math.sin on complex, imaginary part

Percentage Accurate: 54.7% → 99.9%

Time: 11.2s

Alternatives: 17

Speedup: 2.4×

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

Specification

Math

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

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: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

Alternative 2: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := \sinh \left(-im\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))))
        (t_1 (sinh (- im))))
   (if (<= t_0 -4e-8)
     t_1
     (if (<= t_0 5e-11)
       (* (- (cos re)) im)
       (* t_1 (fma (* re re) -0.5 1.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double t_1 = sinh(-im);
	double tmp;
	if (t_0 <= -4e-8) {
		tmp = t_1;
	} else if (t_0 <= 5e-11) {
		tmp = -cos(re) * im;
	} else {
		tmp = t_1 * fma((re * re), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = sinh(Float64(-im))
	tmp = 0.0
	if (t_0 <= -4e-8)
		tmp = t_1;
	elseif (t_0 <= 5e-11)
		tmp = Float64(Float64(-cos(re)) * im);
	else
		tmp = Float64(t_1 * fma(Float64(re * re), -0.5, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sinh[(-im)], $MachinePrecision]}, If[LessEqual[t$95$0, -4e-8], t$95$1, If[LessEqual[t$95$0, 5e-11], N[((-N[Cos[re], $MachinePrecision]) * im), $MachinePrecision], N[(t$95$1 * N[(N[(re * re), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -4.0000000000000001e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 77.8%

      \[\leadsto \sinh \left(-im\right) \]

   if -4.0000000000000001e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 5.00000000000000018e-11

   1. Initial program 6.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5.00000000000000018e-11 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 76.4

         \[\leadsto \left(\mathsf{fma}\left(-0.5, \color{blue}{re \cdot re}, 1\right) \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5 \]

   7. Applied rewrites 76.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(-0.5, re \cdot re, 1\right)} \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 76.4

          \[\leadsto \color{blue}{\sinh \left(-im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)} \]

   9. Applied rewrites 76.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (- (exp (- im)) (exp im)))))
   (if (<= t_0 -4e-8)
     (sinh (- im))
     (if (<= t_0 5e-11)
       (* (- (cos re)) im)
       (*
        (fma
         (-
          (*
           (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re)
           re)
          0.25)
         (* re re)
         0.5)
        (*
         (-
          (*
           (-
            (*
             (*
              (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
              im)
             im)
            0.3333333333333333)
           (* im im))
          2.0)
         im))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) - exp(im));
	double tmp;
	if (t_0 <= -4e-8) {
		tmp = sinh(-im);
	} else if (t_0 <= 5e-11) {
		tmp = -cos(re) * im;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im)))
	tmp = 0.0
	if (t_0 <= -4e-8)
		tmp = sinh(Float64(-im));
	elseif (t_0 <= 5e-11)
		tmp = Float64(Float64(-cos(re)) * im);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-8], N[Sinh[(-im)], $MachinePrecision], If[LessEqual[t$95$0, 5e-11], N[((-N[Cos[re], $MachinePrecision]) * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -4.0000000000000001e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 77.8%

      \[\leadsto \sinh \left(-im\right) \]

   if -4.0000000000000001e-8 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 5.00000000000000018e-11

   1. Initial program 6.9%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 5.00000000000000018e-11 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 99.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      15. lower-*.f64 65.4

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   11. Applied rewrites 69.4%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\left(-\cos re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (sinh (- im))
   (*
    (fma
     (-
      (* (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re) re)
      0.25)
     (* re re)
     0.5)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = sinh(-im);
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = sinh(Float64(-im));
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Sinh[(-im)], $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.2

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

   5. Applied rewrites 30.2%

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

   7. Applied rewrites 66.2%

      \[\leadsto \sinh \left(-im\right) \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      15. lower-*.f64 66.4

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   11. Applied rewrites 70.3%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\sinh \left(-im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    (-
     (*
      (*
       (-
        (*
         (* (- (* (* im im) -0.0001984126984126984) 0.008333333333333333) im)
         im)
        0.16666666666666666)
       im)
      im)
     1.0)
    im)
   (*
    (fma
     (-
      (* (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re) re)
      0.25)
     (* re re)
     0.5)
    (*
     (-
      (*
       (-
        (*
         (* (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666) im)
         im)
        0.3333333333333333)
       (* im im))
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * ((((((((-0.0003968253968253968 * (im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666) * im) * im) - 0.3333333333333333) * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.1%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      15. lower-*.f64 66.4

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right) \cdot im\right)} \]

   11. Applied rewrites 70.3%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \color{blue}{\left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666\right) \cdot im\right) \cdot im - 0.3333333333333333\right) \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    (-
     (*
      (*
       (-
        (*
         (* (- (* (* im im) -0.0001984126984126984) 0.008333333333333333) im)
         im)
        0.16666666666666666)
       im)
      im)
     1.0)
    im)
   (*
    (fma
     (-
      (* (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re) re)
      0.25)
     (* re re)
     0.5)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.1%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      15. lower-*.f64 66.4

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    (-
     (*
      (*
       (-
        (*
         (* (- (* (* im im) -0.0001984126984126984) 0.008333333333333333) im)
         im)
        0.16666666666666666)
       im)
      im)
     1.0)
    im)
   (*
    (fma
     (-
      (* (* (fma -0.0006944444444444445 (* re re) 0.020833333333333332) re) re)
      0.25)
     (* re re)
     0.5)
    (* (- (* -0.3333333333333333 (* im im)) 2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) - exp(im))) <= 0.0) {
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im;
	} else {
		tmp = fma((((fma(-0.0006944444444444445, (re * re), 0.020833333333333332) * re) * re) - 0.25), (re * re), 0.5) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) - exp(im))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im);
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332) * re) * re) - 0.25), Float64(re * re), 0.5) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.1%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im} \]

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right) \cdot re} - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot re\right)} \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)} \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      15. lower-*.f64 66.4

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, \color{blue}{re \cdot re}, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 66.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(im \cdot \left(\frac{-1}{3} \cdot {im}^{2} - 2\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {im}^{2} - 2\right) \cdot im\right)} \]

       3. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\color{blue}{\left(\frac{-1}{3} \cdot {im}^{2} - 2\right)} \cdot im\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\color{blue}{\frac{-1}{3} \cdot {im}^{2}} - 2\right) \cdot im\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right) \cdot re\right) \cdot re - \frac{1}{4}, re \cdot re, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

       6. lower-*.f64 58.1

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

   11. Applied rewrites 58.1%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right) \cdot re\right) \cdot re - 0.25, re \cdot re, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(im \cdot im\right) \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)}{im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (- (exp (- im)) (exp im))) 0.0)
   (*
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0)
    im)
   (/ (* (* im im) (fma (* 0.5 re) re -1.0)) im)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.0

   1. Initial program 38.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 30.2

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

   5. Applied rewrites 30.2%

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

   7. Applied rewrites 66.2%

      \[\leadsto \sinh \left(-im\right) \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 60.6%

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

   if 0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 97.4%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 9.8

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

   5. Applied rewrites 9.8%

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

   7. Applied rewrites 44.4%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 43.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right) \leq 0:\\ \;\;\;\;\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(im \cdot im\right) \cdot \mathsf{fma}\left(0.5 \cdot re, re, -1\right)}{im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right)\\ \mathbf{elif}\;t\_0 \leq 0.498:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right) \cdot re\right) \cdot re - 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= t_0 -0.005)
     (fma (* im (* 0.5 re)) re (- im))
     (if (<= t_0 0.498)
       (* (- (* (* (fma -0.041666666666666664 (* re re) 0.5) re) re) 1.0) im)
       (* (- (* -0.16666666666666666 (* im im)) 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = fma((im * (0.5 * re)), re, -im);
	} else if (t_0 <= 0.498) {
		tmp = (((fma(-0.041666666666666664, (re * re), 0.5) * re) * re) - 1.0) * im;
	} else {
		tmp = ((-0.16666666666666666 * (im * im)) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = fma(Float64(im * Float64(0.5 * re)), re, Float64(-im));
	elseif (t_0 <= 0.498)
		tmp = Float64(Float64(Float64(Float64(fma(-0.041666666666666664, Float64(re * re), 0.5) * re) * re) - 1.0) * im);
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(im * im)) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision], If[LessEqual[t$95$0, 0.498], N[(N[(N[(N[(N[(-0.041666666666666664 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.7%

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

   10. Applied rewrites 34.8%

       \[\leadsto \mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < 0.498

   1. Initial program 55.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 50.5

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 50.6%

      \[\leadsto \left(\left(\mathsf{fma}\left(-0.041666666666666664, re \cdot re, 0.5\right) \cdot re\right) \cdot re - 1\right) \cdot im \]

   if 0.498 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 52.6%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 80.1%

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

Alternative 10: 72.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.005:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.005)
   (*
    (* (* re re) -0.25)
    (*
     (-
      (* (* (- (* -0.016666666666666666 (* im im)) 0.3333333333333333) im) im)
      2.0)
     im))
   (*
    (-
     (*
      (*
       (-
        (*
         (* (- (* (* im im) -0.0001984126984126984) 0.008333333333333333) im)
         im)
        0.16666666666666666)
       im)
      im)
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.005) {
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * cos(re)) <= (-0.005d0)) then
        tmp = ((re * re) * (-0.25d0)) * (((((((-0.016666666666666666d0) * (im * im)) - 0.3333333333333333d0) * im) * im) - 2.0d0) * im)
    else
        tmp = (((((((((im * im) * (-0.0001984126984126984d0)) - 0.008333333333333333d0) * im) * im) - 0.16666666666666666d0) * im) * im) - 1.0d0) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.005) {
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	} else {
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.005:
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im)
	else:
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.005)
		tmp = Float64(Float64(Float64(re * re) * -0.25) * Float64(Float64(Float64(Float64(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.005)
		tmp = ((re * re) * -0.25) * ((((((-0.016666666666666666 * (im * im)) - 0.3333333333333333) * im) * im) - 2.0) * im);
	else
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 50.0%

       \[\leadsto \left(\left(re \cdot re\right) \cdot \color{blue}{-0.25}\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.5%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.5%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.005)
   (*
    (fma (* re re) -0.25 0.5)
    (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
   (*
    (-
     (*
      (*
       (-
        (*
         (* (- (* (* im im) -0.0001984126984126984) 0.008333333333333333) im)
         im)
        0.16666666666666666)
       im)
      im)
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.005) {
		tmp = fma((re * re), -0.25, 0.5) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else {
		tmp = (((((((((im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.005)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(im * im) * -0.0001984126984126984) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

       6. lower-*.f64 45.0

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

   11. Applied rewrites 45.0%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.5%

      \[\leadsto \left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \color{blue}{im} \]
   9. Step-by-step derivation

   10. Applied rewrites 81.5%

       \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.005)
   (*
    (fma (* re re) -0.25 0.5)
    (* (- (* -0.3333333333333333 (* im im)) 2.0) im))
   (*
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.005) {
		tmp = fma((re * re), -0.25, 0.5) * (((-0.3333333333333333 * (im * im)) - 2.0) * im);
	} else {
		tmp = (((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.005)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(Float64(Float64(-0.3333333333333333 * Float64(im * im)) - 2.0) * im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\left(\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

      5. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(\left(\left(\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333\right) \cdot im\right) \cdot im - 2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \cdot \left(\left(\frac{-1}{3} \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

       6. lower-*.f64 45.0

          \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\left(-0.3333333333333333 \cdot \color{blue}{\left(im \cdot im\right)} - 2\right) \cdot im\right) \]

   11. Applied rewrites 45.0%

       \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \color{blue}{\left(\left(-0.3333333333333333 \cdot \left(im \cdot im\right) - 2\right) \cdot im\right)} \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 53.1

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

   5. Applied rewrites 53.1%

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

   7. Applied rewrites 87.7%

      \[\leadsto \sinh \left(-im\right) \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 77.3%

       \[\leadsto \left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.005)
   (fma (* im (* 0.5 re)) re (- im))
   (*
    (-
     (* (* (- (* -0.008333333333333333 (* im im)) 0.16666666666666666) im) im)
     1.0)
    im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.005) {
		tmp = fma((im * (0.5 * re)), re, -im);
	} else {
		tmp = (((((-0.008333333333333333 * (im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.005)
		tmp = fma(Float64(im * Float64(0.5 * re)), re, Float64(-im));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-0.008333333333333333 * Float64(im * im)) - 0.16666666666666666) * im) * im) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.7%

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

   10. Applied rewrites 34.8%

       \[\leadsto \mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 53.1

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

   5. Applied rewrites 53.1%

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

   7. Applied rewrites 87.7%

      \[\leadsto \sinh \left(-im\right) \]

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 77.3%

       \[\leadsto \left(\left(\left(-0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666\right) \cdot im\right) \cdot im - 1\right) \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(im \cdot im\right) - 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.005)
   (fma (* im (* 0.5 re)) re (- im))
   (* (- (* -0.16666666666666666 (* im im)) 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.005) {
		tmp = fma((im * (0.5 * re)), re, -im);
	} else {
		tmp = ((-0.16666666666666666 * (im * im)) - 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.005)
		tmp = fma(Float64(im * Float64(0.5 * re)), re, Float64(-im));
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * Float64(im * im)) - 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.7%

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

   10. Applied rewrites 34.8%

       \[\leadsto \mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-exp.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 68.9%

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

Alternative 15: 38.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.005) (fma (* im (* 0.5 re)) re (- im)) (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.005) {
		tmp = fma((im * (0.5 * re)), re, -im);
	} else {
		tmp = -im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.005)
		tmp = fma(Float64(im * Float64(0.5 * re)), re, Float64(-im));
	else
		tmp = Float64(-im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(N[(im * N[(0.5 * re), $MachinePrecision]), $MachinePrecision] * re + (-im)), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.7%

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

   10. Applied rewrites 34.8%

       \[\leadsto \mathsf{fma}\left(im \cdot \left(0.5 \cdot re\right), re, -im\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 41.0%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 38.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.005:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.005) (* im (* (* re re) 0.5)) (- im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.005) {
		tmp = im * ((re * re) * 0.5);
	} else {
		tmp = -im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * cos(re)) <= (-0.005d0)) then
        tmp = im * ((re * re) * 0.5d0)
    else
        tmp = -im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.005) {
		tmp = im * ((re * re) * 0.5);
	} else {
		tmp = -im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.005:
		tmp = im * ((re * re) * 0.5)
	else:
		tmp = -im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.005)
		tmp = Float64(im * Float64(Float64(re * re) * 0.5));
	else
		tmp = Float64(-im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.005)
		tmp = im * ((re * re) * 0.5);
	else
		tmp = -im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(im * N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], (-im)]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0050000000000000001

   1. Initial program 58.2%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 34.7%

      \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot re, re, -1\right)} \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 34.7%

       \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right) \]

   if -0.0050000000000000001 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 53.3%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-cos.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 41.0%

      \[\leadsto -im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.6% accurate, 105.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return -im

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-cos.f64 51.9

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

5. Applied rewrites 51.9%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.3%

   \[\leadsto -im \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× 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 2024332 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

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

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