math.sin on complex, imaginary part

Percentage Accurate: 55.6% → 99.4%

Time: 12.9s

Alternatives: 10

Speedup: 1.3×

• 51.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  0.5
  (*
   im
   (log1p
    (expm1
     (*
      (cos re)
      (fma
       (pow im 2.0)
       (fma (pow im 2.0) -0.016666666666666666 -0.3333333333333333)
       -2.0)))))))

C

double code(double re, double im) {
	return 0.5 * (im * log1p(expm1((cos(re) * fma(pow(im, 2.0), fma(pow(im, 2.0), -0.016666666666666666, -0.3333333333333333), -2.0)))));
}

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * log1p(expm1(Float64(cos(re) * fma((im ^ 2.0), fma((im ^ 2.0), -0.016666666666666666, -0.3333333333333333), -2.0))))))
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * N[Log[1 + N[(Exp[N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 2.0], $MachinePrecision] * N[(N[Power[im, 2.0], $MachinePrecision] * -0.016666666666666666 + -0.3333333333333333), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. /-rgt-identity 53.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 53.8%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 53.8%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 53.8%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 53.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 53.8%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 92.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 92.1%

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

   2. *-commutative 92.1%

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

   3. associate-*r* 92.1%

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

   4. distribute-rgt-out 92.1%

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

   5. +-commutative 92.1%

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

   6. metadata-eval 92.1%

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

   7. sub-neg 92.1%

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

   8. associate-*l* 92.1%

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

   9. *-commutative 92.1%

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

   10. distribute-lft-out 92.1%

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

   11. +-commutative 92.1%

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

   12. fma-define 92.1%

       \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333, -2\right)}\right)\right) \]

7. Simplified 92.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]
8. Step-by-step derivation

   1. log1p-expm1-u 99.5%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)}\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)}\right) \]
10. Add Preprocessing

Alternative 2: 50.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ t_1 := 0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ t_2 := {im}^{5} \cdot -0.008333333333333333\\ \mathbf{if}\;\cos re \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.65:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.73:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.765:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.82:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.826:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.85:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.9:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.91:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.912:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.92:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.94:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.942:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.96:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.965:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.97:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.99:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.9995:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* 0.5 (+ (* im -2.0) (* im (* -0.08333333333333333 (pow re 4.0))))))
        (t_1 (* 0.5 (* (cos re) (* im -2.0))))
        (t_2 (* (pow im 5.0) -0.008333333333333333)))
   (if (<= (cos re) 0.5)
     t_1
     (if (<= (cos re) 0.65)
       (* 0.5 (* im (- (* (pow im 2.0) -0.3333333333333333) 2.0)))
       (if (<= (cos re) 0.7)
         t_1
         (if (<= (cos re) 0.73)
           t_2
           (if (<= (cos re) 0.765)
             t_1
             (if (<= (cos re) 0.82)
               t_0
               (if (<= (cos re) 0.826)
                 t_1
                 (if (<= (cos re) 0.85)
                   t_2
                   (if (<= (cos re) 0.9)
                     t_1
                     (if (<= (cos re) 0.91)
                       t_0
                       (if (<= (cos re) 0.912)
                         t_1
                         (if (<= (cos re) 0.92)
                           t_0
                           (if (<= (cos re) 0.94)
                             t_1
                             (if (<= (cos re) 0.942)
                               t_2
                               (if (<= (cos re) 0.96)
                                 t_1
                                 (if (<= (cos re) 0.965)
                                   t_2
                                   (if (<= (cos re) 0.97)
                                     t_1
                                     (if (<= (cos re) 0.99)
                                       t_2
                                       (if (<= (cos re) 0.999)
                                         t_1
                                         (if (<= (cos re) 0.9995)
                                           t_2
                                           (if (<= (cos re) 1.0)
                                             t_1
                                             (*
                                              0.5
                                              (log1p
                                               (expm1
                                                (*
                                                 im
                                                 -2.0)))))))))))))))))))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * pow(re, 4.0))));
	double t_1 = 0.5 * (cos(re) * (im * -2.0));
	double t_2 = pow(im, 5.0) * -0.008333333333333333;
	double tmp;
	if (cos(re) <= 0.5) {
		tmp = t_1;
	} else if (cos(re) <= 0.65) {
		tmp = 0.5 * (im * ((pow(im, 2.0) * -0.3333333333333333) - 2.0));
	} else if (cos(re) <= 0.7) {
		tmp = t_1;
	} else if (cos(re) <= 0.73) {
		tmp = t_2;
	} else if (cos(re) <= 0.765) {
		tmp = t_1;
	} else if (cos(re) <= 0.82) {
		tmp = t_0;
	} else if (cos(re) <= 0.826) {
		tmp = t_1;
	} else if (cos(re) <= 0.85) {
		tmp = t_2;
	} else if (cos(re) <= 0.9) {
		tmp = t_1;
	} else if (cos(re) <= 0.91) {
		tmp = t_0;
	} else if (cos(re) <= 0.912) {
		tmp = t_1;
	} else if (cos(re) <= 0.92) {
		tmp = t_0;
	} else if (cos(re) <= 0.94) {
		tmp = t_1;
	} else if (cos(re) <= 0.942) {
		tmp = t_2;
	} else if (cos(re) <= 0.96) {
		tmp = t_1;
	} else if (cos(re) <= 0.965) {
		tmp = t_2;
	} else if (cos(re) <= 0.97) {
		tmp = t_1;
	} else if (cos(re) <= 0.99) {
		tmp = t_2;
	} else if (cos(re) <= 0.999) {
		tmp = t_1;
	} else if (cos(re) <= 0.9995) {
		tmp = t_2;
	} else if (cos(re) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * Math.pow(re, 4.0))));
	double t_1 = 0.5 * (Math.cos(re) * (im * -2.0));
	double t_2 = Math.pow(im, 5.0) * -0.008333333333333333;
	double tmp;
	if (Math.cos(re) <= 0.5) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.65) {
		tmp = 0.5 * (im * ((Math.pow(im, 2.0) * -0.3333333333333333) - 2.0));
	} else if (Math.cos(re) <= 0.7) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.73) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.765) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.82) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.826) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.85) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.9) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.91) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.912) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.92) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.94) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.942) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.96) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.965) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.97) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.99) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.999) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.9995) {
		tmp = t_2;
	} else if (Math.cos(re) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * math.pow(re, 4.0))))
	t_1 = 0.5 * (math.cos(re) * (im * -2.0))
	t_2 = math.pow(im, 5.0) * -0.008333333333333333
	tmp = 0
	if math.cos(re) <= 0.5:
		tmp = t_1
	elif math.cos(re) <= 0.65:
		tmp = 0.5 * (im * ((math.pow(im, 2.0) * -0.3333333333333333) - 2.0))
	elif math.cos(re) <= 0.7:
		tmp = t_1
	elif math.cos(re) <= 0.73:
		tmp = t_2
	elif math.cos(re) <= 0.765:
		tmp = t_1
	elif math.cos(re) <= 0.82:
		tmp = t_0
	elif math.cos(re) <= 0.826:
		tmp = t_1
	elif math.cos(re) <= 0.85:
		tmp = t_2
	elif math.cos(re) <= 0.9:
		tmp = t_1
	elif math.cos(re) <= 0.91:
		tmp = t_0
	elif math.cos(re) <= 0.912:
		tmp = t_1
	elif math.cos(re) <= 0.92:
		tmp = t_0
	elif math.cos(re) <= 0.94:
		tmp = t_1
	elif math.cos(re) <= 0.942:
		tmp = t_2
	elif math.cos(re) <= 0.96:
		tmp = t_1
	elif math.cos(re) <= 0.965:
		tmp = t_2
	elif math.cos(re) <= 0.97:
		tmp = t_1
	elif math.cos(re) <= 0.99:
		tmp = t_2
	elif math.cos(re) <= 0.999:
		tmp = t_1
	elif math.cos(re) <= 0.9995:
		tmp = t_2
	elif math.cos(re) <= 1.0:
		tmp = t_1
	else:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(im * Float64(-0.08333333333333333 * (re ^ 4.0)))))
	t_1 = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)))
	t_2 = Float64((im ^ 5.0) * -0.008333333333333333)
	tmp = 0.0
	if (cos(re) <= 0.5)
		tmp = t_1;
	elseif (cos(re) <= 0.65)
		tmp = Float64(0.5 * Float64(im * Float64(Float64((im ^ 2.0) * -0.3333333333333333) - 2.0)));
	elseif (cos(re) <= 0.7)
		tmp = t_1;
	elseif (cos(re) <= 0.73)
		tmp = t_2;
	elseif (cos(re) <= 0.765)
		tmp = t_1;
	elseif (cos(re) <= 0.82)
		tmp = t_0;
	elseif (cos(re) <= 0.826)
		tmp = t_1;
	elseif (cos(re) <= 0.85)
		tmp = t_2;
	elseif (cos(re) <= 0.9)
		tmp = t_1;
	elseif (cos(re) <= 0.91)
		tmp = t_0;
	elseif (cos(re) <= 0.912)
		tmp = t_1;
	elseif (cos(re) <= 0.92)
		tmp = t_0;
	elseif (cos(re) <= 0.94)
		tmp = t_1;
	elseif (cos(re) <= 0.942)
		tmp = t_2;
	elseif (cos(re) <= 0.96)
		tmp = t_1;
	elseif (cos(re) <= 0.965)
		tmp = t_2;
	elseif (cos(re) <= 0.97)
		tmp = t_1;
	elseif (cos(re) <= 0.99)
		tmp = t_2;
	elseif (cos(re) <= 0.999)
		tmp = t_1;
	elseif (cos(re) <= 0.9995)
		tmp = t_2;
	elseif (cos(re) <= 1.0)
		tmp = t_1;
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(im * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], 0.5], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.65], N[(0.5 * N[(im * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.7], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.73], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.765], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.82], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.826], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.85], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.9], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.91], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.912], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.92], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.94], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.942], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.96], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.965], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.97], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.99], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.999], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.9995], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 1.0], t$95$1, N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (cos.f64 re) < 0.5 or 0.650000000000000022 < (cos.f64 re) < 0.69999999999999996 or 0.72999999999999998 < (cos.f64 re) < 0.765000000000000013 or 0.819999999999999951 < (cos.f64 re) < 0.825999999999999956 or 0.849999999999999978 < (cos.f64 re) < 0.900000000000000022 or 0.910000000000000031 < (cos.f64 re) < 0.912000000000000033 or 0.92000000000000004 < (cos.f64 re) < 0.93999999999999995 or 0.94199999999999995 < (cos.f64 re) < 0.95999999999999996 or 0.964999999999999969 < (cos.f64 re) < 0.96999999999999997 or 0.98999999999999999 < (cos.f64 re) < 0.998999999999999999 or 0.99950000000000006 < (cos.f64 re) < 1

   1. Initial program 49.8%

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

      1. /-rgt-identity 49.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 49.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 49.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 49.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 49.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 49.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 49.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 49.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 49.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 49.8%

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

      11. cos-neg 49.8%

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

   3. Simplified 49.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 56.8%

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

   if 0.5 < (cos.f64 re) < 0.650000000000000022

   1. Initial program 71.9%

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

      1. /-rgt-identity 71.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 71.9%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 71.9%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 71.9%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 71.9%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 71.9%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 71.9%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 71.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 71.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 71.9%

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

      11. cos-neg 71.9%

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

   3. Simplified 71.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 99.7%

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

   6. Taylor expanded in re around 0 75.7%

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

   if 0.69999999999999996 < (cos.f64 re) < 0.72999999999999998 or 0.825999999999999956 < (cos.f64 re) < 0.849999999999999978 or 0.93999999999999995 < (cos.f64 re) < 0.94199999999999995 or 0.95999999999999996 < (cos.f64 re) < 0.964999999999999969 or 0.96999999999999997 < (cos.f64 re) < 0.98999999999999999 or 0.998999999999999999 < (cos.f64 re) < 0.99950000000000006

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

      12. fma-define 100.0%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333, -2\right)}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in re around 0 100.0%

       \[\leadsto \color{blue}{-0.008333333333333333 \cdot {im}^{5}} \]
   12. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]

   if 0.765000000000000013 < (cos.f64 re) < 0.819999999999999951 or 0.900000000000000022 < (cos.f64 re) < 0.910000000000000031 or 0.912000000000000033 < (cos.f64 re) < 0.92000000000000004

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 4.3%

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

   6. Taylor expanded in re around 0 100.0%

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

   7. Taylor expanded in re around inf 100.0%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}}\right) \]

      2. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

      3. associate-*l* 100.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   9. Simplified 100.0%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   if 1 < (cos.f64 re)

   1. Initial program 53.8%

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

      1. /-rgt-identity 53.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 53.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 53.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 53.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 53.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 53.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 53.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 53.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 53.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 53.8%

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

      11. cos-neg 53.8%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 52.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. pow2 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*l* 25.7%

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

   7. Applied egg-rr 25.7%

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

      1. log1p-expm1-u 49.0%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt{im \cdot \left(-2 \cdot \cos re\right)}\right)}^{2}\right)\right)} \]

      2. unpow2 49.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{im \cdot \left(-2 \cdot \cos re\right)} \cdot \sqrt{im \cdot \left(-2 \cdot \cos re\right)}}\right)\right) \]

      3. add-sqr-sqrt 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

      4. *-commutative 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(-2 \cdot \cos re\right) \cdot im}\right)\right) \]

      5. *-commutative 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\cos re \cdot -2\right)} \cdot im\right)\right) \]

      6. associate-*l* 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos re \cdot \left(-2 \cdot im\right)}\right)\right) \]

      7. *-commutative 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \color{blue}{\left(im \cdot -2\right)}\right)\right) \]

   9. Applied egg-rr 99.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(im \cdot -2\right)\right)\right)} \]

   10. Taylor expanded in re around 0 62.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-2 \cdot im}\right)\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.65:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.7:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.73:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.765:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.82:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.826:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.85:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.9:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.91:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.912:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.92:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.94:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.942:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.96:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.965:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.97:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.99:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.9995:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 1:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ t_1 := 0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ t_2 := {im}^{5} \cdot -0.008333333333333333\\ t_3 := 0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \mathbf{if}\;\cos re \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.65:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\cos re \leq 0.7:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.73:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.765:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.82:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.826:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.85:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.9:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.91:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.912:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.92:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.94:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.942:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.96:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.965:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.97:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.99:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.9995:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 1:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* 0.5 (+ (* im -2.0) (* im (* -0.08333333333333333 (pow re 4.0))))))
        (t_1 (* 0.5 (* (cos re) (* im -2.0))))
        (t_2 (* (pow im 5.0) -0.008333333333333333))
        (t_3 (* 0.5 (* im (- (* (pow im 2.0) -0.3333333333333333) 2.0)))))
   (if (<= (cos re) 0.5)
     t_1
     (if (<= (cos re) 0.65)
       t_3
       (if (<= (cos re) 0.7)
         t_1
         (if (<= (cos re) 0.73)
           t_2
           (if (<= (cos re) 0.765)
             t_1
             (if (<= (cos re) 0.82)
               t_0
               (if (<= (cos re) 0.826)
                 t_1
                 (if (<= (cos re) 0.85)
                   t_2
                   (if (<= (cos re) 0.9)
                     t_1
                     (if (<= (cos re) 0.91)
                       t_0
                       (if (<= (cos re) 0.912)
                         t_1
                         (if (<= (cos re) 0.92)
                           t_0
                           (if (<= (cos re) 0.94)
                             t_1
                             (if (<= (cos re) 0.942)
                               t_2
                               (if (<= (cos re) 0.96)
                                 t_1
                                 (if (<= (cos re) 0.965)
                                   t_2
                                   (if (<= (cos re) 0.97)
                                     t_1
                                     (if (<= (cos re) 0.99)
                                       t_2
                                       (if (<= (cos re) 0.999)
                                         t_1
                                         (if (<= (cos re) 0.9995)
                                           t_2
                                           (if (<= (cos re) 1.0)
                                             t_1
                                             t_3)))))))))))))))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * pow(re, 4.0))));
	double t_1 = 0.5 * (cos(re) * (im * -2.0));
	double t_2 = pow(im, 5.0) * -0.008333333333333333;
	double t_3 = 0.5 * (im * ((pow(im, 2.0) * -0.3333333333333333) - 2.0));
	double tmp;
	if (cos(re) <= 0.5) {
		tmp = t_1;
	} else if (cos(re) <= 0.65) {
		tmp = t_3;
	} else if (cos(re) <= 0.7) {
		tmp = t_1;
	} else if (cos(re) <= 0.73) {
		tmp = t_2;
	} else if (cos(re) <= 0.765) {
		tmp = t_1;
	} else if (cos(re) <= 0.82) {
		tmp = t_0;
	} else if (cos(re) <= 0.826) {
		tmp = t_1;
	} else if (cos(re) <= 0.85) {
		tmp = t_2;
	} else if (cos(re) <= 0.9) {
		tmp = t_1;
	} else if (cos(re) <= 0.91) {
		tmp = t_0;
	} else if (cos(re) <= 0.912) {
		tmp = t_1;
	} else if (cos(re) <= 0.92) {
		tmp = t_0;
	} else if (cos(re) <= 0.94) {
		tmp = t_1;
	} else if (cos(re) <= 0.942) {
		tmp = t_2;
	} else if (cos(re) <= 0.96) {
		tmp = t_1;
	} else if (cos(re) <= 0.965) {
		tmp = t_2;
	} else if (cos(re) <= 0.97) {
		tmp = t_1;
	} else if (cos(re) <= 0.99) {
		tmp = t_2;
	} else if (cos(re) <= 0.999) {
		tmp = t_1;
	} else if (cos(re) <= 0.9995) {
		tmp = t_2;
	} else if (cos(re) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 * ((im * (-2.0d0)) + (im * ((-0.08333333333333333d0) * (re ** 4.0d0))))
    t_1 = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    t_2 = (im ** 5.0d0) * (-0.008333333333333333d0)
    t_3 = 0.5d0 * (im * (((im ** 2.0d0) * (-0.3333333333333333d0)) - 2.0d0))
    if (cos(re) <= 0.5d0) then
        tmp = t_1
    else if (cos(re) <= 0.65d0) then
        tmp = t_3
    else if (cos(re) <= 0.7d0) then
        tmp = t_1
    else if (cos(re) <= 0.73d0) then
        tmp = t_2
    else if (cos(re) <= 0.765d0) then
        tmp = t_1
    else if (cos(re) <= 0.82d0) then
        tmp = t_0
    else if (cos(re) <= 0.826d0) then
        tmp = t_1
    else if (cos(re) <= 0.85d0) then
        tmp = t_2
    else if (cos(re) <= 0.9d0) then
        tmp = t_1
    else if (cos(re) <= 0.91d0) then
        tmp = t_0
    else if (cos(re) <= 0.912d0) then
        tmp = t_1
    else if (cos(re) <= 0.92d0) then
        tmp = t_0
    else if (cos(re) <= 0.94d0) then
        tmp = t_1
    else if (cos(re) <= 0.942d0) then
        tmp = t_2
    else if (cos(re) <= 0.96d0) then
        tmp = t_1
    else if (cos(re) <= 0.965d0) then
        tmp = t_2
    else if (cos(re) <= 0.97d0) then
        tmp = t_1
    else if (cos(re) <= 0.99d0) then
        tmp = t_2
    else if (cos(re) <= 0.999d0) then
        tmp = t_1
    else if (cos(re) <= 0.9995d0) then
        tmp = t_2
    else if (cos(re) <= 1.0d0) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * Math.pow(re, 4.0))));
	double t_1 = 0.5 * (Math.cos(re) * (im * -2.0));
	double t_2 = Math.pow(im, 5.0) * -0.008333333333333333;
	double t_3 = 0.5 * (im * ((Math.pow(im, 2.0) * -0.3333333333333333) - 2.0));
	double tmp;
	if (Math.cos(re) <= 0.5) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.65) {
		tmp = t_3;
	} else if (Math.cos(re) <= 0.7) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.73) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.765) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.82) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.826) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.85) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.9) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.91) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.912) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.92) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.94) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.942) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.96) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.965) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.97) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.99) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.999) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.9995) {
		tmp = t_2;
	} else if (Math.cos(re) <= 1.0) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * math.pow(re, 4.0))))
	t_1 = 0.5 * (math.cos(re) * (im * -2.0))
	t_2 = math.pow(im, 5.0) * -0.008333333333333333
	t_3 = 0.5 * (im * ((math.pow(im, 2.0) * -0.3333333333333333) - 2.0))
	tmp = 0
	if math.cos(re) <= 0.5:
		tmp = t_1
	elif math.cos(re) <= 0.65:
		tmp = t_3
	elif math.cos(re) <= 0.7:
		tmp = t_1
	elif math.cos(re) <= 0.73:
		tmp = t_2
	elif math.cos(re) <= 0.765:
		tmp = t_1
	elif math.cos(re) <= 0.82:
		tmp = t_0
	elif math.cos(re) <= 0.826:
		tmp = t_1
	elif math.cos(re) <= 0.85:
		tmp = t_2
	elif math.cos(re) <= 0.9:
		tmp = t_1
	elif math.cos(re) <= 0.91:
		tmp = t_0
	elif math.cos(re) <= 0.912:
		tmp = t_1
	elif math.cos(re) <= 0.92:
		tmp = t_0
	elif math.cos(re) <= 0.94:
		tmp = t_1
	elif math.cos(re) <= 0.942:
		tmp = t_2
	elif math.cos(re) <= 0.96:
		tmp = t_1
	elif math.cos(re) <= 0.965:
		tmp = t_2
	elif math.cos(re) <= 0.97:
		tmp = t_1
	elif math.cos(re) <= 0.99:
		tmp = t_2
	elif math.cos(re) <= 0.999:
		tmp = t_1
	elif math.cos(re) <= 0.9995:
		tmp = t_2
	elif math.cos(re) <= 1.0:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(Float64(im * -2.0) + Float64(im * Float64(-0.08333333333333333 * (re ^ 4.0)))))
	t_1 = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)))
	t_2 = Float64((im ^ 5.0) * -0.008333333333333333)
	t_3 = Float64(0.5 * Float64(im * Float64(Float64((im ^ 2.0) * -0.3333333333333333) - 2.0)))
	tmp = 0.0
	if (cos(re) <= 0.5)
		tmp = t_1;
	elseif (cos(re) <= 0.65)
		tmp = t_3;
	elseif (cos(re) <= 0.7)
		tmp = t_1;
	elseif (cos(re) <= 0.73)
		tmp = t_2;
	elseif (cos(re) <= 0.765)
		tmp = t_1;
	elseif (cos(re) <= 0.82)
		tmp = t_0;
	elseif (cos(re) <= 0.826)
		tmp = t_1;
	elseif (cos(re) <= 0.85)
		tmp = t_2;
	elseif (cos(re) <= 0.9)
		tmp = t_1;
	elseif (cos(re) <= 0.91)
		tmp = t_0;
	elseif (cos(re) <= 0.912)
		tmp = t_1;
	elseif (cos(re) <= 0.92)
		tmp = t_0;
	elseif (cos(re) <= 0.94)
		tmp = t_1;
	elseif (cos(re) <= 0.942)
		tmp = t_2;
	elseif (cos(re) <= 0.96)
		tmp = t_1;
	elseif (cos(re) <= 0.965)
		tmp = t_2;
	elseif (cos(re) <= 0.97)
		tmp = t_1;
	elseif (cos(re) <= 0.99)
		tmp = t_2;
	elseif (cos(re) <= 0.999)
		tmp = t_1;
	elseif (cos(re) <= 0.9995)
		tmp = t_2;
	elseif (cos(re) <= 1.0)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * ((im * -2.0) + (im * (-0.08333333333333333 * (re ^ 4.0))));
	t_1 = 0.5 * (cos(re) * (im * -2.0));
	t_2 = (im ^ 5.0) * -0.008333333333333333;
	t_3 = 0.5 * (im * (((im ^ 2.0) * -0.3333333333333333) - 2.0));
	tmp = 0.0;
	if (cos(re) <= 0.5)
		tmp = t_1;
	elseif (cos(re) <= 0.65)
		tmp = t_3;
	elseif (cos(re) <= 0.7)
		tmp = t_1;
	elseif (cos(re) <= 0.73)
		tmp = t_2;
	elseif (cos(re) <= 0.765)
		tmp = t_1;
	elseif (cos(re) <= 0.82)
		tmp = t_0;
	elseif (cos(re) <= 0.826)
		tmp = t_1;
	elseif (cos(re) <= 0.85)
		tmp = t_2;
	elseif (cos(re) <= 0.9)
		tmp = t_1;
	elseif (cos(re) <= 0.91)
		tmp = t_0;
	elseif (cos(re) <= 0.912)
		tmp = t_1;
	elseif (cos(re) <= 0.92)
		tmp = t_0;
	elseif (cos(re) <= 0.94)
		tmp = t_1;
	elseif (cos(re) <= 0.942)
		tmp = t_2;
	elseif (cos(re) <= 0.96)
		tmp = t_1;
	elseif (cos(re) <= 0.965)
		tmp = t_2;
	elseif (cos(re) <= 0.97)
		tmp = t_1;
	elseif (cos(re) <= 0.99)
		tmp = t_2;
	elseif (cos(re) <= 0.999)
		tmp = t_1;
	elseif (cos(re) <= 0.9995)
		tmp = t_2;
	elseif (cos(re) <= 1.0)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[(im * -2.0), $MachinePrecision] + N[(im * N[(-0.08333333333333333 * N[Power[re, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[(im * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], 0.5], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.65], t$95$3, If[LessEqual[N[Cos[re], $MachinePrecision], 0.7], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.73], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.765], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.82], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.826], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.85], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.9], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.91], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.912], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.92], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.94], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.942], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.96], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.965], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.97], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.99], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.999], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.9995], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 1.0], t$95$1, t$95$3]]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 re) < 0.5 or 0.650000000000000022 < (cos.f64 re) < 0.69999999999999996 or 0.72999999999999998 < (cos.f64 re) < 0.765000000000000013 or 0.819999999999999951 < (cos.f64 re) < 0.825999999999999956 or 0.849999999999999978 < (cos.f64 re) < 0.900000000000000022 or 0.910000000000000031 < (cos.f64 re) < 0.912000000000000033 or 0.92000000000000004 < (cos.f64 re) < 0.93999999999999995 or 0.94199999999999995 < (cos.f64 re) < 0.95999999999999996 or 0.964999999999999969 < (cos.f64 re) < 0.96999999999999997 or 0.98999999999999999 < (cos.f64 re) < 0.998999999999999999 or 0.99950000000000006 < (cos.f64 re) < 1

   1. Initial program 49.8%

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

      1. /-rgt-identity 49.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 49.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 49.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 49.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 49.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 49.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 49.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 49.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 49.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 49.8%

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

      11. cos-neg 49.8%

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

   3. Simplified 49.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 56.8%

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

   if 0.5 < (cos.f64 re) < 0.650000000000000022 or 1 < (cos.f64 re)

   1. Initial program 71.9%

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

      1. /-rgt-identity 71.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 71.9%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 71.9%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 71.9%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 71.9%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 71.9%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 71.9%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 71.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 71.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 71.9%

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

      11. cos-neg 71.9%

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

   3. Simplified 71.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 99.7%

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

   6. Taylor expanded in re around 0 75.7%

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

   if 0.69999999999999996 < (cos.f64 re) < 0.72999999999999998 or 0.825999999999999956 < (cos.f64 re) < 0.849999999999999978 or 0.93999999999999995 < (cos.f64 re) < 0.94199999999999995 or 0.95999999999999996 < (cos.f64 re) < 0.964999999999999969 or 0.96999999999999997 < (cos.f64 re) < 0.98999999999999999 or 0.998999999999999999 < (cos.f64 re) < 0.99950000000000006

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

      12. fma-define 100.0%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333, -2\right)}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in re around 0 100.0%

       \[\leadsto \color{blue}{-0.008333333333333333 \cdot {im}^{5}} \]
   12. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]

   if 0.765000000000000013 < (cos.f64 re) < 0.819999999999999951 or 0.900000000000000022 < (cos.f64 re) < 0.910000000000000031 or 0.912000000000000033 < (cos.f64 re) < 0.92000000000000004

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 4.3%

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

   6. Taylor expanded in re around 0 100.0%

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

   7. Taylor expanded in re around inf 100.0%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{-0.08333333333333333 \cdot \left(im \cdot {re}^{4}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(-0.08333333333333333 \cdot im\right) \cdot {re}^{4}}\right) \]

      2. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{\left(im \cdot -0.08333333333333333\right)} \cdot {re}^{4}\right) \]

      3. associate-*l* 100.0%

         \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]

   9. Simplified 100.0%

      \[\leadsto 0.5 \cdot \left(-2 \cdot im + \color{blue}{im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.65:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.7:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.73:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.765:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.82:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.826:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.85:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.9:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.91:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.912:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.92:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2 + im \cdot \left(-0.08333333333333333 \cdot {re}^{4}\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.94:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.942:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.96:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.965:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.97:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.99:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.9995:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 1:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ t_1 := {im}^{5} \cdot -0.008333333333333333\\ t_2 := 0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \mathbf{if}\;\cos re \leq 0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.65:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\cos re \leq 0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.73:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.765:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.826:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.85:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.91:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.912:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.92:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.94:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.942:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.96:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.965:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.97:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.99:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\cos re \leq 0.9995:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\cos re \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (cos re) (* im -2.0))))
        (t_1 (* (pow im 5.0) -0.008333333333333333))
        (t_2 (* 0.5 (* im (- (* (pow im 2.0) -0.3333333333333333) 2.0)))))
   (if (<= (cos re) 0.5)
     t_0
     (if (<= (cos re) 0.65)
       t_2
       (if (<= (cos re) 0.7)
         t_0
         (if (<= (cos re) 0.73)
           t_1
           (if (<= (cos re) 0.765)
             t_0
             (if (<= (cos re) 0.82)
               t_1
               (if (<= (cos re) 0.826)
                 t_0
                 (if (<= (cos re) 0.85)
                   t_1
                   (if (<= (cos re) 0.9)
                     t_0
                     (if (<= (cos re) 0.91)
                       t_1
                       (if (<= (cos re) 0.912)
                         t_0
                         (if (<= (cos re) 0.92)
                           t_1
                           (if (<= (cos re) 0.94)
                             t_0
                             (if (<= (cos re) 0.942)
                               t_1
                               (if (<= (cos re) 0.96)
                                 t_0
                                 (if (<= (cos re) 0.965)
                                   t_1
                                   (if (<= (cos re) 0.97)
                                     t_0
                                     (if (<= (cos re) 0.99)
                                       t_1
                                       (if (<= (cos re) 0.999)
                                         t_0
                                         (if (<= (cos re) 0.9995)
                                           t_1
                                           (if (<= (cos re) 1.0)
                                             t_0
                                             t_2)))))))))))))))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (cos(re) * (im * -2.0));
	double t_1 = pow(im, 5.0) * -0.008333333333333333;
	double t_2 = 0.5 * (im * ((pow(im, 2.0) * -0.3333333333333333) - 2.0));
	double tmp;
	if (cos(re) <= 0.5) {
		tmp = t_0;
	} else if (cos(re) <= 0.65) {
		tmp = t_2;
	} else if (cos(re) <= 0.7) {
		tmp = t_0;
	} else if (cos(re) <= 0.73) {
		tmp = t_1;
	} else if (cos(re) <= 0.765) {
		tmp = t_0;
	} else if (cos(re) <= 0.82) {
		tmp = t_1;
	} else if (cos(re) <= 0.826) {
		tmp = t_0;
	} else if (cos(re) <= 0.85) {
		tmp = t_1;
	} else if (cos(re) <= 0.9) {
		tmp = t_0;
	} else if (cos(re) <= 0.91) {
		tmp = t_1;
	} else if (cos(re) <= 0.912) {
		tmp = t_0;
	} else if (cos(re) <= 0.92) {
		tmp = t_1;
	} else if (cos(re) <= 0.94) {
		tmp = t_0;
	} else if (cos(re) <= 0.942) {
		tmp = t_1;
	} else if (cos(re) <= 0.96) {
		tmp = t_0;
	} else if (cos(re) <= 0.965) {
		tmp = t_1;
	} else if (cos(re) <= 0.97) {
		tmp = t_0;
	} else if (cos(re) <= 0.99) {
		tmp = t_1;
	} else if (cos(re) <= 0.999) {
		tmp = t_0;
	} else if (cos(re) <= 0.9995) {
		tmp = t_1;
	} else if (cos(re) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (cos(re) * (im * (-2.0d0)))
    t_1 = (im ** 5.0d0) * (-0.008333333333333333d0)
    t_2 = 0.5d0 * (im * (((im ** 2.0d0) * (-0.3333333333333333d0)) - 2.0d0))
    if (cos(re) <= 0.5d0) then
        tmp = t_0
    else if (cos(re) <= 0.65d0) then
        tmp = t_2
    else if (cos(re) <= 0.7d0) then
        tmp = t_0
    else if (cos(re) <= 0.73d0) then
        tmp = t_1
    else if (cos(re) <= 0.765d0) then
        tmp = t_0
    else if (cos(re) <= 0.82d0) then
        tmp = t_1
    else if (cos(re) <= 0.826d0) then
        tmp = t_0
    else if (cos(re) <= 0.85d0) then
        tmp = t_1
    else if (cos(re) <= 0.9d0) then
        tmp = t_0
    else if (cos(re) <= 0.91d0) then
        tmp = t_1
    else if (cos(re) <= 0.912d0) then
        tmp = t_0
    else if (cos(re) <= 0.92d0) then
        tmp = t_1
    else if (cos(re) <= 0.94d0) then
        tmp = t_0
    else if (cos(re) <= 0.942d0) then
        tmp = t_1
    else if (cos(re) <= 0.96d0) then
        tmp = t_0
    else if (cos(re) <= 0.965d0) then
        tmp = t_1
    else if (cos(re) <= 0.97d0) then
        tmp = t_0
    else if (cos(re) <= 0.99d0) then
        tmp = t_1
    else if (cos(re) <= 0.999d0) then
        tmp = t_0
    else if (cos(re) <= 0.9995d0) then
        tmp = t_1
    else if (cos(re) <= 1.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.cos(re) * (im * -2.0));
	double t_1 = Math.pow(im, 5.0) * -0.008333333333333333;
	double t_2 = 0.5 * (im * ((Math.pow(im, 2.0) * -0.3333333333333333) - 2.0));
	double tmp;
	if (Math.cos(re) <= 0.5) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.65) {
		tmp = t_2;
	} else if (Math.cos(re) <= 0.7) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.73) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.765) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.82) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.826) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.85) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.9) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.91) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.912) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.92) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.94) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.942) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.96) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.965) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.97) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.99) {
		tmp = t_1;
	} else if (Math.cos(re) <= 0.999) {
		tmp = t_0;
	} else if (Math.cos(re) <= 0.9995) {
		tmp = t_1;
	} else if (Math.cos(re) <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.cos(re) * (im * -2.0))
	t_1 = math.pow(im, 5.0) * -0.008333333333333333
	t_2 = 0.5 * (im * ((math.pow(im, 2.0) * -0.3333333333333333) - 2.0))
	tmp = 0
	if math.cos(re) <= 0.5:
		tmp = t_0
	elif math.cos(re) <= 0.65:
		tmp = t_2
	elif math.cos(re) <= 0.7:
		tmp = t_0
	elif math.cos(re) <= 0.73:
		tmp = t_1
	elif math.cos(re) <= 0.765:
		tmp = t_0
	elif math.cos(re) <= 0.82:
		tmp = t_1
	elif math.cos(re) <= 0.826:
		tmp = t_0
	elif math.cos(re) <= 0.85:
		tmp = t_1
	elif math.cos(re) <= 0.9:
		tmp = t_0
	elif math.cos(re) <= 0.91:
		tmp = t_1
	elif math.cos(re) <= 0.912:
		tmp = t_0
	elif math.cos(re) <= 0.92:
		tmp = t_1
	elif math.cos(re) <= 0.94:
		tmp = t_0
	elif math.cos(re) <= 0.942:
		tmp = t_1
	elif math.cos(re) <= 0.96:
		tmp = t_0
	elif math.cos(re) <= 0.965:
		tmp = t_1
	elif math.cos(re) <= 0.97:
		tmp = t_0
	elif math.cos(re) <= 0.99:
		tmp = t_1
	elif math.cos(re) <= 0.999:
		tmp = t_0
	elif math.cos(re) <= 0.9995:
		tmp = t_1
	elif math.cos(re) <= 1.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)))
	t_1 = Float64((im ^ 5.0) * -0.008333333333333333)
	t_2 = Float64(0.5 * Float64(im * Float64(Float64((im ^ 2.0) * -0.3333333333333333) - 2.0)))
	tmp = 0.0
	if (cos(re) <= 0.5)
		tmp = t_0;
	elseif (cos(re) <= 0.65)
		tmp = t_2;
	elseif (cos(re) <= 0.7)
		tmp = t_0;
	elseif (cos(re) <= 0.73)
		tmp = t_1;
	elseif (cos(re) <= 0.765)
		tmp = t_0;
	elseif (cos(re) <= 0.82)
		tmp = t_1;
	elseif (cos(re) <= 0.826)
		tmp = t_0;
	elseif (cos(re) <= 0.85)
		tmp = t_1;
	elseif (cos(re) <= 0.9)
		tmp = t_0;
	elseif (cos(re) <= 0.91)
		tmp = t_1;
	elseif (cos(re) <= 0.912)
		tmp = t_0;
	elseif (cos(re) <= 0.92)
		tmp = t_1;
	elseif (cos(re) <= 0.94)
		tmp = t_0;
	elseif (cos(re) <= 0.942)
		tmp = t_1;
	elseif (cos(re) <= 0.96)
		tmp = t_0;
	elseif (cos(re) <= 0.965)
		tmp = t_1;
	elseif (cos(re) <= 0.97)
		tmp = t_0;
	elseif (cos(re) <= 0.99)
		tmp = t_1;
	elseif (cos(re) <= 0.999)
		tmp = t_0;
	elseif (cos(re) <= 0.9995)
		tmp = t_1;
	elseif (cos(re) <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (cos(re) * (im * -2.0));
	t_1 = (im ^ 5.0) * -0.008333333333333333;
	t_2 = 0.5 * (im * (((im ^ 2.0) * -0.3333333333333333) - 2.0));
	tmp = 0.0;
	if (cos(re) <= 0.5)
		tmp = t_0;
	elseif (cos(re) <= 0.65)
		tmp = t_2;
	elseif (cos(re) <= 0.7)
		tmp = t_0;
	elseif (cos(re) <= 0.73)
		tmp = t_1;
	elseif (cos(re) <= 0.765)
		tmp = t_0;
	elseif (cos(re) <= 0.82)
		tmp = t_1;
	elseif (cos(re) <= 0.826)
		tmp = t_0;
	elseif (cos(re) <= 0.85)
		tmp = t_1;
	elseif (cos(re) <= 0.9)
		tmp = t_0;
	elseif (cos(re) <= 0.91)
		tmp = t_1;
	elseif (cos(re) <= 0.912)
		tmp = t_0;
	elseif (cos(re) <= 0.92)
		tmp = t_1;
	elseif (cos(re) <= 0.94)
		tmp = t_0;
	elseif (cos(re) <= 0.942)
		tmp = t_1;
	elseif (cos(re) <= 0.96)
		tmp = t_0;
	elseif (cos(re) <= 0.965)
		tmp = t_1;
	elseif (cos(re) <= 0.97)
		tmp = t_0;
	elseif (cos(re) <= 0.99)
		tmp = t_1;
	elseif (cos(re) <= 0.999)
		tmp = t_0;
	elseif (cos(re) <= 0.9995)
		tmp = t_1;
	elseif (cos(re) <= 1.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(im * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], 0.5], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.65], t$95$2, If[LessEqual[N[Cos[re], $MachinePrecision], 0.7], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.73], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.765], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.82], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.826], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.85], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.9], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.91], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.912], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.92], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.94], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.942], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.96], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.965], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.97], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.99], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 0.999], t$95$0, If[LessEqual[N[Cos[re], $MachinePrecision], 0.9995], t$95$1, If[LessEqual[N[Cos[re], $MachinePrecision], 1.0], t$95$0, t$95$2]]]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < 0.5 or 0.650000000000000022 < (cos.f64 re) < 0.69999999999999996 or 0.72999999999999998 < (cos.f64 re) < 0.765000000000000013 or 0.819999999999999951 < (cos.f64 re) < 0.825999999999999956 or 0.849999999999999978 < (cos.f64 re) < 0.900000000000000022 or 0.910000000000000031 < (cos.f64 re) < 0.912000000000000033 or 0.92000000000000004 < (cos.f64 re) < 0.93999999999999995 or 0.94199999999999995 < (cos.f64 re) < 0.95999999999999996 or 0.964999999999999969 < (cos.f64 re) < 0.96999999999999997 or 0.98999999999999999 < (cos.f64 re) < 0.998999999999999999 or 0.99950000000000006 < (cos.f64 re) < 1

   1. Initial program 49.8%

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

      1. /-rgt-identity 49.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 49.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 49.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 49.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 49.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 49.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 49.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 49.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 49.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 49.8%

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

      11. cos-neg 49.8%

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

   3. Simplified 49.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 56.8%

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

   if 0.5 < (cos.f64 re) < 0.650000000000000022 or 1 < (cos.f64 re)

   1. Initial program 71.9%

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

      1. /-rgt-identity 71.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 71.9%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 71.9%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 71.9%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 71.9%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 71.9%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 71.9%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 71.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 71.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 71.9%

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

      11. cos-neg 71.9%

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

   3. Simplified 71.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 99.7%

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

   6. Taylor expanded in re around 0 75.7%

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

   if 0.69999999999999996 < (cos.f64 re) < 0.72999999999999998 or 0.765000000000000013 < (cos.f64 re) < 0.819999999999999951 or 0.825999999999999956 < (cos.f64 re) < 0.849999999999999978 or 0.900000000000000022 < (cos.f64 re) < 0.910000000000000031 or 0.912000000000000033 < (cos.f64 re) < 0.92000000000000004 or 0.93999999999999995 < (cos.f64 re) < 0.94199999999999995 or 0.95999999999999996 < (cos.f64 re) < 0.964999999999999969 or 0.96999999999999997 < (cos.f64 re) < 0.98999999999999999 or 0.998999999999999999 < (cos.f64 re) < 0.99950000000000006

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 76.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

      4. distribute-rgt-out 76.0%

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

      5. +-commutative 76.0%

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

      6. metadata-eval 76.0%

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

      7. sub-neg 76.0%

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

      8. associate-*l* 76.0%

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

      9. *-commutative 76.0%

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

      10. distribute-lft-out 76.0%

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

      11. +-commutative 76.0%

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

      12. fma-define 76.0%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333, -2\right)}\right)\right) \]

   7. Simplified 76.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]

   8. Taylor expanded in im around inf 76.0%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 76.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 76.0%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in re around 0 76.0%

       \[\leadsto \color{blue}{-0.008333333333333333 \cdot {im}^{5}} \]
   12. Step-by-step derivation

       1. *-commutative 76.0%

          \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]

   13. Simplified 76.0%

       \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 0.5:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.65:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.7:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.73:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.765:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.82:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.826:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.85:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.9:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.91:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.912:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.92:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.94:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.942:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.96:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.965:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.97:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.99:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;\cos re \leq 0.9995:\\ \;\;\;\;{im}^{5} \cdot -0.008333333333333333\\ \mathbf{elif}\;\cos re \leq 1:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq 1:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) 1.0)
   (* 0.5 (* im (* (cos re) (- (* (pow im 2.0) -0.3333333333333333) 2.0))))
   (* 0.5 (log1p (expm1 (* im -2.0))))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= 1.0) {
		tmp = 0.5 * (im * (cos(re) * ((pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else {
		tmp = 0.5 * log1p(expm1((im * -2.0)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.cos(re) <= 1.0) {
		tmp = 0.5 * (im * (Math.cos(re) * ((Math.pow(im, 2.0) * -0.3333333333333333) - 2.0)));
	} else {
		tmp = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.cos(re) <= 1.0:
		tmp = 0.5 * (im * (math.cos(re) * ((math.pow(im, 2.0) * -0.3333333333333333) - 2.0)))
	else:
		tmp = 0.5 * math.log1p(math.expm1((im * -2.0)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= 1.0)
		tmp = Float64(0.5 * Float64(im * Float64(cos(re) * Float64(Float64((im ^ 2.0) * -0.3333333333333333) - 2.0))));
	else
		tmp = Float64(0.5 * log1p(expm1(Float64(im * -2.0))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], 1.0], N[(0.5 * N[(im * N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < 1

   1. Initial program 53.8%

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

      1. /-rgt-identity 53.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 53.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 53.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 53.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 53.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 53.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 53.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 53.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 53.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 53.8%

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

      11. cos-neg 53.8%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 92.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 92.1%

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

      2. *-commutative 92.1%

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

      3. associate-*r* 92.1%

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

      4. distribute-rgt-out 92.1%

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

      5. +-commutative 92.1%

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

      6. metadata-eval 92.1%

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

      7. sub-neg 92.1%

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

      8. associate-*l* 92.1%

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

      9. *-commutative 92.1%

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

      10. distribute-lft-out 92.1%

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

      11. +-commutative 92.1%

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

      12. fma-define 92.1%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333, -2\right)}\right)\right) \]

   7. Simplified 92.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]

   8. Taylor expanded in im around 0 85.2%

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

   if 1 < (cos.f64 re)

   1. Initial program 53.8%

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

      1. /-rgt-identity 53.8%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 53.8%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 53.8%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 53.8%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 53.8%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 53.8%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 53.8%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 53.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 53.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 53.8%

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

      11. cos-neg 53.8%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 52.7%

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

      1. add-sqr-sqrt 25.7%

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

      2. pow2 25.7%

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

      3. *-commutative 25.7%

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

      4. associate-*l* 25.7%

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

   7. Applied egg-rr 25.7%

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

      1. log1p-expm1-u 49.0%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt{im \cdot \left(-2 \cdot \cos re\right)}\right)}^{2}\right)\right)} \]

      2. unpow2 49.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{im \cdot \left(-2 \cdot \cos re\right)} \cdot \sqrt{im \cdot \left(-2 \cdot \cos re\right)}}\right)\right) \]

      3. add-sqr-sqrt 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

      4. *-commutative 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(-2 \cdot \cos re\right) \cdot im}\right)\right) \]

      5. *-commutative 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\cos re \cdot -2\right)} \cdot im\right)\right) \]

      6. associate-*l* 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos re \cdot \left(-2 \cdot im\right)}\right)\right) \]

      7. *-commutative 99.3%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \color{blue}{\left(im \cdot -2\right)}\right)\right) \]

   9. Applied egg-rr 99.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(im \cdot -2\right)\right)\right)} \]

   10. Taylor expanded in re around 0 62.8%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-2 \cdot im}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq 1:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\cos re \cdot \left({im}^{2} \cdot -0.3333333333333333 - 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(\cos re \cdot -2\right)\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (log1p (expm1 (* im (* (cos re) -2.0))))))

C

double code(double re, double im) {
	return 0.5 * log1p(expm1((im * (cos(re) * -2.0))));
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.log1p(Math.expm1((im * (Math.cos(re) * -2.0))));
}

Python

def code(re, im):
	return 0.5 * math.log1p(math.expm1((im * (math.cos(re) * -2.0))))

Julia

function code(re, im)
	return Float64(0.5 * log1p(expm1(Float64(im * Float64(cos(re) * -2.0)))))
end

Wolfram

code[re_, im_] := N[(0.5 * N[Log[1 + N[(Exp[N[(im * N[(N[Cos[re], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. /-rgt-identity 53.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 53.8%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 53.8%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 53.8%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 53.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 53.8%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 52.7%

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

   1. log1p-expm1-u 99.3%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-2 \cdot im\right) \cdot \cos re\right)\right)} \]

   2. *-commutative 99.3%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(im \cdot -2\right)} \cdot \cos re\right)\right) \]

   3. associate-*l* 99.3%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(-2 \cdot \cos re\right)\right)\right)} \]

8. Final simplification 99.3%

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot \left(\cos re \cdot -2\right)\right)\right) \]
9. Add Preprocessing

Alternative 7: 73.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ t_1 := 0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;im \leq 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (log1p (expm1 (* im -2.0)))))
        (t_1 (* 0.5 (* im (+ -2.0 (pow re 2.0))))))
   (if (<= im 480.0)
     (* 0.5 (* (cos re) (* im -2.0)))
     (if (<= im 2.3e+18)
       t_0
       (if (<= im 7.8e+18)
         t_1
         (if (<= im 9.5e+34)
           t_0
           (if (<= im 1e+35)
             t_1
             (if (<= im 4.5e+61)
               t_0
               (* (cos re) (* (pow im 5.0) -0.008333333333333333))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * log1p(expm1((im * -2.0)));
	double t_1 = 0.5 * (im * (-2.0 + pow(re, 2.0)));
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (cos(re) * (im * -2.0));
	} else if (im <= 2.3e+18) {
		tmp = t_0;
	} else if (im <= 7.8e+18) {
		tmp = t_1;
	} else if (im <= 9.5e+34) {
		tmp = t_0;
	} else if (im <= 1e+35) {
		tmp = t_1;
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = cos(re) * (pow(im, 5.0) * -0.008333333333333333);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.log1p(Math.expm1((im * -2.0)));
	double t_1 = 0.5 * (im * (-2.0 + Math.pow(re, 2.0)));
	double tmp;
	if (im <= 480.0) {
		tmp = 0.5 * (Math.cos(re) * (im * -2.0));
	} else if (im <= 2.3e+18) {
		tmp = t_0;
	} else if (im <= 7.8e+18) {
		tmp = t_1;
	} else if (im <= 9.5e+34) {
		tmp = t_0;
	} else if (im <= 1e+35) {
		tmp = t_1;
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = Math.cos(re) * (Math.pow(im, 5.0) * -0.008333333333333333);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.log1p(math.expm1((im * -2.0)))
	t_1 = 0.5 * (im * (-2.0 + math.pow(re, 2.0)))
	tmp = 0
	if im <= 480.0:
		tmp = 0.5 * (math.cos(re) * (im * -2.0))
	elif im <= 2.3e+18:
		tmp = t_0
	elif im <= 7.8e+18:
		tmp = t_1
	elif im <= 9.5e+34:
		tmp = t_0
	elif im <= 1e+35:
		tmp = t_1
	elif im <= 4.5e+61:
		tmp = t_0
	else:
		tmp = math.cos(re) * (math.pow(im, 5.0) * -0.008333333333333333)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * log1p(expm1(Float64(im * -2.0))))
	t_1 = Float64(0.5 * Float64(im * Float64(-2.0 + (re ^ 2.0))))
	tmp = 0.0
	if (im <= 480.0)
		tmp = Float64(0.5 * Float64(cos(re) * Float64(im * -2.0)));
	elseif (im <= 2.3e+18)
		tmp = t_0;
	elseif (im <= 7.8e+18)
		tmp = t_1;
	elseif (im <= 9.5e+34)
		tmp = t_0;
	elseif (im <= 1e+35)
		tmp = t_1;
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = Float64(cos(re) * Float64((im ^ 5.0) * -0.008333333333333333));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Log[1 + N[(Exp[N[(im * -2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(im * N[(-2.0 + N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 480.0], N[(0.5 * N[(N[Cos[re], $MachinePrecision] * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.3e+18], t$95$0, If[LessEqual[im, 7.8e+18], t$95$1, If[LessEqual[im, 9.5e+34], t$95$0, If[LessEqual[im, 1e+35], t$95$1, If[LessEqual[im, 4.5e+61], t$95$0, N[(N[Cos[re], $MachinePrecision] * N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 480

   1. Initial program 39.1%

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

      1. /-rgt-identity 39.1%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 39.1%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 39.1%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 39.1%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 39.1%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 39.1%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 39.1%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 39.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 39.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 39.1%

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

      11. cos-neg 39.1%

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

   3. Simplified 39.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 67.8%

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

   if 480 < im < 2.3e18 or 7.8e18 < im < 9.4999999999999999e34 or 9.9999999999999997e34 < im < 4.5e61

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 3.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*l* 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. log1p-expm1-u 0.0%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\sqrt{im \cdot \left(-2 \cdot \cos re\right)}\right)}^{2}\right)\right)} \]

      2. unpow2 0.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{im \cdot \left(-2 \cdot \cos re\right)} \cdot \sqrt{im \cdot \left(-2 \cdot \cos re\right)}}\right)\right) \]

      3. add-sqr-sqrt 100.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{im \cdot \left(-2 \cdot \cos re\right)}\right)\right) \]

      4. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(-2 \cdot \cos re\right) \cdot im}\right)\right) \]

      5. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\cos re \cdot -2\right)} \cdot im\right)\right) \]

      6. associate-*l* 100.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos re \cdot \left(-2 \cdot im\right)}\right)\right) \]

      7. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \color{blue}{\left(im \cdot -2\right)}\right)\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos re \cdot \left(im \cdot -2\right)\right)\right)} \]

   10. Taylor expanded in re around 0 100.0%

       \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-2 \cdot im}\right)\right) \]

   if 2.3e18 < im < 7.8e18 or 9.4999999999999999e34 < im < 9.9999999999999997e34

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 3.3%

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

   6. Taylor expanded in re around 0 68.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im + im \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 68.1%

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

      2. distribute-lft-out 68.1%

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

   8. Simplified 68.1%

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

   if 4.5e61 < im

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      2. exp-0 100.0%

         \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      4. cos-neg 100.0%

         \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

      6. associate-*r/ 100.0%

         \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

      7. exp-0 100.0%

         \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

      8. /-rgt-identity 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

      9. *-commutative 100.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

      10. neg-sub0 100.0%

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

      11. cos-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in im around 0 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. +-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. associate-*l* 100.0%

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

      9. *-commutative 100.0%

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

      10. distribute-lft-out 100.0%

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

      11. +-commutative 100.0%

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

      12. fma-define 100.0%

          \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333, -2\right)}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]

   8. Taylor expanded in im around inf 100.0%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

   11. Taylor expanded in im around 0 100.0%

       \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \cos re\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto \color{blue}{\left(-0.008333333333333333 \cdot {im}^{5}\right) \cdot \cos re} \]

       2. *-commutative 100.0%

          \[\leadsto \color{blue}{\left({im}^{5} \cdot -0.008333333333333333\right)} \cdot \cos re \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\left({im}^{5} \cdot -0.008333333333333333\right) \cdot \cos re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 480:\\ \;\;\;\;0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 7.8 \cdot 10^{+18}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{+34}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{elif}\;im \leq 10^{+35}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(-2 + {re}^{2}\right)\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(im \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{5} \cdot -0.008333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 0.5 * (Math.cos(re) * (im * -2.0));
}

Python

def code(re, im):
	return 0.5 * (math.cos(re) * (im * -2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. /-rgt-identity 53.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 53.8%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 53.8%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 53.8%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 53.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 53.8%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 52.7%

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

6. Final simplification 52.7%

   \[\leadsto 0.5 \cdot \left(\cos re \cdot \left(im \cdot -2\right)\right) \]
7. Add Preprocessing

Alternative 9: 34.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {im}^{5} \cdot -0.008333333333333333 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (pow im 5.0) -0.008333333333333333))

C

double code(double re, double im) {
	return pow(im, 5.0) * -0.008333333333333333;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (im ** 5.0d0) * (-0.008333333333333333d0)
end function

Java

public static double code(double re, double im) {
	return Math.pow(im, 5.0) * -0.008333333333333333;
}

Python

def code(re, im):
	return math.pow(im, 5.0) * -0.008333333333333333

Julia

function code(re, im)
	return Float64((im ^ 5.0) * -0.008333333333333333)
end

MATLAB

function tmp = code(re, im)
	tmp = (im ^ 5.0) * -0.008333333333333333;
end

Wolfram

code[re_, im_] := N[(N[Power[im, 5.0], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

   1. /-rgt-identity 53.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 53.8%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 53.8%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 53.8%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 53.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 53.8%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 92.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(-2 \cdot \cos re + {im}^{2} \cdot \left(-0.3333333333333333 \cdot \cos re + -0.016666666666666666 \cdot \left({im}^{2} \cdot \cos re\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 92.1%

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

   2. *-commutative 92.1%

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

   3. associate-*r* 92.1%

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

   4. distribute-rgt-out 92.1%

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

   5. +-commutative 92.1%

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

   6. metadata-eval 92.1%

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

   7. sub-neg 92.1%

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

   8. associate-*l* 92.1%

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

   9. *-commutative 92.1%

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

   10. distribute-lft-out 92.1%

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

   11. +-commutative 92.1%

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

   12. fma-define 92.1%

       \[\leadsto 0.5 \cdot \left(im \cdot \left(\cos re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, -0.016666666666666666 \cdot {im}^{2} - 0.3333333333333333, -2\right)}\right)\right) \]

7. Simplified 92.1%

   \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(\cos re \cdot \mathsf{fma}\left({im}^{2}, \mathsf{fma}\left({im}^{2}, -0.016666666666666666, -0.3333333333333333\right), -2\right)\right)\right)} \]

8. Taylor expanded in im around inf 44.8%

   \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(-0.016666666666666666 \cdot \left({im}^{4} \cdot \cos re\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* 44.8%

      \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

10. Simplified 44.8%

    \[\leadsto 0.5 \cdot \left(im \cdot \color{blue}{\left(\left(-0.016666666666666666 \cdot {im}^{4}\right) \cdot \cos re\right)}\right) \]

11. Taylor expanded in re around 0 34.1%

    \[\leadsto \color{blue}{-0.008333333333333333 \cdot {im}^{5}} \]
12. Step-by-step derivation

    1. *-commutative 34.1%

       \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]

13. Simplified 34.1%

    \[\leadsto \color{blue}{{im}^{5} \cdot -0.008333333333333333} \]
14. Add Preprocessing

Alternative 10: 29.2% accurate, 61.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. /-rgt-identity 53.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \cos re}{1}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. exp-0 53.8%

      \[\leadsto \frac{0.5 \cdot \cos re}{\color{blue}{e^{0}}} \cdot \left(e^{0 - im} - e^{im}\right) \]

   3. associate-*l/ 53.8%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   4. cos-neg 53.8%

      \[\leadsto \frac{\left(0.5 \cdot \color{blue}{\cos \left(-re\right)}\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}} \]

   5. associate-*l* 53.8%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)}}{e^{0}} \]

   6. associate-*r/ 53.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{e^{0}}} \]

   7. exp-0 53.8%

      \[\leadsto 0.5 \cdot \frac{\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)}{\color{blue}{1}} \]

   8. /-rgt-identity 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\cos \left(-re\right) \cdot \left(e^{0 - im} - e^{im}\right)\right)} \]

   9. *-commutative 53.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \cos \left(-re\right)\right)} \]

   10. neg-sub0 53.8%

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

   11. cos-neg 53.8%

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

3. Simplified 53.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot \cos re\right)} \]
4. Add Preprocessing

5. Taylor expanded in im around 0 52.7%

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

6. Taylor expanded in re around 0 27.7%

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

   1. *-commutative 27.7%

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

8. Simplified 27.7%

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

Developer target: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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