math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

Alternatives: 18

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.999999999997596:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cosh im \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma im im 2.0)
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5))
     (if (<= t_0 0.999999999997596) (cos re) (* 0.5 (* (cosh im) 2.0))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else if (t_0 <= 0.999999999997596) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (cosh(im) * 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	elseif (t_0 <= 0.999999999997596)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(cosh(im) * 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.999999999997596], N[Cos[re], $MachinePrecision], N[(0.5 * N[(N[Cosh[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999997596034

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.999999999997596034 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.999999999997596:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\cosh im \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma im im 2.0)
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5))
     (if (<= t_0 2.0) (cos re) (* 0.5 (+ 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else if (t_0 <= 2.0) {
		tmp = cos(re);
	} else {
		tmp = 0.5 * (1.0 + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	elseif (t_0 <= 2.0)
		tmp = cos(re);
	else
		tmp = Float64(0.5 * Float64(1.0 + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[Cos[re], $MachinePrecision], N[(0.5 * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 99.1

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

   8. Applied rewrites 99.1%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.9%

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

4. Final simplification 78.8%

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

Alternative 4: 93.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.999999999997596:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma im im 2.0)
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5))
     (if (<= t_0 0.999999999997596)
       (cos re)
       (*
        (*
         (fma
          (fma
           (fma 0.001388888888888889 (* im im) 0.041666666666666664)
           (* im im)
           0.5)
          (* im im)
          1.0)
         2.0)
        0.5)))))

C

double code(double re, double im) {
	double t_0 = (exp(im) + exp(-im)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else if (t_0 <= 0.999999999997596) {
		tmp = cos(re);
	} else {
		tmp = (fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	elseif (t_0 <= 0.999999999997596)
		tmp = cos(re);
	else
		tmp = Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.999999999997596], N[Cos[re], $MachinePrecision], N[(N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999997596034

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\cos re} \]
   7. Step-by-step derivation

      1. lower-cos.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.999999999997596034 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot 2\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot 2\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      13. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      14. lower-*.f64 92.5

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 92.5%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.999999999997596:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) -0.05)
   (* (* -0.25 (* re re)) (fma im im 2.0))
   (*
    (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) 2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= -0.05) {
		tmp = (-0.25 * (re * re)) * fma(im, im, 2.0);
	} else {
		tmp = (fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5)) <= -0.05)
		tmp = Float64(Float64(-0.25 * Float64(re * re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 56.7%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 80.9

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 80.9%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) -0.05)
   (* (* -0.25 (* re re)) (fma im im 2.0))
   (* (* (fma (* 0.041666666666666664 (* im im)) (* im im) 1.0) 2.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= -0.05) {
		tmp = (-0.25 * (re * re)) * fma(im, im, 2.0);
	} else {
		tmp = (fma((0.041666666666666664 * (im * im)), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5)) <= -0.05)
		tmp = Float64(Float64(-0.25 * Float64(re * re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(fma(Float64(0.041666666666666664 * Float64(im * im)), Float64(im * im), 1.0) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 56.7%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 80.9

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 80.9%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot 0.5 \]

   11. Taylor expanded in im around inf

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2}, \color{blue}{im} \cdot im, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 80.4%

       \[\leadsto \left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), \color{blue}{im} \cdot im, 1\right) \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) 2.0)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (* (* (* (* (fma 0.041666666666666664 (* im im) 0.5) im) im) 2.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= 2.0) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else {
		tmp = (((fma(0.041666666666666664, (im * im), 0.5) * im) * im) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5)) <= 2.0)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.041666666666666664, Float64(im * im), 0.5) * im) * im) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 89.0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.6

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

   8. Applied rewrites 69.6%

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

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 79.7

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 79.7%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot 0.5 \]

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 79.7%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot \color{blue}{im}\right) \cdot 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) -0.05)
   (* (* -0.25 (* re re)) (fma im im 2.0))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= -0.05) {
		tmp = (-0.25 * (re * re)) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5)) <= -0.05)
		tmp = Float64(Float64(-0.25 * Float64(re * re)) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 56.7%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;2 \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) -0.05)
   (* 2.0 (* -0.25 (* re re)))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= -0.05) {
		tmp = 2.0 * (-0.25 * (re * re));
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5)) <= -0.05)
		tmp = Float64(2.0 * Float64(-0.25 * Float64(re * re)));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.05], N[(2.0 * N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.3

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 29.3%

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

   if -0.050000000000000003 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 68.1%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.05:\\ \;\;\;\;2 \cdot \left(-0.25 \cdot \left(re \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;2 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) 2.0)
   (* 2.0 0.5)
   (* (* im im) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= 2.0) {
		tmp = 2.0 * 0.5;
	} else {
		tmp = (im * im) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((exp(im) + exp(-im)) * (cos(re) * 0.5d0)) <= 2.0d0) then
        tmp = 2.0d0 * 0.5d0
    else
        tmp = (im * im) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (((Math.exp(im) + Math.exp(-im)) * (Math.cos(re) * 0.5)) <= 2.0) {
		tmp = 2.0 * 0.5;
	} else {
		tmp = (im * im) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if ((math.exp(im) + math.exp(-im)) * (math.cos(re) * 0.5)) <= 2.0:
		tmp = 2.0 * 0.5
	else:
		tmp = (im * im) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(exp(im) + exp(Float64(-im))) * Float64(cos(re) * 0.5)) <= 2.0)
		tmp = Float64(2.0 * 0.5);
	else
		tmp = Float64(Float64(im * im) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((exp(im) + exp(-im)) * (cos(re) * 0.5)) <= 2.0)
		tmp = 2.0 * 0.5;
	else
		tmp = (im * im) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
   7. Step-by-step derivation

   8. Applied rewrites 48.1%

      \[\leadsto \color{blue}{0.5} \cdot 2 \]

   if 2 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 56.4

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

   10. Applied rewrites 56.4%

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

   11. Taylor expanded in im around inf

       \[\leadsto {im}^{\color{blue}{2}} \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.4%

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

4. Final simplification 51.5%

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

Alternative 11: 68.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\\ \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot t\_0\\ \mathbf{elif}\;\cos re \leq 0.999999999997596:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) 2.0)))
   (if (<= (cos re) -0.02)
     (* (fma (* re re) -0.25 0.5) t_0)
     (if (<= (cos re) 0.999999999997596)
       (*
        (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5)
        (fma im im 2.0))
       (* t_0 0.5)))))

C

double code(double re, double im) {
	double t_0 = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * 2.0;
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma((re * re), -0.25, 0.5) * t_0;
	} else if (cos(re) <= 0.999999999997596) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * fma(im, im, 2.0);
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * 2.0)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * t_0);
	elseif (cos(re) <= 0.999999999997596)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.999999999997596], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 0.7

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 0.7%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot 0.5 \]

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \]

       5. lower-*.f64 58.2

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \]

   13. Applied rewrites 58.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \]

   if -0.0200000000000000004 < (cos.f64 re) < 0.999999999997596034

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   if 0.999999999997596034 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 90.7

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 90.7%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right)\\ \mathbf{elif}\;\cos re \leq 0.999999999997596:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.999999999997596:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (* -0.25 (* re re)) (fma im im 2.0))
   (if (<= (cos re) 0.999999999997596)
     (*
      (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5)
      (fma im im 2.0))
     (*
      (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) 2.0)
      0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = (-0.25 * (re * re)) * fma(im, im, 2.0);
	} else if (cos(re) <= 0.999999999997596) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * fma(im, im, 2.0);
	} else {
		tmp = (fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(Float64(-0.25 * Float64(re * re)) * fma(im, im, 2.0));
	elseif (cos(re) <= 0.999999999997596)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.999999999997596], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 56.7%

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

   if -0.0200000000000000004 < (cos.f64 re) < 0.999999999997596034

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 69.5

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.0

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

   8. Applied rewrites 63.0%

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

   if 0.999999999997596034 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 100.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 90.7

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 90.7%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.999999999997596:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(1 + e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (+ 1.0 (exp im)) (* (cos re) 0.5)))

C

double code(double re, double im) {
	return (1.0 + exp(im)) * (cos(re) * 0.5);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (1.0d0 + exp(im)) * (cos(re) * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return (1.0 + Math.exp(im)) * (Math.cos(re) * 0.5);
}

Python

def code(re, im):
	return (1.0 + math.exp(im)) * (math.cos(re) * 0.5)

Julia

function code(re, im)
	return Float64(Float64(1.0 + exp(im)) * Float64(cos(re) * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = (1.0 + exp(im)) * (cos(re) * 0.5);
end

Wolfram

code[re_, im_] := N[(N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 71.6%

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

6. Final simplification 71.6%

   \[\leadsto \left(1 + e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) \]
7. Add Preprocessing

Alternative 14: 71.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (*
    (fma im im 2.0)
    (fma
     (fma
      (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
      (* re re)
      -0.25)
     (* re re)
     0.5))
   (*
    (*
     (fma
      (fma
       (fma 0.001388888888888889 (* im im) 0.041666666666666664)
       (* im im)
       0.5)
      (* im im)
      1.0)
     2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else {
		tmp = (fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(im, im, 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	else
		tmp = Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 59.4

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

   8. Applied rewrites 59.4%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot 2\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot 2\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      13. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      14. lower-*.f64 85.3

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 85.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (*
    (fma (* re re) -0.25 0.5)
    (* (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0) 2.0))
   (*
    (*
     (fma
      (fma
       (fma 0.001388888888888889 (* im im) 0.041666666666666664)
       (* im im)
       0.5)
      (* im im)
      1.0)
     2.0)
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = fma((re * re), -0.25, 0.5) * (fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * 2.0);
	} else {
		tmp = (fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * 2.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. lower-*.f64 0.7

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 0.7%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot 0.5 \]

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \]

       4. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \]

       5. lower-*.f64 58.2

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \]

   13. Applied rewrites 58.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \]

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 92.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. cosh-undef N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cosh.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot 2\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot 2\right) \cdot \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      13. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \frac{1}{2} \]

      14. lower-*.f64 85.3

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot 0.5 \]

   10. Applied rewrites 85.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + 1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.02)
   (* (* -0.25 (* re re)) (fma im im 2.0))
   (* (+ (fma (fma (fma 0.16666666666666666 im 0.5) im 1.0) im 1.0) 1.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.02) {
		tmp = (-0.25 * (re * re)) * fma(im, im, 2.0);
	} else {
		tmp = (fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0) + 1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.02)
		tmp = Float64(Float64(-0.25 * Float64(re * re)) * fma(im, im, 2.0));
	else
		tmp = Float64(Float64(fma(fma(fma(0.16666666666666666, im, 0.5), im, 1.0), im, 1.0) + 1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.02], N[(N[(-0.25 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * im + 0.5), $MachinePrecision] * im + 1.0), $MachinePrecision] * im + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0200000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 56.7%

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

   if -0.0200000000000000004 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 63.8%

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

   9. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(1 + \color{blue}{\mathsf{fma}\left(1 + im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right), im, 1\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(1 + \mathsf{fma}\left(\color{blue}{im \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot im\right) + 1}, im, 1\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(1 + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot im\right) \cdot im} + 1, im, 1\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(1 + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot im, im, 1\right)}, im, 1\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot im + \frac{1}{2}}, im, 1\right), im, 1\right)\right) \]

       8. lower-fma.f64 56.0

          \[\leadsto 0.5 \cdot \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, im, 0.5\right)}, im, 1\right), im, 1\right)\right) \]

   11. Applied rewrites 56.0%

       \[\leadsto 0.5 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.02:\\ \;\;\;\;\left(-0.25 \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, im, 0.5\right), im, 1\right), im, 1\right) + 1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.3% accurate, 26.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 75.5

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

5. Applied rewrites 75.5%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 51.8%

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

Alternative 18: 28.8% accurate, 52.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
7. Step-by-step derivation

8. Applied rewrites 29.5%

   \[\leadsto \color{blue}{0.5} \cdot 2 \]

9. Final simplification 29.5%

   \[\leadsto 2 \cdot 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))