math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 10.1s

Alternatives: 24

Speedup: 1.5×

• 50.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := e^{-im\_m}\\ \left(\frac{1}{t\_0} + t\_0\right) \cdot \left(\cos re \cdot 0.5\right) \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (exp (- im_m)))) (* (+ (/ 1.0 t_0) t_0) (* (cos re) 0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = exp(-im_m);
	return ((1.0 / t_0) + t_0) * (cos(re) * 0.5);
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    t_0 = exp(-im_m)
    code = ((1.0d0 / t_0) + t_0) * (cos(re) * 0.5d0)
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = Math.exp(-im_m);
	return ((1.0 / t_0) + t_0) * (Math.cos(re) * 0.5);
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = math.exp(-im_m)
	return ((1.0 / t_0) + t_0) * (math.cos(re) * 0.5)

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = exp(Float64(-im_m))
	return Float64(Float64(Float64(1.0 / t_0) + t_0) * Float64(cos(re) * 0.5))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	t_0 = exp(-im_m);
	tmp = ((1.0 / t_0) + t_0) * (cos(re) * 0.5);
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[Exp[(-im$95$m)], $MachinePrecision]}, N[(N[(N[(1.0 / t$95$0), $MachinePrecision] + t$95$0), $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. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lift-exp.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \cos re \cdot 0.5\\ t_1 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778\right) \cdot im\_m\right) \cdot im\_m, im\_m \cdot im\_m, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999986131997715:\\ \;\;\;\;\mathsf{fma}\left(im\_m, im\_m, 2\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (cos re) 0.5)) (t_1 (* (+ (exp im_m) (exp (- im_m))) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (* (* (* (* im_m im_m) 0.002777777777777778) im_m) im_m)
       (* im_m im_m)
       2.0)
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5))
     (if (<= t_1 0.9999986131997715)
       (* (fma im_m im_m 2.0) t_0)
       (* 1.0 (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = cos(re) * 0.5;
	double t_1 = (exp(im_m) + exp(-im_m)) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((((im_m * im_m) * 0.002777777777777778) * im_m) * im_m), (im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else if (t_1 <= 0.9999986131997715) {
		tmp = fma(im_m, im_m, 2.0) * t_0;
	} else {
		tmp = 1.0 * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(cos(re) * 0.5)
	t_1 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im_m * im_m) * 0.002777777777777778) * im_m) * im_m), Float64(im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	elseif (t_1 <= 0.9999986131997715)
		tmp = Float64(fma(im_m, im_m, 2.0) * t_0);
	else
		tmp = Float64(1.0 * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999986131997715], N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 80.3

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 80.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      15. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), re \cdot re, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{360} \cdot {im}^{4}, \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im, \color{blue}{im} \cdot im, 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999998613199771458

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.999998613199771458 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right) \cdot im\right) \cdot im, im \cdot im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9999986131997715:\\ \;\;\;\;\mathsf{fma}\left(im, im, 2\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778\right) \cdot im\_m\right) \cdot im\_m, im\_m \cdot im\_m, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999986131997715:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* (* (* (* im_m im_m) 0.002777777777777778) im_m) im_m)
       (* im_m im_m)
       2.0)
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5))
     (if (<= t_0 0.9999986131997715) (cos re) (* 1.0 (cosh im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((((im_m * im_m) * 0.002777777777777778) * im_m) * im_m), (im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else if (t_0 <= 0.9999986131997715) {
		tmp = cos(re);
	} else {
		tmp = 1.0 * cosh(im_m);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im_m * im_m) * 0.002777777777777778) * im_m) * im_m), Float64(im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	elseif (t_0 <= 0.9999986131997715)
		tmp = cos(re);
	else
		tmp = Float64(1.0 * cosh(im_m));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999986131997715], N[Cos[re], $MachinePrecision], N[(1.0 * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 80.3

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 80.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      15. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), re \cdot re, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{360} \cdot {im}^{4}, \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im, \color{blue}{im} \cdot im, 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999998613199771458

   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

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

      1. lower-cos.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.999998613199771458 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. cosh-undef N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right) \cdot im\right) \cdot im, im \cdot im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9999986131997715:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778\right) \cdot im\_m\right) \cdot im\_m, im\_m \cdot im\_m, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999986131997715:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right), im\_m \cdot im\_m, 1\right) \cdot im\_m\right) \cdot im\_m + 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* (* (* (* im_m im_m) 0.002777777777777778) im_m) im_m)
       (* im_m im_m)
       2.0)
      (fma
       (fma
        (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
        (* re re)
        -0.25)
       (* re re)
       0.5))
     (if (<= t_0 0.9999986131997715)
       (cos re)
       (*
        (+
         (*
          (*
           (fma
            (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333)
            (* im_m im_m)
            1.0)
           im_m)
          im_m)
         2.0)
        0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((((im_m * im_m) * 0.002777777777777778) * im_m) * im_m), (im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else if (t_0 <= 0.9999986131997715) {
		tmp = cos(re);
	} else {
		tmp = (((fma(fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333), (im_m * im_m), 1.0) * im_m) * im_m) + 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im_m * im_m) * 0.002777777777777778) * im_m) * im_m), Float64(im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	elseif (t_0 <= 0.9999986131997715)
		tmp = cos(re);
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333), Float64(im_m * im_m), 1.0) * im_m) * im_m) + 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999986131997715], N[Cos[re], $MachinePrecision], N[(N[(N[(N[(N[(N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 80.3

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 80.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      15. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), re \cdot re, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{360} \cdot {im}^{4}, \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im, \color{blue}{im} \cdot im, 2\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999998613199771458

   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

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

      1. lower-cos.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.999998613199771458 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 91.1

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 91.1%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.1%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.1%

       \[\leadsto 0.5 \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + \color{blue}{2}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right) \cdot im\right) \cdot im, im \cdot im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{elif}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq 0.9999986131997715:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right) \cdot im\_m\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5))))
   (if (<= t_0 -0.05)
     (fma -0.5 (* re re) 1.0)
     (if (<= t_0 2.0)
       (* 0.5 (fma im_m im_m 2.0))
       (* (* (fma (* im_m im_m) 0.041666666666666664 0.5) im_m) im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 0.5 * fma(im_m, im_m, 2.0);
	} else {
		tmp = (fma((im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(0.5 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 73.4%

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

4. Final simplification 59.2%

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

Alternative 6: 62.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right) \cdot im\_m\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5))))
   (if (<= t_0 -0.05)
     (fma -0.5 (* re re) 1.0)
     (if (<= t_0 2.0)
       (* 0.5 (fma im_m im_m 2.0))
       (* (* (* 0.041666666666666664 (* im_m im_m)) im_m) im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = (exp(im_m) + exp(-im_m)) * (cos(re) * 0.5);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 0.5 * fma(im_m, im_m, 2.0);
	} else {
		tmp = ((0.041666666666666664 * (im_m * im_m)) * im_m) * im_m;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(0.5 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(Float64(0.041666666666666664 * Float64(im_m * im_m)) * im_m) * im_m);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $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))) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 67.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{24} \cdot {im}^{\color{blue}{4}} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.4%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \cdot im\right) \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.2%

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

Alternative 7: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right) \cdot im\_m\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778, im\_m \cdot im\_m, 1\right), im\_m \cdot im\_m, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.09)
   (*
    (* (* (fma (* im_m im_m) 0.041666666666666664 0.5) im_m) im_m)
    (fma -0.5 (* re re) 1.0))
   (*
    (fma
     (fma (* (* im_m im_m) 0.002777777777777778) (* im_m im_m) 1.0)
     (* im_m im_m)
     2.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.09) {
		tmp = ((fma((im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = fma(fma(((im_m * im_m) * 0.002777777777777778), (im_m * im_m), 1.0), (im_m * im_m), 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.09)
		tmp = Float64(Float64(Float64(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(fma(fma(Float64(Float64(im_m * im_m) * 0.002777777777777778), Float64(im_m * im_m), 1.0), Float64(im_m * im_m), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.09], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $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))) < -0.089999999999999997

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 43.7%

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

   if -0.089999999999999997 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.4%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right), im \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 77.3%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot \left(im \cdot im\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right) \cdot im\_m\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right) \cdot im\_m\right) \cdot im\_m, im\_m \cdot im\_m, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.09)
   (*
    (* (* (fma (* im_m im_m) 0.041666666666666664 0.5) im_m) im_m)
    (fma -0.5 (* re re) 1.0))
   (*
    (fma
     (*
      (* (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333) im_m)
      im_m)
     (* im_m im_m)
     2.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.09) {
		tmp = ((fma((im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = fma(((fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333) * im_m) * im_m), (im_m * im_m), 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.09)
		tmp = Float64(Float64(Float64(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(fma(Float64(Float64(fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333) * im_m) * im_m), Float64(im_m * im_m), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.09], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $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))) < -0.089999999999999997

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 43.7%

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

   if -0.089999999999999997 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.4%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left({im}^{4} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{im}^{2}}\right), \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 77.2%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right) \cdot im\right) \cdot im, \color{blue}{im} \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right) \cdot im\right) \cdot im, im \cdot im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right) \cdot im\_m\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\left(\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778\right) \cdot im\_m\right) \cdot im\_m, im\_m \cdot im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.09)
   (*
    (* (* (fma (* im_m im_m) 0.041666666666666664 0.5) im_m) im_m)
    (fma -0.5 (* re re) 1.0))
   (*
    0.5
    (fma
     (* (* (* (* im_m im_m) 0.002777777777777778) im_m) im_m)
     (* im_m im_m)
     2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.09) {
		tmp = ((fma((im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = 0.5 * fma(((((im_m * im_m) * 0.002777777777777778) * im_m) * im_m), (im_m * im_m), 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.09)
		tmp = Float64(Float64(Float64(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5) * im_m) * im_m) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(0.5 * fma(Float64(Float64(Float64(Float64(im_m * im_m) * 0.002777777777777778) * im_m) * im_m), Float64(im_m * im_m), 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.09], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.089999999999999997

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 43.7%

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

   if -0.089999999999999997 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.4%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\frac{1}{360} \cdot {im}^{4}, \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 77.2%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im, \color{blue}{im} \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right) \cdot im\right) \cdot im, im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\left(0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right) \cdot im\_m\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\left(\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778\right) \cdot im\_m\right) \cdot im\_m, im\_m \cdot im\_m, 2\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.09)
   (*
    (* (* (* 0.041666666666666664 (* im_m im_m)) im_m) im_m)
    (fma -0.5 (* re re) 1.0))
   (*
    0.5
    (fma
     (* (* (* (* im_m im_m) 0.002777777777777778) im_m) im_m)
     (* im_m im_m)
     2.0))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.09) {
		tmp = (((0.041666666666666664 * (im_m * im_m)) * im_m) * im_m) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = 0.5 * fma(((((im_m * im_m) * 0.002777777777777778) * im_m) * im_m), (im_m * im_m), 2.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.09)
		tmp = Float64(Float64(Float64(Float64(0.041666666666666664 * Float64(im_m * im_m)) * im_m) * im_m) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(0.5 * fma(Float64(Float64(Float64(Float64(im_m * im_m) * 0.002777777777777778) * im_m) * im_m), Float64(im_m * im_m), 2.0));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.09], N[(N[(N[(N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.089999999999999997

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 43.2%

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

   if -0.089999999999999997 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 77.4%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\frac{1}{360} \cdot {im}^{4}, \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 77.2%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im, \color{blue}{im} \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right) \cdot im\right) \cdot im, im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\left(e^{im\_m} + e^{-im\_m}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\left(0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right)\right) \cdot im\_m\right) \cdot im\_m\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im\_m, im\_m, 0.5\right) \cdot im\_m, im\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.09)
   (*
    (* (* (* 0.041666666666666664 (* im_m im_m)) im_m) im_m)
    (fma -0.5 (* re re) 1.0))
   (fma (* (fma (* 0.041666666666666664 im_m) im_m 0.5) im_m) im_m 1.0)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (((exp(im_m) + exp(-im_m)) * (cos(re) * 0.5)) <= -0.09) {
		tmp = (((0.041666666666666664 * (im_m * im_m)) * im_m) * im_m) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = fma((fma((0.041666666666666664 * im_m), im_m, 0.5) * im_m), im_m, 1.0);
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (Float64(Float64(exp(im_m) + exp(Float64(-im_m))) * Float64(cos(re) * 0.5)) <= -0.09)
		tmp = Float64(Float64(Float64(Float64(0.041666666666666664 * Float64(im_m * im_m)) * im_m) * im_m) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = fma(Float64(fma(Float64(0.041666666666666664 * im_m), im_m, 0.5) * im_m), im_m, 1.0);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.09], N[(N[(N[(N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * im$95$m), $MachinePrecision] * im$95$m + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.089999999999999997

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 43.2%

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

   if -0.089999999999999997 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 69.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), \color{blue}{im \cdot im}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 69.9%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{im} + e^{-im}\right) \cdot \left(\cos re \cdot 0.5\right) \leq -0.09:\\ \;\;\;\;\left(\left(\left(0.041666666666666664 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.1% accurate, 0.9× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.05)
   (* (fma (* re re) -0.25 0.5) (fma im_m im_m 2.0))
   (fma (* (fma (* 0.041666666666666664 im_m) im_m 0.5) im_m) im_m 1.0)))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * im$95$m), $MachinePrecision] * im$95$m + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), \color{blue}{im \cdot im}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 70.6%

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

4. Final simplification 62.2%

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

Alternative 13: 52.8% accurate, 0.9× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (* (+ (exp im_m) (exp (- im_m))) (* (cos re) 0.5)) -0.05)
   (fma -0.5 (* re re) 1.0)
   (* 0.5 (fma im_m im_m 2.0))))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(N[Exp[im$95$m], $MachinePrecision] + N[Exp[(-im$95$m)], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 59.3%

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

4. Final simplification 50.9%

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

Alternative 14: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \cosh im\_m \cdot \cos re \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (cosh im_m) (cos re)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[Cosh[im$95$m], $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. cosh-undef N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 100.0

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

6. Applied rewrites 100.0%

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

Alternative 15: 71.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778\right) \cdot im\_m\right) \cdot im\_m, im\_m \cdot im\_m, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right), im\_m \cdot im\_m, 1\right) \cdot im\_m\right) \cdot im\_m + 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05)
   (*
    (fma
     (* (* (* (* im_m im_m) 0.002777777777777778) im_m) im_m)
     (* im_m im_m)
     2.0)
    (fma
     (fma
      (fma -0.0006944444444444445 (* re re) 0.020833333333333332)
      (* re re)
      -0.25)
     (* re re)
     0.5))
   (*
    (+
     (*
      (*
       (fma
        (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333)
        (* im_m im_m)
        1.0)
       im_m)
      im_m)
     2.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(((((im_m * im_m) * 0.002777777777777778) * im_m) * im_m), (im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, (re * re), 0.020833333333333332), (re * re), -0.25), (re * re), 0.5);
	} else {
		tmp = (((fma(fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333), (im_m * im_m), 1.0) * im_m) * im_m) + 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(im_m * im_m) * 0.002777777777777778) * im_m) * im_m), Float64(im_m * im_m), 2.0) * fma(fma(fma(-0.0006944444444444445, Float64(re * re), 0.020833333333333332), Float64(re * re), -0.25), Float64(re * re), 0.5));
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333), Float64(im_m * im_m), 1.0) * im_m) * im_m) + 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(N[(-0.0006944444444444445 * N[(re * re), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 91.7

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 91.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + {re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({re}^{2} \cdot \left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}\right) \cdot {re}^{2}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) - \frac{1}{4}, {re}^{2}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}\right) \cdot {re}^{2} + \color{blue}{\frac{-1}{4}}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{1440} \cdot {re}^{2}, {re}^{2}, \frac{-1}{4}\right)}, {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{1440} \cdot {re}^{2} + \frac{1}{48}}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{1440}, {re}^{2}, \frac{1}{48}\right)}, {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, \color{blue}{re \cdot re}, \frac{1}{48}\right), {re}^{2}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), \color{blue}{re \cdot re}, \frac{-1}{4}\right), {re}^{2}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      15. lower-*.f64 44.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), \color{blue}{re \cdot re}, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{1440}, re \cdot re, \frac{1}{48}\right), re \cdot re, \frac{-1}{4}\right), re \cdot re, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{360} \cdot {im}^{4}, \color{blue}{im} \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 44.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(im \cdot im\right)\right) \cdot im\right) \cdot im, \color{blue}{im} \cdot im, 2\right) \]

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 78.2%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto 0.5 \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + \color{blue}{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot 0.002777777777777778\right) \cdot im\right) \cdot im, im \cdot im, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0006944444444444445, re \cdot re, 0.020833333333333332\right), re \cdot re, -0.25\right), re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right), im\_m \cdot im\_m, 1\right)\\ \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(t\_0, im\_m \cdot im\_m, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 \cdot im\_m\right) \cdot im\_m + 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0
         (fma
          (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333)
          (* im_m im_m)
          1.0)))
   (if (<= (cos re) -0.05)
     (* (fma t_0 (* im_m im_m) 2.0) (fma (* re re) -0.25 0.5))
     (* (+ (* (* t_0 im_m) im_m) 2.0) 0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = fma(fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333), (im_m * im_m), 1.0);
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(t_0, (im_m * im_m), 2.0) * fma((re * re), -0.25, 0.5);
	} else {
		tmp = (((t_0 * im_m) * im_m) + 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = fma(fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333), Float64(im_m * im_m), 1.0)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(t_0, Float64(im_m * im_m), 2.0) * fma(Float64(re * re), -0.25, 0.5));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * im_m) * im_m) + 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(t$95$0 * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 91.7

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 91.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{4}, \frac{1}{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{4}, \frac{1}{2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. lower-*.f64 43.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot re}, -0.25, 0.5\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   8. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 78.2%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto 0.5 \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + \color{blue}{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \cdot \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right), im\_m \cdot im\_m, 1\right) \cdot im\_m\right) \cdot im\_m + 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05)
   (*
    (fma (fma (* im_m im_m) 0.041666666666666664 0.5) (* im_m im_m) 1.0)
    (fma -0.5 (* re re) 1.0))
   (*
    (+
     (*
      (*
       (fma
        (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333)
        (* im_m im_m)
        1.0)
       im_m)
      im_m)
     2.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(fma((im_m * im_m), 0.041666666666666664, 0.5), (im_m * im_m), 1.0) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = (((fma(fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333), (im_m * im_m), 1.0) * im_m) * im_m) + 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5), Float64(im_m * im_m), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333), Float64(im_m * im_m), 1.0) * im_m) * im_m) + 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 78.2%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto 0.5 \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + \color{blue}{2}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im\right) \cdot im + 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 70.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im\_m \cdot im\_m, 0.08333333333333333\right), im\_m \cdot im\_m, 1\right) \cdot im\_m, im\_m, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05)
   (*
    (fma (fma (* im_m im_m) 0.041666666666666664 0.5) (* im_m im_m) 1.0)
    (fma -0.5 (* re re) 1.0))
   (*
    (fma
     (*
      (fma
       (fma 0.002777777777777778 (* im_m im_m) 0.08333333333333333)
       (* im_m im_m)
       1.0)
      im_m)
     im_m
     2.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(fma((im_m * im_m), 0.041666666666666664, 0.5), (im_m * im_m), 1.0) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = fma((fma(fma(0.002777777777777778, (im_m * im_m), 0.08333333333333333), (im_m * im_m), 1.0) * im_m), im_m, 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5), Float64(im_m * im_m), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(fma(Float64(fma(fma(0.002777777777777778, Float64(im_m * im_m), 0.08333333333333333), Float64(im_m * im_m), 1.0) * im_m), im_m, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.002777777777777778 * N[(im$95$m * im$95$m), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 78.2%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 78.2%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, \color{blue}{im}, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right) \cdot im, im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 70.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im\_m \cdot im\_m, 0.041666666666666664, 0.5\right), im\_m \cdot im\_m, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(im\_m \cdot im\_m\right) \cdot 0.002777777777777778, im\_m \cdot im\_m, 1\right), im\_m \cdot im\_m, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05)
   (*
    (fma (fma (* im_m im_m) 0.041666666666666664 0.5) (* im_m im_m) 1.0)
    (fma -0.5 (* re re) 1.0))
   (*
    (fma
     (fma (* (* im_m im_m) 0.002777777777777778) (* im_m im_m) 1.0)
     (* im_m im_m)
     2.0)
    0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(fma((im_m * im_m), 0.041666666666666664, 0.5), (im_m * im_m), 1.0) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = fma(fma(((im_m * im_m) * 0.002777777777777778), (im_m * im_m), 1.0), (im_m * im_m), 2.0) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(fma(fma(Float64(im_m * im_m), 0.041666666666666664, 0.5), Float64(im_m * im_m), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(fma(fma(Float64(Float64(im_m * im_m) * 0.002777777777777778), Float64(im_m * im_m), 1.0), Float64(im_m * im_m), 2.0) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 43.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 92.8

          \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   5. Applied rewrites 92.8%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 78.2%

      \[\leadsto \color{blue}{0.5} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {im}^{2}, im \cdot im, 1\right), im \cdot im, 2\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 78.1%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778 \cdot \left(im \cdot im\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.002777777777777778, im \cdot im, 1\right), im \cdot im, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im\_m, im\_m, 0.5\right) \cdot im\_m, im\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma -0.5 (* re re) 1.0)
   (fma (* (fma (* 0.041666666666666664 im_m) im_m 0.5) im_m) im_m 1.0)))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * im$95$m), $MachinePrecision] * im$95$m + 0.5), $MachinePrecision] * im$95$m), $MachinePrecision] * im$95$m + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), \color{blue}{im \cdot im}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 70.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 70.6%

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

Alternative 21: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im\_m \cdot im\_m\right), im\_m \cdot im\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05)
   (fma -0.5 (* re re) 1.0)
   (fma (* 0.041666666666666664 (* im_m im_m)) (* im_m im_m) 1.0)))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.041666666666666664 * N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision] * N[(im$95$m * im$95$m), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 70.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 70.5%

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

4. Final simplification 59.2%

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

Alternative 22: 34.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05) (fma -0.5 (* re re) 1.0) 1.0))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = fma(-0.5, (re * re), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = fma(-0.5, Float64(re * re), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 32.2%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 34.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;\cos re \leq -0.05:\\ \;\;\;\;-0.5 \cdot \left(re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (cos re) -0.05) (* -0.5 (* re re)) 1.0))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (cos(re) <= -0.05) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (cos(re) <= (-0.05d0)) then
        tmp = (-0.5d0) * (re * re)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (Math.cos(re) <= -0.05) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if math.cos(re) <= -0.05:
		tmp = -0.5 * (re * re)
	else:
		tmp = 1.0
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (cos(re) <= -0.05)
		tmp = Float64(-0.5 * Float64(re * re));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (cos(re) <= -0.05)
		tmp = -0.5 * (re * re);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[Cos[re], $MachinePrecision], -0.05], N[(-0.5 * N[(re * re), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 re) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 28.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 28.3%

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

   if -0.050000000000000003 < (cos.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-cos.f64 46.8

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 32.2%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 28.1% accurate, 316.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 1 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 1.0)

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 1.0;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 1.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 1.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 1.0

Julia

im_m = abs(im)
function code(re, im_m)
	return 1.0
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 1.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. lower-cos.f64 49.9

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

5. Applied rewrites 49.9%

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

6. Taylor expanded in re around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 23.8%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))