

      14. sinh-lowering-sinh.f64 100.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{sinh.f64}\left(im\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(im\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 99.9%

      \[\leadsto -\color{blue}{1} \cdot \sinh im \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.983:\\ \;\;\;\;\left(\cos re \cdot 0.5\right) \cdot \left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \sinh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 0.983:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \sinh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 0.983)
   (*
    im
    (*
     (cos re)
     (+
      -1.0
      (*
       (* im im)
       (+ -0.16666666666666666 (* (* im im) -0.008333333333333333))))))
   (- 0.0 (sinh im))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 0.983) {
		tmp = im * (cos(re) * (-1.0 + ((im * im) * (-0.16666666666666666 + ((im * im) * -0.008333333333333333)))));
	} else {
		tmp = 0.0 - sinh(im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (cos(re) <= 0.983d0) then
        tmp = im * (cos(re) * ((-1.0d0) + ((im * im) * ((-0.16666666666666666d0) + ((im * im) * (-0.008333333333333333d0))))))
    else
        tmp = 0.0d0 - sinh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 0.983) {
		tmp = im * (Math.cos(re) * (-1.0 + ((im * im) * (-0.16666666666666666 + ((im * im) * -0.008333333333333333)))));
	} else {
		tmp = 0.0 - Math.sinh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 0.983:
		tmp = im * (math.cos(re) * (-1.0 + ((im * im) * (-0.16666666666666666 + ((im * im) * -0.008333333333333333)))))
	else:
		tmp = 0.0 - math.sinh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 0.983)
		tmp = Float64(im * Float64(cos(re) * Float64(-1.0 + Float64(Float64(im * im) * Float64(-0.16666666666666666 + Float64(Float64(im * im) * -0.008333333333333333))))));
	else
		tmp = Float64(0.0 - sinh(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (cos(re) <= 0.983)
		tmp = im * (cos(re) * (-1.0 + ((im * im) * (-0.16666666666666666 + ((im * im) * -0.008333333333333333)))));
	else
		tmp = 0.0 - sinh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 0.983], N[(im * N[(N[Cos[re], $MachinePrecision] * N[(-1.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(im * im), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sinh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 0.982999999999999985

   1. Initial program 55.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(im, \color{blue}{\left(-1 \cdot \cos re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \color{blue}{{im}^{2}} \cdot \left(\frac{-1}{6} \cdot \cos re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + {im}^{2} \cdot \left(\frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{\frac{-1}{6} \cdot \cos re}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot \left({im}^{2} \cdot \cos re\right)\right) + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{6} \cdot \cos re\right)}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left({im}^{2} \cdot \left(\left(\frac{-1}{120} \cdot {im}^{2}\right) \cdot \cos re\right) + {im}^{\color{blue}{2}} \cdot \left(\frac{-1}{6} \cdot \cos re\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \color{blue}{{im}^{2}} \cdot \left(\frac{-1}{6} \cdot \cos re\right)\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \left({im}^{2} \cdot \frac{-1}{6}\right) \cdot \color{blue}{\cos re}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \left(\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right)\right) \cdot \cos re + \left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \cos \color{blue}{re}\right)\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot -1 + \cos re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)}\right)\right) \]

      10. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(im, \left(\cos re \cdot \color{blue}{\left(-1 + \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2}\right) + \frac{-1}{6} \cdot {im}^{2}\right)\right)}\right)\right) \]

   5. Simplified 95.4%

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

   if 0.982999999999999985 < (cos.f64 re)

   1. Initial program 56.1%

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

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   4. Applied egg-rr 100.0%

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

      1. remove-double-div N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{\frac{1}{-2 \cdot \sinh im}}\right) \cdot \frac{1}{2} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos re}{\frac{1}{-2 \cdot \sinh im}} \cdot \frac{1}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{\cos re}{\frac{\frac{1}{\sinh im}}{-2}} \cdot \frac{1}{2} \]

      4. associate-/r/ N/A

         \[\leadsto \left(\frac{\cos re}{\frac{1}{\sinh im}} \cdot -2\right) \cdot \frac{1}{2} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot \color{blue}{\left(-2 \cdot \frac{1}{2}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot -1 \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\cos re}{\frac{1}{\sinh im}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\cos re}{\frac{1}{\sinh im}}\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{\frac{1}{\sinh im}}\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{1} \cdot \sinh im\right)\right) \]

      11. /-rgt-identity N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\cos re \cdot \sinh im\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\cos re, \sinh im\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \sinh im\right)\right) \]

      14. sinh-lowering-sinh.f64 100.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{sinh.f64}\left(im\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(im\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 99.9%

      \[\leadsto -\color{blue}{1} \cdot \sinh im \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.983:\\ \;\;\;\;im \cdot \left(\cos re \cdot \left(-1 + \left(im \cdot im\right) \cdot \left(-0.16666666666666666 + \left(im \cdot im\right) \cdot -0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \sinh im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot \left(im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{if}\;im \leq 0.00013:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0 - \sinh im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) (* im (+ -1.0 (* (* im im) -0.16666666666666666))))))
   (if (<= im 0.00013) t_0 (if (<= im 1.05e+103) (- 0.0 (sinh im)) t_0))))

C

double code(double re, double im) {
	double t_0 = cos(re) * (im * (-1.0 + ((im * im) * -0.16666666666666666)));
	double tmp;
	if (im <= 0.00013) {
		tmp = t_0;
	} else if (im <= 1.05e+103) {
		tmp = 0.0 - sinh(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 = cos(re) * (im * ((-1.0d0) + ((im * im) * (-0.16666666666666666d0))))
    if (im <= 0.00013d0) then
        tmp = t_0
    else if (im <= 1.05d+103) then
        tmp = 0.0d0 - sinh(im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.cos(re) * (im * (-1.0 + ((im * im) * -0.16666666666666666)));
	double tmp;
	if (im <= 0.00013) {
		tmp = t_0;
	} else if (im <= 1.05e+103) {
		tmp = 0.0 - Math.sinh(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.cos(re) * (im * (-1.0 + ((im * im) * -0.16666666666666666)))
	tmp = 0
	if im <= 0.00013:
		tmp = t_0
	elif im <= 1.05e+103:
		tmp = 0.0 - math.sinh(im)
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(cos(re) * Float64(im * Float64(-1.0 + Float64(Float64(im * im) * -0.16666666666666666))))
	tmp = 0.0
	if (im <= 0.00013)
		tmp = t_0;
	elseif (im <= 1.05e+103)
		tmp = Float64(0.0 - sinh(im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = cos(re) * (im * (-1.0 + ((im * im) * -0.16666666666666666)));
	tmp = 0.0;
	if (im <= 0.00013)
		tmp = t_0;
	elseif (im <= 1.05e+103)
		tmp = 0.0 - sinh(im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * N[(im * N[(-1.0 + N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.00013], t$95$0, If[LessEqual[im, 1.05e+103], N[(0.0 - N[Sinh[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.29999999999999989e-4 or 1.0500000000000001e103 < im

   1. Initial program 51.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \cos re + \frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \cos re\right) + \color{blue}{-1 \cdot \cos re}\right) \]

      2. *-commutative N/A

         \[\leadsto im \cdot \left(\frac{-1}{6} \cdot \left(\cos re \cdot {im}^{2}\right) + -1 \cdot \cos re\right) \]

      3. associate-*r* N/A

         \[\leadsto im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2} + \color{blue}{-1} \cdot \cos re\right) \]

      4. +-commutative N/A

         \[\leadsto im \cdot \left(-1 \cdot \cos re + \color{blue}{\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto im \cdot \left(-1 \cdot \cos re\right) + \color{blue}{im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(im \cdot -1\right) \cdot \cos re + \color{blue}{im} \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(-1 \cdot im\right) \cdot \cos re + im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot \cos re + im \cdot \left(\left(\frac{-1}{6} \cdot \cos re\right) \cdot {im}^{2}\right) \]

      9. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot \cos re + im \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\cos re \cdot {im}^{2}\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot \cos re + im \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \color{blue}{\cos re}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot \cos re + im \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{\cos re}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(\mathsf{neg}\left(im\right)\right) \cdot \cos re + \left(im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right) \cdot \color{blue}{\cos re} \]

      13. distribute-rgt-out N/A

          \[\leadsto \cos re \cdot \color{blue}{\left(\left(\mathsf{neg}\left(im\right)\right) + im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos re, \color{blue}{\left(\left(\mathsf{neg}\left(im\right)\right) + im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right)}\right) \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(\color{blue}{\left(\mathsf{neg}\left(im\right)\right)} + im \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      16. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(-1 \cdot im + \color{blue}{im} \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \left(im \cdot -1 + \color{blue}{im} \cdot \left(\frac{-1}{6} \cdot {im}^{2}\right)\right)\right) \]

   5. Simplified 94.6%

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

   if 1.29999999999999989e-4 < im < 1.0500000000000001e103

   1. Initial program 100.0%

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

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   4. Applied egg-rr 100.0%

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

      1. remove-double-div N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{\frac{1}{-2 \cdot \sinh im}}\right) \cdot \frac{1}{2} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos re}{\frac{1}{-2 \cdot \sinh im}} \cdot \frac{1}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{\cos re}{\frac{\frac{1}{\sinh im}}{-2}} \cdot \frac{1}{2} \]

      4. associate-/r/ N/A

         \[\leadsto \left(\frac{\cos re}{\frac{1}{\sinh im}} \cdot -2\right) \cdot \frac{1}{2} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot \color{blue}{\left(-2 \cdot \frac{1}{2}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot -1 \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\cos re}{\frac{1}{\sinh im}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\cos re}{\frac{1}{\sinh im}}\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{\frac{1}{\sinh im}}\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{1} \cdot \sinh im\right)\right) \]

      11. /-rgt-identity N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\cos re \cdot \sinh im\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\cos re, \sinh im\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \sinh im\right)\right) \]

      14. sinh-lowering-sinh.f64 100.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{sinh.f64}\left(im\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(im\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 64.3%

      \[\leadsto -\color{blue}{1} \cdot \sinh im \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00013:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;0 - \sinh im\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left(im \cdot \left(-1 + \left(im \cdot im\right) \cdot -0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0 - \cos re \cdot im\\ \mathbf{else}:\\ \;\;\;\;0 - \sinh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00012) (- 0.0 (* (cos re) im)) (- 0.0 (sinh im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.0 - (cos(re) * im);
	} else {
		tmp = 0.0 - sinh(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 <= 0.00012d0) then
        tmp = 0.0d0 - (cos(re) * im)
    else
        tmp = 0.0d0 - sinh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00012) {
		tmp = 0.0 - (Math.cos(re) * im);
	} else {
		tmp = 0.0 - Math.sinh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00012:
		tmp = 0.0 - (math.cos(re) * im)
	else:
		tmp = 0.0 - math.sinh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00012)
		tmp = Float64(0.0 - Float64(cos(re) * im));
	else
		tmp = Float64(0.0 - sinh(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00012)
		tmp = 0.0 - (cos(re) * im);
	else
		tmp = 0.0 - sinh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00012], N[(0.0 - N[(N[Cos[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Sinh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.20000000000000003e-4

   1. Initial program 40.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 65.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 65.9%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(im \cdot \cos re\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\cos re \cdot im\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\cos re, im\right)\right) \]

      5. cos-lowering-cos.f64 65.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), im\right)\right) \]

   7. Applied egg-rr 65.9%

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

   if 1.20000000000000003e-4 < im

   1. Initial program 100.0%

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

      1. associate-*l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right), \color{blue}{\frac{1}{2}}\right) \]

   4. Applied egg-rr 100.0%

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

      1. remove-double-div N/A

         \[\leadsto \left(\cos re \cdot \frac{1}{\frac{1}{-2 \cdot \sinh im}}\right) \cdot \frac{1}{2} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos re}{\frac{1}{-2 \cdot \sinh im}} \cdot \frac{1}{2} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{\cos re}{\frac{\frac{1}{\sinh im}}{-2}} \cdot \frac{1}{2} \]

      4. associate-/r/ N/A

         \[\leadsto \left(\frac{\cos re}{\frac{1}{\sinh im}} \cdot -2\right) \cdot \frac{1}{2} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot \color{blue}{\left(-2 \cdot \frac{1}{2}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot -1 \]

      7. *-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\cos re}{\frac{1}{\sinh im}}} \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\cos re}{\frac{1}{\sinh im}}\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{\frac{1}{\sinh im}}\right)\right) \]

      10. associate-/r/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{1} \cdot \sinh im\right)\right) \]

      11. /-rgt-identity N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\cos re \cdot \sinh im\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\cos re, \sinh im\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \sinh im\right)\right) \]

      14. sinh-lowering-sinh.f64 100.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{sinh.f64}\left(im\right)\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(im\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 77.5%

      \[\leadsto -\color{blue}{1} \cdot \sinh im \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00012:\\ \;\;\;\;0 - \cos re \cdot im\\ \mathbf{else}:\\ \;\;\;\;0 - \sinh im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 0 - \sinh im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (- 0.0 (sinh im)))

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 0.0 - Math.sinh(im);
}

Python

def code(re, im):
	return 0.0 - math.sinh(im)

Julia

function code(re, im)
	return Float64(0.0 - sinh(im))
end

MATLAB

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

Wolfram

code[re_, im_] := N[(0.0 - N[Sinh[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. associate-*l* N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\cos re \cdot \left(e^{0 - im} - e^{im}\right)\right), \color{blue}{\frac{1}{2}}\right) \]

4. Applied egg-rr 99.9%

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

   1. remove-double-div N/A

      \[\leadsto \left(\cos re \cdot \frac{1}{\frac{1}{-2 \cdot \sinh im}}\right) \cdot \frac{1}{2} \]

   2. un-div-inv N/A

      \[\leadsto \frac{\cos re}{\frac{1}{-2 \cdot \sinh im}} \cdot \frac{1}{2} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\cos re}{\frac{\frac{1}{\sinh im}}{-2}} \cdot \frac{1}{2} \]

   4. associate-/r/ N/A

      \[\leadsto \left(\frac{\cos re}{\frac{1}{\sinh im}} \cdot -2\right) \cdot \frac{1}{2} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot \color{blue}{\left(-2 \cdot \frac{1}{2}\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\cos re}{\frac{1}{\sinh im}} \cdot -1 \]

   7. *-commutative N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\cos re}{\frac{1}{\sinh im}}} \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\cos re}{\frac{1}{\sinh im}}\right) \]

   9. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{\frac{1}{\sinh im}}\right)\right) \]

   10. associate-/r/ N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\cos re}{1} \cdot \sinh im\right)\right) \]

   11. /-rgt-identity N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\cos re \cdot \sinh im\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\cos re, \sinh im\right)\right) \]

   13. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \sinh im\right)\right) \]

   14. sinh-lowering-sinh.f64 99.9%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(re\right), \mathsf{sinh.f64}\left(im\right)\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Taylor expanded in re around 0

   \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(im\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 64.9%

   \[\leadsto -\color{blue}{1} \cdot \sinh im \]

10. Final simplification 64.9%

    \[\leadsto 0 - \sinh im \]
11. Add Preprocessing

Alternative 7: 54.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\\ t_1 := 0.5 \cdot t\_0\\ \mathbf{if}\;im \leq 5.6 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+132}:\\ \;\;\;\;t\_0 \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (+ -2.0 (* im (* im -0.3333333333333333)))))
        (t_1 (* 0.5 t_0)))
   (if (<= im 5.6e+20)
     t_1
     (if (<= im 3e+132) (* t_0 (+ 0.5 (* (* re re) -0.25))) t_1))))

C

double code(double re, double im) {
	double t_0 = im * (-2.0 + (im * (im * -0.3333333333333333)));
	double t_1 = 0.5 * t_0;
	double tmp;
	if (im <= 5.6e+20) {
		tmp = t_1;
	} else if (im <= 3e+132) {
		tmp = t_0 * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = im * ((-2.0d0) + (im * (im * (-0.3333333333333333d0))))
    t_1 = 0.5d0 * t_0
    if (im <= 5.6d+20) then
        tmp = t_1
    else if (im <= 3d+132) then
        tmp = t_0 * (0.5d0 + ((re * re) * (-0.25d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = im * (-2.0 + (im * (im * -0.3333333333333333)));
	double t_1 = 0.5 * t_0;
	double tmp;
	if (im <= 5.6e+20) {
		tmp = t_1;
	} else if (im <= 3e+132) {
		tmp = t_0 * (0.5 + ((re * re) * -0.25));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = im * (-2.0 + (im * (im * -0.3333333333333333)))
	t_1 = 0.5 * t_0
	tmp = 0
	if im <= 5.6e+20:
		tmp = t_1
	elif im <= 3e+132:
		tmp = t_0 * (0.5 + ((re * re) * -0.25))
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(im * Float64(-2.0 + Float64(im * Float64(im * -0.3333333333333333))))
	t_1 = Float64(0.5 * t_0)
	tmp = 0.0
	if (im <= 5.6e+20)
		tmp = t_1;
	elseif (im <= 3e+132)
		tmp = Float64(t_0 * Float64(0.5 + Float64(Float64(re * re) * -0.25)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = im * (-2.0 + (im * (im * -0.3333333333333333)));
	t_1 = 0.5 * t_0;
	tmp = 0.0;
	if (im <= 5.6e+20)
		tmp = t_1;
	elseif (im <= 3e+132)
		tmp = t_0 * (0.5 + ((re * re) * -0.25));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[(-2.0 + N[(im * N[(im * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * t$95$0), $MachinePrecision]}, If[LessEqual[im, 5.6e+20], t$95$1, If[LessEqual[im, 3e+132], N[(t$95$0 * N[(0.5 + N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if im < 5.6e20 or 2.9999999999999998e132 < im

   1. Initial program 51.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 93.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{3}}\right)\right)\right)\right)\right) \]

   5. Simplified 93.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 59.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]

   if 5.6e20 < im < 2.9999999999999998e132

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 25.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{3}}\right)\right)\right)\right)\right) \]

   5. Simplified 25.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{4} \cdot {re}^{2}\right)\right), \mathsf{*.f64}\left(\color{blue}{im}, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left({re}^{2} \cdot \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({re}^{2}\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 50.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{4}\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]

   8. Simplified 50.4%

      \[\leadsto \color{blue}{\left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 5.6 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 3 \cdot 10^{+132}:\\ \;\;\;\;\left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \cdot \left(0.5 + \left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.2% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \mathbf{if}\;im \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \frac{-1}{re \cdot re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* im (+ -2.0 (* im (* im -0.3333333333333333)))))))
   (if (<= im 1.75e+30)
     t_0
     (if (<= im 6e+103) (* (* re re) (* im (+ 0.5 (/ -1.0 (* re re))))) t_0))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))));
	double tmp;
	if (im <= 1.75e+30) {
		tmp = t_0;
	} else if (im <= 6e+103) {
		tmp = (re * re) * (im * (0.5 + (-1.0 / (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (im * ((-2.0d0) + (im * (im * (-0.3333333333333333d0)))))
    if (im <= 1.75d+30) then
        tmp = t_0
    else if (im <= 6d+103) then
        tmp = (re * re) * (im * (0.5d0 + ((-1.0d0) / (re * re))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))));
	double tmp;
	if (im <= 1.75e+30) {
		tmp = t_0;
	} else if (im <= 6e+103) {
		tmp = (re * re) * (im * (0.5 + (-1.0 / (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))))
	tmp = 0
	if im <= 1.75e+30:
		tmp = t_0
	elif im <= 6e+103:
		tmp = (re * re) * (im * (0.5 + (-1.0 / (re * re))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(im * Float64(-2.0 + Float64(im * Float64(im * -0.3333333333333333)))))
	tmp = 0.0
	if (im <= 1.75e+30)
		tmp = t_0;
	elseif (im <= 6e+103)
		tmp = Float64(Float64(re * re) * Float64(im * Float64(0.5 + Float64(-1.0 / Float64(re * re)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))));
	tmp = 0.0;
	if (im <= 1.75e+30)
		tmp = t_0;
	elseif (im <= 6e+103)
		tmp = (re * re) * (im * (0.5 + (-1.0 / (re * re))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(im * N[(-2.0 + N[(im * N[(im * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 1.75e+30], t$95$0, If[LessEqual[im, 6e+103], N[(N[(re * re), $MachinePrecision] * N[(im * N[(0.5 + N[(-1.0 / N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 1.75000000000000011e30 or 6e103 < im

   1. Initial program 52.7%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 92.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{3}}\right)\right)\right)\right)\right) \]

   5. Simplified 92.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 59.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]

   if 1.75000000000000011e30 < im < 6e103

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 3.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 3.8%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 31.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   8. Simplified 31.7%

      \[\leadsto 0 - im \cdot \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \]

   9. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({re}^{2}\right), \color{blue}{\left(-1 \cdot \frac{im}{{re}^{2}} - \frac{-1}{2} \cdot im\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(re \cdot re\right), \left(\color{blue}{-1 \cdot \frac{im}{{re}^{2}}} - \frac{-1}{2} \cdot im\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\color{blue}{-1 \cdot \frac{im}{{re}^{2}}} - \frac{-1}{2} \cdot im\right)\right) \]

       4. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(-1 \cdot \frac{im}{{re}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(-1 \cdot \frac{im}{{re}^{2}} + \frac{1}{2} \cdot im\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{-1 \cdot im}{{re}^{2}} + \color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(\frac{im \cdot -1}{{re}^{2}} + \frac{1}{2} \cdot im\right)\right) \]

       8. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \frac{-1}{{re}^{2}} + \color{blue}{\frac{1}{2}} \cdot im\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \frac{-1}{{re}^{2}} + im \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

       10. distribute-lft-out N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \left(im \cdot \color{blue}{\left(\frac{-1}{{re}^{2}} + \frac{1}{2}\right)}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{{re}^{2}} + \frac{1}{2}\right)}\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\left(\frac{-1}{{re}^{2}}\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left({re}^{2}\right)\right), \frac{1}{2}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left(re \cdot re\right)\right), \frac{1}{2}\right)\right)\right) \]

       15. *-lowering-*.f64 42.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(re, re\right)\right), \frac{1}{2}\right)\right)\right) \]

   11. Simplified 42.6%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot \left(\frac{-1}{re \cdot re} + 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;im \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot \left(0.5 + \frac{-1}{re \cdot re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.2% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right) \cdot \left(im \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (+
   -2.0
   (*
    (* im im)
    (+
     -0.3333333333333333
     (*
      (* im im)
      (+ -0.016666666666666666 (* (* im im) -0.0003968253968253968))))))
  (* im 0.5)))

C

double code(double re, double im) {
	return (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))) * (im * 0.5);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((-2.0d0) + ((im * im) * ((-0.3333333333333333d0) + ((im * im) * ((-0.016666666666666666d0) + ((im * im) * (-0.0003968253968253968d0))))))) * (im * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))) * (im * 0.5);
}

Python

def code(re, im):
	return (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))) * (im * 0.5)

Julia

function code(re, im)
	return Float64(Float64(-2.0 + Float64(Float64(im * im) * Float64(-0.3333333333333333 + Float64(Float64(im * im) * Float64(-0.016666666666666666 + Float64(Float64(im * im) * -0.0003968253968253968)))))) * Float64(im * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = (-2.0 + ((im * im) * (-0.3333333333333333 + ((im * im) * (-0.016666666666666666 + ((im * im) * -0.0003968253968253968)))))) * (im * 0.5);
end

Wolfram

code[re_, im_] := N[(N[(-2.0 + N[(N[(im * im), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(im * im), $MachinePrecision] * N[(-0.016666666666666666 + N[(N[(im * im), $MachinePrecision] * -0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) + -2\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{{im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)} - \frac{1}{3}\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) + \frac{-1}{3}\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{3} + \color{blue}{{im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \color{blue}{\left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)\right)}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right)}\right)\right)\right)\right)\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{-1}{2520} \cdot {im}^{2}} - \frac{1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{60}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{2520} \cdot {im}^{2} + \frac{-1}{60}\right)\right)\right)\right)\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{-1}{60} + \color{blue}{\frac{-1}{2520} \cdot {im}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \color{blue}{\left(\frac{-1}{2520} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \left({im}^{2} \cdot \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{-1}{2520}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   22. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

   23. *-lowering-*.f64 94.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{-1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{-1}{2520}\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 94.7%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{2} \cdot im\right) \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)} \]

   2. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right) \cdot \left(\color{blue}{{im}^{2}} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \]

   3. *-commutative N/A

      \[\leadsto \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot im\right)}\right) \]

8. Simplified 60.4%

   \[\leadsto \color{blue}{\left(-2 + \left(im \cdot im\right) \cdot \left(-0.3333333333333333 + \left(im \cdot im\right) \cdot \left(-0.016666666666666666 + \left(im \cdot im\right) \cdot -0.0003968253968253968\right)\right)\right) \cdot \left(im \cdot 0.5\right)} \]
9. Add Preprocessing

Alternative 10: 33.4% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+22}:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.1e+22) (- 0.0 im) (* im (+ -1.0 (* 0.5 (* re re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+22) {
		tmp = 0.0 - im;
	} else {
		tmp = im * (-1.0 + (0.5 * (re * 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 <= 2.1d+22) then
        tmp = 0.0d0 - im
    else
        tmp = im * ((-1.0d0) + (0.5d0 * (re * re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.1e+22) {
		tmp = 0.0 - im;
	} else {
		tmp = im * (-1.0 + (0.5 * (re * re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.1e+22:
		tmp = 0.0 - im
	else:
		tmp = im * (-1.0 + (0.5 * (re * re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.1e+22)
		tmp = Float64(0.0 - im);
	else
		tmp = Float64(im * Float64(-1.0 + Float64(0.5 * Float64(re * re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.1e+22)
		tmp = 0.0 - im;
	else
		tmp = im * (-1.0 + (0.5 * (re * re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.1e+22], N[(0.0 - im), $MachinePrecision], N[(im * N[(-1.0 + N[(0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.0999999999999998e22

   1. Initial program 41.4%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 64.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 64.9%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 36.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 36.2%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 36.2%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 36.2%

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

   if 2.0999999999999998e22 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 5.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 5.6%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 20.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   8. Simplified 20.5%

      \[\leadsto 0 - im \cdot \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \left(1 + \frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \left(re \cdot re\right)\right) \cdot im\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right)\right) \cdot \color{blue}{im} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right)\right), \color{blue}{im}\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right)\right), im\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right)\right), im\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(re \cdot re\right)\right)\right)\right), im\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\left(re \cdot re\right) \cdot \frac{-1}{2}\right)\right)\right), im\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left(\left(re \cdot re\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)\right)\right), im\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \left(\left(re \cdot re\right) \cdot \frac{1}{2}\right)\right), im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{1}{2}\right)\right), im\right) \]

      12. *-lowering-*.f64 20.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{1}{2}\right)\right), im\right) \]

   10. Applied egg-rr 20.5%

       \[\leadsto \color{blue}{\left(-1 + \left(re \cdot re\right) \cdot 0.5\right) \cdot im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.1 \cdot 10^{+22}:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-1 + 0.5 \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.8% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+32}:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.7e+32) (- 0.0 im) (* 0.5 (* im (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.7e+32) {
		tmp = 0.0 - im;
	} else {
		tmp = 0.5 * (im * (re * 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 <= 3.7d+32) then
        tmp = 0.0d0 - im
    else
        tmp = 0.5d0 * (im * (re * re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.7e+32) {
		tmp = 0.0 - im;
	} else {
		tmp = 0.5 * (im * (re * re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.7e+32:
		tmp = 0.0 - im
	else:
		tmp = 0.5 * (im * (re * re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.7e+32)
		tmp = Float64(0.0 - im);
	else
		tmp = Float64(0.5 * Float64(im * Float64(re * re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.7e+32)
		tmp = 0.0 - im;
	else
		tmp = 0.5 * (im * (re * re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.7e+32], N[(0.0 - im), $MachinePrecision], N[(0.5 * N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.7e32

   1. Initial program 42.6%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 63.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 63.7%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 35.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

   8. Simplified 35.6%

      \[\leadsto \color{blue}{0 - im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      2. neg-lowering-neg.f64 35.6%

         \[\leadsto \mathsf{neg.f64}\left(im\right) \]

   10. Applied egg-rr 35.6%

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

   if 3.7e32 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

      5. cos-lowering-cos.f64 5.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

   5. Simplified 5.7%

      \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot {re}^{2}\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left({re}^{2}\right)}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(re \cdot \color{blue}{re}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 21.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right)\right)\right) \]

   8. Simplified 21.7%

      \[\leadsto 0 - im \cdot \color{blue}{\left(1 + -0.5 \cdot \left(re \cdot re\right)\right)} \]

   9. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(im \cdot {re}^{2}\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left({re}^{2}\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(re \cdot \color{blue}{re}\right)\right)\right) \]

       4. *-lowering-*.f64 19.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(re, \color{blue}{re}\right)\right)\right) \]

   11. Simplified 19.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 3.7 \cdot 10^{+32}:\\ \;\;\;\;0 - im\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (* im (+ -2.0 (* im (* im -0.3333333333333333))))))

C

double code(double re, double im) {
	return 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (im * ((-2.0d0) + (im * (im * (-0.3333333333333333d0)))))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))));
}

Python

def code(re, im):
	return 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))))

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * Float64(-2.0 + Float64(im * Float64(im * -0.3333333333333333)))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * (-2.0 + (im * (im * -0.3333333333333333))));
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[(-2.0 + N[(im * N[(im * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(\frac{-1}{3} \cdot {im}^{2} + -2\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \left(-2 + \color{blue}{\frac{-1}{3} \cdot {im}^{2}}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \color{blue}{\left(\frac{-1}{3} \cdot {im}^{2}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left({im}^{2} \cdot \color{blue}{\frac{-1}{3}}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(\left(im \cdot im\right) \cdot \frac{-1}{3}\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \left(im \cdot \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \frac{-1}{3}\right)}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 87.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(re\right)\right), \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\frac{-1}{3}}\right)\right)\right)\right)\right) \]

5. Simplified 87.1%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(-2, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \frac{-1}{3}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 55.3%

   \[\leadsto \color{blue}{0.5} \cdot \left(im \cdot \left(-2 + im \cdot \left(im \cdot -0.3333333333333333\right)\right)\right) \]
9. Add Preprocessing

Alternative 13: 29.1% accurate, 103.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (- 0.0 im))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return 0.0 - im

Julia

function code(re, im)
	return Float64(0.0 - im)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(0.0 - im), $MachinePrecision]

Derivation

1. Initial program 55.8%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im \cdot \cos re\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{im \cdot \cos re} \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(im \cdot \cos re\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \color{blue}{\cos re}\right)\right) \]

   5. cos-lowering-cos.f64 50.3%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(im, \mathsf{cos.f64}\left(re\right)\right)\right) \]

5. Simplified 50.3%

   \[\leadsto \color{blue}{0 - im \cdot \cos re} \]

6. Taylor expanded in re around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-sub0 N/A

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

   3. --lowering--.f64 28.5%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{im}\right) \]

8. Simplified 28.5%

   \[\leadsto \color{blue}{0 - im} \]
9. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   2. neg-lowering-neg.f64 28.5%

      \[\leadsto \mathsf{neg.f64}\left(im\right) \]

10. Applied egg-rr 28.5%

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

11. Final simplification 28.5%

    \[\leadsto 0 - im \]
12. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024164 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (cos re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (cos re)) (- (exp (- 0 im)) (exp im)))))

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