math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 14.8s

Alternatives: 15

Speedup: 1.5×

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

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(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 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(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 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \cosh im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sin re) (cosh im)))

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return math.sin(re) * math.cosh(im)

Julia

function code(re, im)
	return Float64(sin(re) * cosh(im))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. sub0-neg N/A

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

   7. cosh-undef N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

   10. exp-0 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

   12. exp-0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

   14. sin-lowering-sin.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

4. Applied egg-rr 100.0%

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

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

   5. cosh-lowering-cosh.f64 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
7. Add Preprocessing

Alternative 2: 78.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{-81}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5e-81)
   (* re (cosh im))
   (*
    (sin re)
    (+
     1.0
     (*
      (* im im)
      (+
       0.5
       (*
        (* im im)
        (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5e-81) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5d-81) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5e-81) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5e-81:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5e-81)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5e-81)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5e-81], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.99999999999999981e-81

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 67.3%

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

   if 4.99999999999999981e-81 < re

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}, \mathsf{sin.f64}\left(re\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(\color{blue}{re}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

      13. *-lowering-*.f64 93.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   7. Simplified 93.1%

      \[\leadsto \color{blue}{\left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)} \cdot \sin re \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{-81}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00325:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00325)
   (* (* (sin re) 0.5) (+ (* im im) 2.0))
   (if (<= im 2.6e+77)
     (* re (cosh im))
     (* (sin re) (* im (* im (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00325) {
		tmp = (sin(re) * 0.5) * ((im * im) + 2.0);
	} else if (im <= 2.6e+77) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 0.00325d0) then
        tmp = (sin(re) * 0.5d0) * ((im * im) + 2.0d0)
    else if (im <= 2.6d+77) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (im * (im * ((im * im) * 0.041666666666666664d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00325) {
		tmp = (Math.sin(re) * 0.5) * ((im * im) + 2.0);
	} else if (im <= 2.6e+77) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (im * (im * ((im * im) * 0.041666666666666664)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00325:
		tmp = (math.sin(re) * 0.5) * ((im * im) + 2.0)
	elif im <= 2.6e+77:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (im * (im * ((im * im) * 0.041666666666666664)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00325)
		tmp = Float64(Float64(sin(re) * 0.5) * Float64(Float64(im * im) + 2.0));
	elseif (im <= 2.6e+77)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00325)
		tmp = (sin(re) * 0.5) * ((im * im) + 2.0);
	elseif (im <= 2.6e+77)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (im * (im * ((im * im) * 0.041666666666666664)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00325], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.6e+77], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00324999999999999985

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 83.0%

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

   5. Simplified 83.0%

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

   if 0.00324999999999999985 < im < 2.6000000000000002e77

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 59.0%

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

   if 2.6000000000000002e77 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({im}^{4} \cdot \sin re\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{24} \cdot {im}^{4}\right) \cdot \color{blue}{\sin re} \]

      2. *-commutative N/A

         \[\leadsto \sin re \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right) \]

      4. pow-sqr N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \sin re \cdot \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right) \]

      6. *-commutative N/A

         \[\leadsto \sin re \cdot \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00325:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5 \cdot 10^{-81}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5e-81)
   (* re (cosh im))
   (*
    (sin re)
    (+ 1.0 (* (* im im) (+ 0.5 (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5e-81) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 5d-81) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + ((im * im) * (0.5d0 + ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5e-81) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5e-81:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5e-81)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(Float64(im * im) * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5e-81)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + ((im * im) * (0.5 + ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5e-81], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(N[(im * im), $MachinePrecision] * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.99999999999999981e-81

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 67.3%

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

   if 4.99999999999999981e-81 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{if}\;im \leq 0.00325:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* (sin re) 0.5) (+ (* im im) 2.0))))
   (if (<= im 0.00325)
     t_0
     (if (<= im 1.35e+154)
       (*
        re
        (*
         (+
          2.0
          (*
           im
           (*
            im
            (+
             1.0
             (*
              im
              (*
               im
               (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
         (+ 0.5 (* -0.08333333333333333 (* re re)))))
       t_0))))

C

double code(double re, double im) {
	double t_0 = (sin(re) * 0.5) * ((im * im) + 2.0);
	double tmp;
	if (im <= 0.00325) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (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 = (sin(re) * 0.5d0) * ((im * im) + 2.0d0)
    if (im <= 0.00325d0) then
        tmp = t_0
    else if (im <= 1.35d+154) then
        tmp = re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.sin(re) * 0.5) * ((im * im) + 2.0);
	double tmp;
	if (im <= 0.00325) {
		tmp = t_0;
	} else if (im <= 1.35e+154) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.sin(re) * 0.5) * ((im * im) + 2.0)
	tmp = 0
	if im <= 0.00325:
		tmp = t_0
	elif im <= 1.35e+154:
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(sin(re) * 0.5) * Float64(Float64(im * im) + 2.0))
	tmp = 0.0
	if (im <= 0.00325)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (sin(re) * 0.5) * ((im * im) + 2.0);
	tmp = 0.0;
	if (im <= 0.00325)
		tmp = t_0;
	elseif (im <= 1.35e+154)
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 0.00325], t$95$0, If[LessEqual[im, 1.35e+154], N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if im < 0.00324999999999999985 or 1.35000000000000003e154 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 85.9%

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

   5. Simplified 85.9%

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

   if 0.00324999999999999985 < im < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \left({im}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 77.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 77.5%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{-1}{12} \cdot \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}}\right), re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot \color{blue}{{re}^{2}}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 62.5%

      \[\leadsto \color{blue}{re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 0.00325:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot \left(im \cdot im + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;re \cdot \cosh im\\ \mathbf{else}:\\ \;\;\;\;\sin re \cdot \left(1 + im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 9.5e-81)
   (* re (cosh im))
   (* (sin re) (+ 1.0 (* im (* im (* (* im im) 0.041666666666666664)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 9.5e-81) {
		tmp = re * cosh(im);
	} else {
		tmp = sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 9.5d-81) then
        tmp = re * cosh(im)
    else
        tmp = sin(re) * (1.0d0 + (im * (im * ((im * im) * 0.041666666666666664d0))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 9.5e-81) {
		tmp = re * Math.cosh(im);
	} else {
		tmp = Math.sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 9.5e-81:
		tmp = re * math.cosh(im)
	else:
		tmp = math.sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 9.5e-81)
		tmp = Float64(re * cosh(im));
	else
		tmp = Float64(sin(re) * Float64(1.0 + Float64(im * Float64(im * Float64(Float64(im * im) * 0.041666666666666664)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 9.5e-81)
		tmp = re * cosh(im);
	else
		tmp = sin(re) * (1.0 + (im * (im * ((im * im) * 0.041666666666666664))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 9.5e-81], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision], N[(N[Sin[re], $MachinePrecision] * N[(1.0 + N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.49999999999999917e-81

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 67.3%

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

   if 9.49999999999999917e-81 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{24} \cdot {im}^{4}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot {im}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \color{blue}{{im}^{2}}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{{im}^{2}}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \left({im}^{2} \cdot \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 89.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right) \]

   8. Simplified 89.3%

      \[\leadsto \sin re \cdot \left(1 + \color{blue}{im \cdot \left(im \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00165:\\ \;\;\;\;\sin re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \cosh im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00165) (sin re) (* re (cosh im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00165) {
		tmp = sin(re);
	} else {
		tmp = re * cosh(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 <= 0.00165d0) then
        tmp = sin(re)
    else
        tmp = re * cosh(im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00165) {
		tmp = Math.sin(re);
	} else {
		tmp = re * Math.cosh(im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00165:
		tmp = math.sin(re)
	else:
		tmp = re * math.cosh(im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00165)
		tmp = sin(re);
	else
		tmp = Float64(re * cosh(im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00165)
		tmp = sin(re);
	else
		tmp = re * cosh(im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00165], N[Sin[re], $MachinePrecision], N[(re * N[Cosh[im], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 0.00165

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 67.0%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 67.0%

      \[\leadsto \color{blue}{\sin re} \]

   if 0.00165 < im

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 70.3%

      \[\leadsto \color{blue}{re} \cdot \cosh im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00172:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{+142}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \frac{0.5}{im \cdot im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00172)
   (sin re)
   (if (<= im 2.5e+142)
     (*
      re
      (*
       (+
        2.0
        (*
         im
         (*
          im
          (+
           1.0
           (*
            im
            (*
             im
             (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
       (+ 0.5 (* -0.08333333333333333 (* re re)))))
     (*
      re
      (+
       1.0
       (*
        (* (* im im) (* im im))
        (+ 0.041666666666666664 (/ 0.5 (* im im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00172) {
		tmp = sin(re);
	} else if (im <= 2.5e+142) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * 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 <= 0.00172d0) then
        tmp = sin(re)
    else if (im <= 2.5d+142) then
        tmp = re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else
        tmp = re * (1.0d0 + (((im * im) * (im * im)) * (0.041666666666666664d0 + (0.5d0 / (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 0.00172) {
		tmp = Math.sin(re);
	} else if (im <= 2.5e+142) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 0.00172:
		tmp = math.sin(re)
	elif im <= 2.5e+142:
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))))
	else:
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * im)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00172)
		tmp = sin(re);
	elseif (im <= 2.5e+142)
		tmp = Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(0.5 / Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 0.00172)
		tmp = sin(re);
	elseif (im <= 2.5e+142)
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	else
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00172], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.5e+142], N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(0.5 / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00171999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 67.0%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 67.0%

      \[\leadsto \color{blue}{\sin re} \]

   if 0.00171999999999999996 < im < 2.5000000000000001e142

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \left({im}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 76.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 76.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{-1}{12} \cdot \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}}\right), re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot \color{blue}{{re}^{2}}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 64.7%

      \[\leadsto \color{blue}{re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)} \]

   if 2.5000000000000001e142 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{24} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}} + \frac{1}{{im}^{4}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left({im}^{4} \cdot \left(\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) + \color{blue}{\frac{1}{{im}^{4}}}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left({im}^{4} \cdot \left(\frac{1}{{im}^{4}} + \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left({im}^{4} \cdot \frac{1}{{im}^{4}} + \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right) \]

      4. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(1 + \color{blue}{{im}^{4}} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{im}^{2}}}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{\frac{1}{2}}{{\color{blue}{im}}^{2}}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \frac{0.5}{im \cdot im}\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 69.2%

       \[\leadsto \color{blue}{re} \cdot \left(1 + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \frac{0.5}{im \cdot im}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 58.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 10^{+142}:\\ \;\;\;\;re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \frac{0.5}{im \cdot im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1e+142)
   (*
    re
    (*
     (+
      2.0
      (*
       im
       (*
        im
        (+
         1.0
         (*
          im
          (*
           im
           (+ 0.08333333333333333 (* (* im im) 0.002777777777777778))))))))
     (+ 0.5 (* -0.08333333333333333 (* re re)))))
   (*
    re
    (+
     1.0
     (* (* (* im im) (* im im)) (+ 0.041666666666666664 (/ 0.5 (* im im))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1e+142) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * 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 <= 1d+142) then
        tmp = re * ((2.0d0 + (im * (im * (1.0d0 + (im * (im * (0.08333333333333333d0 + ((im * im) * 0.002777777777777778d0)))))))) * (0.5d0 + ((-0.08333333333333333d0) * (re * re))))
    else
        tmp = re * (1.0d0 + (((im * im) * (im * im)) * (0.041666666666666664d0 + (0.5d0 / (im * im)))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1e+142) {
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	} else {
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * im)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1e+142:
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))))
	else:
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * im)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1e+142)
		tmp = Float64(re * Float64(Float64(2.0 + Float64(im * Float64(im * Float64(1.0 + Float64(im * Float64(im * Float64(0.08333333333333333 + Float64(Float64(im * im) * 0.002777777777777778)))))))) * Float64(0.5 + Float64(-0.08333333333333333 * Float64(re * re)))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(Float64(im * im) * Float64(im * im)) * Float64(0.041666666666666664 + Float64(0.5 / Float64(im * im))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1e+142)
		tmp = re * ((2.0 + (im * (im * (1.0 + (im * (im * (0.08333333333333333 + ((im * im) * 0.002777777777777778)))))))) * (0.5 + (-0.08333333333333333 * (re * re))));
	else
		tmp = re * (1.0 + (((im * im) * (im * im)) * (0.041666666666666664 + (0.5 / (im * im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1e+142], N[(re * N[(N[(2.0 + N[(im * N[(im * N[(1.0 + N[(im * N[(im * N[(0.08333333333333333 + N[(N[(im * im), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.08333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.041666666666666664 + N[(0.5 / N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.00000000000000005e142

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{1} + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(im \cdot im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(\color{blue}{\frac{1}{12}} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \color{blue}{\left(\frac{1}{360} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \left({im}^{2} \cdot \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{360}}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 94.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sin.f64}\left(re\right)\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{+.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{360}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 94.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left(im \cdot im\right) \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto re \cdot \left(\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      2. fma-define N/A

         \[\leadsto \mathsf{fma}\left(re, \color{blue}{\frac{-1}{12} \cdot \left({re}^{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \frac{-1}{12} \cdot \left(\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) \cdot \color{blue}{{re}^{2}}\right), re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(re, \left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot \color{blue}{{re}^{2}}, re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

      5. fma-undefine N/A

         \[\leadsto re \cdot \left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2}\right) + \color{blue}{re \cdot \left(\frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto re \cdot \color{blue}{\left(\left(\frac{-1}{12} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right) \cdot {re}^{2} + \frac{1}{2} \cdot \left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)\right)} \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{re \cdot \left(\left(2 + im \cdot \left(im \cdot \left(1 + im \cdot \left(im \cdot \left(0.08333333333333333 + \left(im \cdot im\right) \cdot 0.002777777777777778\right)\right)\right)\right)\right) \cdot \left(0.5 + -0.08333333333333333 \cdot \left(re \cdot re\right)\right)\right)} \]

   if 1.00000000000000005e142 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{24} + \left(\frac{1}{2} \cdot \frac{1}{{im}^{2}} + \frac{1}{{im}^{4}}\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left({im}^{4} \cdot \left(\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right) + \color{blue}{\frac{1}{{im}^{4}}}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left({im}^{4} \cdot \left(\frac{1}{{im}^{4}} + \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left({im}^{4} \cdot \frac{1}{{im}^{4}} + \color{blue}{{im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right) \]

      4. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \left(1 + \color{blue}{{im}^{4}} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{4} \cdot \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{4}\right), \color{blue}{\left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{\left(2 \cdot 2\right)}\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({im}^{2} \cdot {im}^{2}\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({im}^{2}\right), \left({im}^{2}\right)\right), \left(\color{blue}{\frac{1}{24}} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(im \cdot im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left({im}^{2}\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \left(im \cdot im\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \left(\frac{1}{24} + \frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{im}^{2}}\right)}\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{im}^{2}}}\right)\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{\frac{1}{2}}{{\color{blue}{im}}^{2}}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2}\right)}\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{im}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \frac{0.5}{im \cdot im}\right)\right)} \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{re}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \mathsf{*.f64}\left(im, im\right)\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, im\right)\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 69.2%

       \[\leadsto \color{blue}{re} \cdot \left(1 + \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \left(0.041666666666666664 + \frac{0.5}{im \cdot im}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+216}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e+216)
   (*
    re
    (+
     1.0
     (*
      im
      (*
       im
       (+
        0.5
        (*
         im
         (*
          im
          (+ 0.041666666666666664 (* (* im im) 0.001388888888888889)))))))))
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 6.2d+216) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + (im * (im * (0.041666666666666664d0 + ((im * im) * 0.001388888888888889d0))))))))
    else
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e+216:
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))))
	else:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e+216)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(im * Float64(im * Float64(0.041666666666666664 + Float64(Float64(im * im) * 0.001388888888888889)))))))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e+216)
		tmp = re * (1.0 + (im * (im * (0.5 + (im * (im * (0.041666666666666664 + ((im * im) * 0.001388888888888889))))))));
	else
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e+216], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(im * N[(im * N[(0.041666666666666664 + N[(N[(im * im), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.20000000000000007e216

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 60.5%

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

   10. Taylor expanded in im around 0

       \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{24}} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \left({im}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 58.1%

           \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{720}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 58.1%

       \[\leadsto re \cdot \color{blue}{\left(1 + im \cdot \left(im \cdot \left(0.5 + im \cdot \left(im \cdot \left(0.041666666666666664 + \left(im \cdot im\right) \cdot 0.001388888888888889\right)\right)\right)\right)\right)} \]

   if 6.20000000000000007e216 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 41.4%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 41.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 32.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 55.1% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+216}:\\ \;\;\;\;re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e+216)
   (* re (+ 1.0 (* im (* im (+ 0.5 (* (* im im) 0.041666666666666664))))))
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 6.2d+216) then
        tmp = re * (1.0d0 + (im * (im * (0.5d0 + ((im * im) * 0.041666666666666664d0)))))
    else
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e+216:
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))))
	else:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e+216)
		tmp = Float64(re * Float64(1.0 + Float64(im * Float64(im * Float64(0.5 + Float64(Float64(im * im) * 0.041666666666666664))))));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e+216)
		tmp = re * (1.0 + (im * (im * (0.5 + ((im * im) * 0.041666666666666664)))));
	else
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e+216], N[(re * N[(1.0 + N[(im * N[(im * N[(0.5 + N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.20000000000000007e216

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \sin re \cdot 1 + \color{blue}{{im}^{2}} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \]

      2. +-commutative N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \color{blue}{\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\frac{1}{2} \cdot \sin re + \left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \color{blue}{\sin re}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto \sin re \cdot 1 + {im}^{2} \cdot \left(\sin re \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sin re \cdot 1 + \left({im}^{2} \cdot \sin re\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sin re \cdot 1 + \left(\left({im}^{2} \cdot \sin re\right) \cdot \frac{1}{2} + \color{blue}{\left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \sin re \cdot 1 + \left(\left(\sin re \cdot {im}^{2}\right) \cdot \frac{1}{2} + \left(\color{blue}{{im}^{2}} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left({im}^{2} \cdot \frac{1}{2}\right) + \color{blue}{\left({im}^{2} \cdot \sin re\right)} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(\left(im \cdot im\right) \cdot \frac{1}{2}\right) + \left({im}^{2} \cdot \sin \color{blue}{re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(im \cdot \frac{1}{2}\right)\right) + \left({im}^{2} \cdot \color{blue}{\sin re}\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left({im}^{2} \cdot \sin re\right) \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \left(\sin re \cdot {im}^{2}\right) \cdot \left(\color{blue}{\frac{1}{24}} \cdot {im}^{2}\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sin re \cdot 1 + \left(\sin re \cdot \left(im \cdot \left(\frac{1}{2} \cdot im\right)\right) + \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

      14. distribute-lft-out N/A

          \[\leadsto \sin re \cdot 1 + \sin re \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} \cdot im\right) + {im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2}\right)\right)} \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\sin re \cdot \left(1 + \left(im \cdot im\right) \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(re, \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{24} \cdot {im}^{2}\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \left(im \cdot \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \color{blue}{\left(im \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right)}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \left({im}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({im}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(im \cdot im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 56.4%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(im, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(im, im\right), \frac{1}{24}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 56.4%

      \[\leadsto \color{blue}{re \cdot \left(1 + im \cdot \left(im \cdot \left(0.5 + \left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)\right)} \]

   if 6.20000000000000007e216 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 41.4%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 41.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 32.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.9% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+216}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e+216)
   (* re (+ 1.0 (* (* im im) 0.5)))
   (* re (+ 1.0 (* (* re re) -0.16666666666666666)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + ((im * im) * 0.5));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 6.2d+216) then
        tmp = re * (1.0d0 + ((im * im) * 0.5d0))
    else
        tmp = re * (1.0d0 + ((re * re) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + ((im * im) * 0.5));
	} else {
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e+216:
		tmp = re * (1.0 + ((im * im) * 0.5))
	else:
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e+216)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * 0.5)));
	else
		tmp = Float64(re * Float64(1.0 + Float64(Float64(re * re) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e+216)
		tmp = re * (1.0 + ((im * im) * 0.5));
	else
		tmp = re * (1.0 + ((re * re) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e+216], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(1.0 + N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.20000000000000007e216

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 60.5%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 45.7%

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

   12. Simplified 45.7%

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

   if 6.20000000000000007e216 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 41.4%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 41.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 32.5%

      \[\leadsto \color{blue}{re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+216}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(1 + \left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.9% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+216}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e+216)
   (* re (+ 1.0 (* (* im im) 0.5)))
   (* re (* (* re re) -0.16666666666666666))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + ((im * im) * 0.5));
	} else {
		tmp = re * ((re * re) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 6.2d+216) then
        tmp = re * (1.0d0 + ((im * im) * 0.5d0))
    else
        tmp = re * ((re * re) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6.2e+216) {
		tmp = re * (1.0 + ((im * im) * 0.5));
	} else {
		tmp = re * ((re * re) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e+216:
		tmp = re * (1.0 + ((im * im) * 0.5))
	else:
		tmp = re * ((re * re) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e+216)
		tmp = Float64(re * Float64(1.0 + Float64(Float64(im * im) * 0.5)));
	else
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6.2e+216)
		tmp = re * (1.0 + ((im * im) * 0.5));
	else
		tmp = re * ((re * re) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e+216], N[(re * N[(1.0 + N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.20000000000000007e216

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. sub0-neg N/A

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

      7. cosh-undef N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \cosh im\right), \sin re\right) \]

      10. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(e^{0} \cdot \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{0}\right), \cosh im\right), \sin \color{blue}{re}\right) \]

      12. exp-0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \cosh im\right), \sin re\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \sin re\right) \]

      14. sin-lowering-sin.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(1, \mathsf{cosh.f64}\left(im\right)\right), \mathsf{sin.f64}\left(re\right)\right) \]

   4. Applied egg-rr 100.0%

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\sin re, \color{blue}{\cosh im}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(re\right), \cosh \color{blue}{im}\right) \]

      5. cosh-lowering-cosh.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Simplified 60.5%

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

   10. Taylor expanded in im around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 45.7%

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

   12. Simplified 45.7%

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

   if 6.20000000000000007e216 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 41.4%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 41.4%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 32.5%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right) \]

       9. *-lowering-*.f64 32.5%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right) \]

   11. Simplified 32.5%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{+216}:\\ \;\;\;\;re \cdot \left(1 + \left(im \cdot im\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.2% accurate, 25.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.25 \cdot 10^{+125}:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;re \cdot \left(\left(re \cdot re\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.25e+125) re (* re (* (* re re) -0.16666666666666666))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.25e+125) {
		tmp = re;
	} else {
		tmp = re * ((re * re) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.25d+125) then
        tmp = re
    else
        tmp = re * ((re * re) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.25e+125) {
		tmp = re;
	} else {
		tmp = re * ((re * re) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.25e+125:
		tmp = re
	else:
		tmp = re * ((re * re) * -0.16666666666666666)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.25e+125)
		tmp = re;
	else
		tmp = Float64(re * Float64(Float64(re * re) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.25e+125)
		tmp = re;
	else
		tmp = re * ((re * re) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.25e+125], re, N[(re * N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.24999999999999991e125

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 51.3%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{re} \]
   7. Step-by-step derivation

   8. Simplified 25.8%

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

   if 1.24999999999999991e125 < re

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re} \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 44.9%

         \[\leadsto \mathsf{sin.f64}\left(re\right) \]

   5. Simplified 44.9%

      \[\leadsto \color{blue}{\sin re} \]

   6. Taylor expanded in re around 0

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

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

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 31.4%

         \[\leadsto \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right)\right) \]

   8. Simplified 31.4%

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

   9. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {re}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{re}\right) \]

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\left(re \cdot re\right), \frac{-1}{6}\right)\right) \]

       9. *-lowering-*.f64 31.4%

          \[\leadsto \mathsf{*.f64}\left(re, \mathsf{*.f64}\left(\mathsf{*.f64}\left(re, re\right), \frac{-1}{6}\right)\right) \]

   11. Simplified 31.4%

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

Alternative 15: 26.4% accurate, 309.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 re)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

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

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re} \]
4. Step-by-step derivation

   1. sin-lowering-sin.f64 50.4%

      \[\leadsto \mathsf{sin.f64}\left(re\right) \]

5. Simplified 50.4%

   \[\leadsto \color{blue}{\sin re} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{re} \]
7. Step-by-step derivation

8. Simplified 22.5%

   \[\leadsto \color{blue}{re} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))