math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 19

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(2.0 * N[Cosh[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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. cosh-undef N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ t_2 := 2 \cdot \cosh im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0.9998944153177931:\\ \;\;\;\;2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5))
        (t_1 (* (+ (exp im) (exp (- im))) t_0))
        (t_2 (* 2.0 (cosh im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* re re) -0.25 0.5) t_2)
     (if (<= t_1 0.9998944153177931) (* 2.0 t_0) (* 0.5 t_2)))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double t_2 = 2.0 * cosh(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((re * re), -0.25, 0.5) * t_2;
	} else if (t_1 <= 0.9998944153177931) {
		tmp = 2.0 * t_0;
	} else {
		tmp = 0.5 * t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	t_2 = Float64(2.0 * cosh(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * t_2);
	elseif (t_1 <= 0.9998944153177931)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(0.5 * t_2);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.9998944153177931], N[(2.0 * t$95$0), $MachinePrecision], N[(0.5 * t$95$2), $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 re around 0

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 0.0

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99989441531779311

   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 0.99989441531779311 < (*.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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 100.0

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(2 \cdot \cosh im\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9998944153177931:\\ \;\;\;\;2 \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;t\_1 \leq 0.9998944153177931:\\ \;\;\;\;2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma 0.001388888888888889 (* im im) 0.041666666666666664)
         (* im im)
         0.5)
        (* im im)
        1.0)
       2.0)
      (fma (* re re) -0.25 0.5))
     (if (<= t_1 0.9998944153177931) (* 2.0 t_0) (* 0.5 (* 2.0 (cosh im)))))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 2.0) * fma((re * re), -0.25, 0.5);
	} else if (t_1 <= 0.9998944153177931) {
		tmp = 2.0 * t_0;
	} else {
		tmp = 0.5 * (2.0 * cosh(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		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) * fma(Float64(re * re), -0.25, 0.5));
	elseif (t_1 <= 0.9998944153177931)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(0.5 * Float64(2.0 * cosh(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], 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] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9998944153177931], N[(2.0 * t$95$0), $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[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 re around 0

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 0.0

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in im around 0

       \[\leadsto \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       7. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       9. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       13. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       14. lower-*.f64 87.8

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   13. Applied rewrites 87.8%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99989441531779311

   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 0.99989441531779311 < (*.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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 100.0

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

   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 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9998944153177931:\\ \;\;\;\;2 \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := \left(e^{im} + e^{-im}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;t\_1 \leq 0.9998944153177931:\\ \;\;\;\;2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)) (t_1 (* (+ (exp im) (exp (- im))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma 0.001388888888888889 (* im im) 0.041666666666666664)
         (* im im)
         0.5)
        (* im im)
        1.0)
       2.0)
      (fma (* re re) -0.25 0.5))
     (if (<= t_1 0.9998944153177931)
       (* 2.0 t_0)
       (*
        (*
         (fma
          (fma
           (fma (* im im) 0.001388888888888889 0.041666666666666664)
           (* im im)
           0.5)
          (* im im)
          1.0)
         2.0)
        0.5)))))

C

double code(double re, double im) {
	double t_0 = cos(re) * 0.5;
	double t_1 = (exp(im) + exp(-im)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 2.0) * fma((re * re), -0.25, 0.5);
	} else if (t_1 <= 0.9998944153177931) {
		tmp = 2.0 * t_0;
	} else {
		tmp = (fma(fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(Float64(exp(im) + exp(Float64(-im))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		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) * fma(Float64(re * re), -0.25, 0.5));
	elseif (t_1 <= 0.9998944153177931)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 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[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im], $MachinePrecision] + N[Exp[(-im)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], 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] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9998944153177931], N[(2.0 * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 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 re around 0

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 0.0

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in im around 0

       \[\leadsto \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       7. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       9. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       13. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       14. lower-*.f64 87.8

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   13. Applied rewrites 87.8%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99989441531779311

   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 0.99989441531779311 < (*.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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 100.0

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

   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. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 90.5

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

   10. Applied rewrites 90.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9998944153177931:\\ \;\;\;\;2 \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (+ (exp im) (exp (- im))) (* (cos re) 0.5))))
   (if (<= t_0 -0.01)
     (* (fma im im 2.0) (fma (* re re) -0.25 0.5))
     (if (<= t_0 2.0)
       (* (fma im im 2.0) 0.5)
       (*
        (* (* (* (fma (* im im) 0.041666666666666664 0.5) im) im) 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 <= -0.01) {
		tmp = fma(im, im, 2.0) * fma((re * re), -0.25, 0.5);
	} else if (t_0 <= 2.0) {
		tmp = fma(im, im, 2.0) * 0.5;
	} else {
		tmp = (((fma((im * im), 0.041666666666666664, 0.5) * im) * im) * 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 <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), -0.25, 0.5));
	elseif (t_0 <= 2.0)
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(im * im), 0.041666666666666664, 0.5) * im) * im) * 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, -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $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))) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -0.0100000000000000002 < (*.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 75.7%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.6

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

   8. Applied rewrites 75.6%

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

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 100.0

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

   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.2

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

   10. Applied rewrites 70.2%

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

   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 70.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) -0.01)
   (*
    (*
     (fma
      (fma
       (fma 0.001388888888888889 (* im im) 0.041666666666666664)
       (* im im)
       0.5)
      (* im im)
      1.0)
     2.0)
    (fma (* re re) -0.25 0.5))
   (*
    (*
     (fma
      (fma
       (fma (* im im) 0.001388888888888889 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)) <= -0.01) {
		tmp = (fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * 2.0) * fma((re * re), -0.25, 0.5);
	} else {
		tmp = (fma(fma(fma((im * im), 0.001388888888888889, 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)) <= -0.01)
		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) * fma(Float64(re * re), -0.25, 0.5));
	else
		tmp = Float64(Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 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], -0.01], 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] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 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 (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

       \[\leadsto \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       4. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       7. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       9. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       11. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       13. unpow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{4}, \frac{1}{2}\right) \]

       14. lower-*.f64 50.2

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   13. Applied rewrites 50.2%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \]

   if -0.0100000000000000002 < (*.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 87.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 87.6

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

   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. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.6

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

   10. Applied rewrites 79.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (+ (exp im) (exp (- im))) (* (cos re) 0.5)) 2.0)
   (* 0.5 2.0)
   (* (* 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 = 0.5 * 2.0;
	} 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 = 0.5d0 * 2.0d0
    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 = 0.5 * 2.0;
	} 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 = 0.5 * 2.0
	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(0.5 * 2.0);
	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 = 0.5 * 2.0;
	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[(0.5 * 2.0), $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 re around 0

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 48.9%

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

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 100.0

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

   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 49.7

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

   10. Applied rewrites 49.7%

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

   11. Taylor expanded in im around inf

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

   13. Applied rewrites 49.7%

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

4. Final simplification 49.2%

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

Alternative 8: 68.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 2\\ \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0) 2.0)))
   (if (<= (cos re) -0.01)
     (* t_0 (fma (* re re) -0.25 0.5))
     (if (<= (cos re) 0.996)
       (*
        (fma im im 2.0)
        (* (fma (fma 0.041666666666666664 (* re re) -0.5) (* re re) 1.0) 0.5))
       (* t_0 0.5)))))

C

double code(double re, double im) {
	double t_0 = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * 2.0;
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = t_0 * fma((re * re), -0.25, 0.5);
	} else if (cos(re) <= 0.996) {
		tmp = fma(im, im, 2.0) * (fma(fma(0.041666666666666664, (re * re), -0.5), (re * re), 1.0) * 0.5);
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * 2.0)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(t_0 * fma(Float64(re * re), -0.25, 0.5));
	elseif (cos(re) <= 0.996)
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(0.041666666666666664, Float64(re * re), -0.5), Float64(re * re), 1.0) * 0.5));
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(t$95$0 * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.996], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 46.9

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

   13. Applied rewrites 46.9%

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

   if -0.0100000000000000002 < (cos.f64 re) < 0.996

   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 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.6

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

   8. Applied rewrites 52.6%

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

   if 0.996 < (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 97.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 97.7

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.6

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

   10. Applied rewrites 84.6%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* (fma im im 2.0) (fma (* re re) -0.25 0.5))
   (if (<= (cos re) 0.996)
     (*
      (fma im im 2.0)
      (* (fma (fma 0.041666666666666664 (* re re) -0.5) (* re re) 1.0) 0.5))
     (*
      (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0) 2.0)
      0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = fma(im, im, 2.0) * fma((re * re), -0.25, 0.5);
	} else if (cos(re) <= 0.996) {
		tmp = fma(im, im, 2.0) * (fma(fma(0.041666666666666664, (re * re), -0.5), (re * re), 1.0) * 0.5);
	} else {
		tmp = (fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), -0.25, 0.5));
	elseif (cos(re) <= 0.996)
		tmp = Float64(fma(im, im, 2.0) * Float64(fma(fma(0.041666666666666664, Float64(re * re), -0.5), Float64(re * re), 1.0) * 0.5));
	else
		tmp = Float64(Float64(fma(fma(Float64(im * im), 0.041666666666666664, 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], -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.996], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(re * re), $MachinePrecision] + -0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 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) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -0.0100000000000000002 < (cos.f64 re) < 0.996

   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 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.6

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

   8. Applied rewrites 52.6%

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

   if 0.996 < (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 97.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 97.7

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.6

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

   10. Applied rewrites 84.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, -0.5\right), re \cdot re, 1\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* (fma im im 2.0) (fma (* re re) -0.25 0.5))
   (if (<= (cos re) 0.996)
     (* (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5) 2.0)
     (*
      (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0) 2.0)
      0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = fma(im, im, 2.0) * fma((re * re), -0.25, 0.5);
	} else if (cos(re) <= 0.996) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * 2.0;
	} else {
		tmp = (fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), -0.25, 0.5));
	elseif (cos(re) <= 0.996)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * 2.0);
	else
		tmp = Float64(Float64(fma(fma(Float64(im * im), 0.041666666666666664, 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], -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.996], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 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) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -0.0100000000000000002 < (cos.f64 re) < 0.996

   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 61.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 11.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot 2 \]
   10. 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 50.8

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

   11. Applied rewrites 50.8%

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

   if 0.996 < (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 97.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 97.7

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.6

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

   10. Applied rewrites 84.6%

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

Alternative 11: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* (fma im im 2.0) (fma (* re re) -0.25 0.5))
   (if (<= (cos re) 0.996)
     (* (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5) 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 (cos(re) <= -0.01) {
		tmp = fma(im, im, 2.0) * fma((re * re), -0.25, 0.5);
	} else if (cos(re) <= 0.996) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * 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 (cos(re) <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), -0.25, 0.5));
	elseif (cos(re) <= 0.996)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * 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[Cos[re], $MachinePrecision], -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.996], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $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 3 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -0.0100000000000000002 < (cos.f64 re) < 0.996

   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 61.6%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 11.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot 2 \]
   10. 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 50.8

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

   11. Applied rewrites 50.8%

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

   if 0.996 < (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 97.7%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 97.7

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.6

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

   10. Applied rewrites 84.6%

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

   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 84.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.996:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.9998:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* (fma im im 2.0) (fma (* re re) -0.25 0.5))
   (if (<= (cos re) 0.9998)
     (* (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5) 2.0)
     (* (fma im im 2.0) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = fma(im, im, 2.0) * fma((re * re), -0.25, 0.5);
	} else if (cos(re) <= 0.9998) {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * 2.0;
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), -0.25, 0.5));
	elseif (cos(re) <= 0.9998)
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * 2.0);
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.9998], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -0.0100000000000000002 < (cos.f64 re) < 0.99980000000000002

   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 60.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.8%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot 2 \]
   10. 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 49.9

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

   11. Applied rewrites 49.9%

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

   if 0.99980000000000002 < (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 99.5%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.0

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

   8. Applied rewrites 75.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.9998:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* (fma im im 2.0) (fma (* re re) -0.25 0.5))
   (if (<= (cos re) 0.999)
     (* (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5) 2.0)
     (* (fma im im 2.0) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = fma(im, im, 2.0) * fma((re * re), -0.25, 0.5);
	} else if (cos(re) <= 0.999) {
		tmp = fma((0.020833333333333332 * (re * re)), (re * re), 0.5) * 2.0;
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), -0.25, 0.5));
	elseif (cos(re) <= 0.999)
		tmp = Float64(fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5) * 2.0);
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[re], $MachinePrecision], 0.999], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -0.0100000000000000002 < (cos.f64 re) < 0.998999999999999999

   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 61.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 12.0%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{48} \cdot {re}^{2} - \frac{1}{4}\right)\right)} \cdot 2 \]
   10. 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 50.1

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

   11. Applied rewrites 50.1%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 49.7%

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

   if 0.998999999999999999 < (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 98.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.4

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

   8. Applied rewrites 74.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{elif}\;\cos re \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (*
    (* (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0) 2.0)
    (fma (* re re) -0.25 0.5))
   (*
    (*
     (fma
      (fma
       (fma (* im im) 0.001388888888888889 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) <= -0.01) {
		tmp = (fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * 2.0) * fma((re * re), -0.25, 0.5);
	} else {
		tmp = (fma(fma(fma((im * im), 0.001388888888888889, 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) <= -0.01)
		tmp = Float64(Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * 2.0) * fma(Float64(re * re), -0.25, 0.5));
	else
		tmp = Float64(Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 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], -0.01], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 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) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 46.9

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

   13. Applied rewrites 46.9%

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

   if -0.0100000000000000002 < (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 87.6%

      \[\leadsto \color{blue}{0.5} \cdot \left(e^{-im} + e^{im}\right) \]
   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 87.6

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

      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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

   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. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.6

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

   10. Applied rewrites 79.6%

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

Alternative 15: 59.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* (fma im im 2.0) (fma (* re re) -0.25 0.5))
   (* (fma im im 2.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = fma(im, im, 2.0) * fma((re * re), -0.25, 0.5);
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(fma(im, im, 2.0) * fma(Float64(re * re), -0.25, 0.5));
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   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. lift-cosh.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 0.7

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

   7. Applied rewrites 0.7%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.0

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

   10. Applied rewrites 57.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 41.7

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

   13. Applied rewrites 41.7%

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

   if -0.0100000000000000002 < (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 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 62.9

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

   8. Applied rewrites 62.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\left(\left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* (* (* -0.5 (* re re)) 0.5) 2.0)
   (* (fma im im 2.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = ((-0.5 * (re * re)) * 0.5) * 2.0;
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(Float64(Float64(-0.5 * Float64(re * re)) * 0.5) * 2.0);
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(N[(N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   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 46.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 21.3

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

   8. Applied rewrites 21.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 21.3%

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

   if -0.0100000000000000002 < (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 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 62.9

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

   8. Applied rewrites 62.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;\left(\left(-0.5 \cdot \left(re \cdot re\right)\right) \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos re) -0.01)
   (* 2.0 (fma (* re re) -0.25 0.5))
   (* (fma im im 2.0) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (cos(re) <= -0.01) {
		tmp = 2.0 * fma((re * re), -0.25, 0.5);
	} else {
		tmp = fma(im, im, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(re) <= -0.01)
		tmp = Float64(2.0 * fma(Float64(re * re), -0.25, 0.5));
	else
		tmp = Float64(fma(im, im, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.01], N[(2.0 * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(im * im + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.0100000000000000002

   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. Taylor expanded in im around 0

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

   8. Applied rewrites 1.1%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot 2 \]
   10. 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 21.3

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

   11. Applied rewrites 21.3%

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

   if -0.0100000000000000002 < (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 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 62.9

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

   8. Applied rewrites 62.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.01:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 47.7% accurate, 26.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(N[(im * im + 2.0), $MachinePrecision] * 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 re around 0

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

5. Applied rewrites 68.6%

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

6. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 49.3

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

8. Applied rewrites 49.3%

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

9. Final simplification 49.3%

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

Alternative 19: 28.4% 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 re around 0

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

5. Applied rewrites 68.6%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 31.4%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))