math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 12

Speedup: 1.5×

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

Specification

Math

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

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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[Cosh[im], $MachinePrecision] * N[Cos[re], $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. Step-by-step derivation

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lift-exp.f64 N/A

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

   4. exp-neg N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. pow-flip N/A

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

   8. lift-neg.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. pow-exp N/A

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

   8. mul-1-neg N/A

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

   9. lift-neg.f64 N/A

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

   10. remove-double-neg N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. associate-*l* N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445 \cdot \left(re \cdot re\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 0.9997279655134533:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma (* -0.0006944444444444445 (* re re)) (* re re) -0.25)
       (* re re)
       0.5)
      2.0)
     (if (<= t_1 0.9997279655134533) (* t_0 2.0) (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma((-0.0006944444444444445 * (re * re)), (re * re), -0.25), (re * re), 0.5) * 2.0;
	} else if (t_1 <= 0.9997279655134533) {
		tmp = t_0 * 2.0;
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(-0.0006944444444444445 * Float64(re * re)), Float64(re * re), -0.25), Float64(re * re), 0.5) * 2.0);
	elseif (t_1 <= 0.9997279655134533)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.9997279655134533], N[(t$95$0 * 2.0), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 94.6

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 94.6%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999727965513453332

   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 100.0%

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

   if 0.999727965513453332 < (*.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. Step-by-step derivation

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. pow-flip N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. pow-exp N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.6%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.002)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $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))) < -2e-3

   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 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 55.3

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

   8. Applied rewrites 55.3%

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

   if -2e-3 < (*.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. Step-by-step derivation

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. pow-flip N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. pow-exp N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 84.4%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.002)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], 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] * 1.0), $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))) < -2e-3

   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 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 55.3

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

   8. Applied rewrites 55.3%

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

   if -2e-3 < (*.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. Step-by-step derivation

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. pow-flip N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. pow-exp N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 84.4%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 76.5

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

   12. Applied rewrites 76.5%

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

Alternative 5: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.002)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.002) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.002)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 1.0), $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))) < -2e-3

   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 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 55.3

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

   8. Applied rewrites 55.3%

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

   if -2e-3 < (*.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. Step-by-step derivation

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-exp.f64 N/A

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

      4. exp-neg N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. pow-flip N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. pow-exp N/A

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

      8. mul-1-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. associate-*l* N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 84.4%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]

   10. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 71.3

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

   12. Applied rewrites 71.3%

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

Alternative 6: 57.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im), $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 69.6

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

   5. Applied rewrites 69.6%

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

   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 56.1

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

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 55.7%

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

   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 75.3

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

   5. Applied rewrites 75.3%

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

   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 59.7%

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

Alternative 7: 46.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (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 79.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.0%

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

   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 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 46.2

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

   5. Applied rewrites 46.2%

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

   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 46.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 46.2%

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

Alternative 8: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;\cos re \leq 0.9995:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.0002)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (if (<= (cos re) 0.9995)
     (* (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5) 2.0)
     (* 0.5 (fma im im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.0002) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else if (cos(re) <= 0.9995) {
		tmp = fma((0.020833333333333332 * (re * re)), (re * re), 0.5) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.0002)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	elseif (cos(re) <= 0.9995)
		tmp = Float64(fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.0002], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.9995], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -2.0000000000000001e-4

   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 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 55.3

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

   8. Applied rewrites 55.3%

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

   if -2.0000000000000001e-4 < (cos.f64 re) < 0.99950000000000006

   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 51.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 43.8

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

   8. Applied rewrites 43.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 43.1%

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

   if 0.99950000000000006 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   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 74.2%

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

Alternative 9: 57.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.0002)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.0002) {
		tmp = fma((re * re), -0.25, 0.5) * 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 (cos(re) <= -0.0002)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * 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[Cos[re], $MachinePrecision], -0.0002], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $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 (cos.f64 re) < -2.0000000000000001e-4

   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 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   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 55.3

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

   8. Applied rewrites 55.3%

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

   if -2.0000000000000001e-4 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   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 60.0%

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

Alternative 10: 53.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.0002:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.0002)
   (* (* (* re re) -0.25) 2.0)
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.0002) {
		tmp = ((re * re) * -0.25) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.0002)
		tmp = Float64(Float64(Float64(re * re) * -0.25) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.0002], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -2.0000000000000001e-4

   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 45.1%

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

   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 37.9

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

   8. Applied rewrites 37.9%

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

   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 37.9%

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

   if -2.0000000000000001e-4 < (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

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   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 75.2

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

   5. Applied rewrites 75.2%

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

   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 60.0%

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

Alternative 11: 46.4% 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 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

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

8. Applied rewrites 46.1%

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

Alternative 12: 27.8% accurate, 52.7× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * 2.0), $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 53.2%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 31.2%

   \[\leadsto \color{blue}{0.5} \cdot 2 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024319 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))