math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 15

Speedup: 1.2×

• 49.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-cos.f64 100.0

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

   6. lift-+.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. +-commutative N/A

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

   11. cosh-undef N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cosh.f64 100.0

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

3. Applied rewrites 100.0%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999931:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cosh im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (* 2.0 (cosh im)) (* (* re re) -0.25))
     (if (<= t_1 0.9999999999999931)
       (* t_0 (fma im im 2.0))
       (* 2.0 (* (cosh im) 0.5))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (2.0 * cosh(im)) * ((re * re) * -0.25);
	} else if (t_1 <= 0.9999999999999931) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = 2.0 * (cosh(im) * 0.5);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 * cosh(im)) * Float64(Float64(re * re) * -0.25));
	elseif (t_1 <= 0.9999999999999931)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(2.0 * Float64(cosh(im) * 0.5));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999931], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cosh[im], $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.8

         \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right) \]

   7. Applied rewrites 12.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999312

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.1

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

   4. Applied rewrites 76.1%

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

   if 0.99999999999999312 < (*.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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999931:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cosh im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (* 2.0 (cosh im)) (* (* re re) -0.25))
     (if (<= t_0 0.9999999999999931) (cos re) (* 2.0 (* (cosh im) 0.5))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (2.0 * cosh(im)) * ((re * re) * -0.25);
	} else if (t_0 <= 0.9999999999999931) {
		tmp = cos(re);
	} else {
		tmp = 2.0 * (cosh(im) * 0.5);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (2.0 * Math.cosh(im)) * ((re * re) * -0.25);
	} else if (t_0 <= 0.9999999999999931) {
		tmp = Math.cos(re);
	} else {
		tmp = 2.0 * (Math.cosh(im) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (2.0 * math.cosh(im)) * ((re * re) * -0.25)
	elif t_0 <= 0.9999999999999931:
		tmp = math.cos(re)
	else:
		tmp = 2.0 * (math.cosh(im) * 0.5)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 * cosh(im)) * Float64(Float64(re * re) * -0.25));
	elseif (t_0 <= 0.9999999999999931)
		tmp = cos(re);
	else
		tmp = Float64(2.0 * Float64(cosh(im) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (2.0 * cosh(im)) * ((re * re) * -0.25);
	elseif (t_0 <= 0.9999999999999931)
		tmp = cos(re);
	else
		tmp = 2.0 * (cosh(im) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999931], N[Cos[re], $MachinePrecision], N[(2.0 * N[(N[Cosh[im], $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.8

         \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right) \]

   7. Applied rewrites 12.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999999999999312

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{1}{4} \cdot {im}^{2} + \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left({im}^{2} \cdot \frac{1}{4} + \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.9

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

   7. Applied rewrites 46.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 56.5

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

   10. Applied rewrites 56.5%

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

   11. Taylor expanded in im around 0

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

       1. lower-cos.f64 51.1

          \[\leadsto \cos re \]

   13. Applied rewrites 51.1%

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

   if 0.99999999999999312 < (*.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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

Alternative 5: 76.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cosh[im], $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

Alternative 6: 76.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im) + exp(im))) <= (-0.05d0)) then
        tmp = (2.0d0 * cosh(im)) * ((re * re) * (-0.25d0))
    else
        tmp = 2.0d0 * (cosh(im) * 0.5d0)
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))) <= -0.05:
		tmp = (2.0 * math.cosh(im)) * ((re * re) * -0.25)
	else:
		tmp = 2.0 * (math.cosh(im) * 0.5)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.05)
		tmp = (2.0 * cosh(im)) * ((re * re) * -0.25);
	else
		tmp = 2.0 * (cosh(im) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cosh[im], $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.8

         \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.25\right) \]

   7. Applied rewrites 12.8%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

Alternative 7: 75.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cosh[im], $MachinePrecision] * 0.5), $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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 48.2

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

   7. Applied rewrites 48.2%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

Alternative 8: 67.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(im * im + 2.0), $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 48.2

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

   7. Applied rewrites 48.2%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{1}{4} \cdot {im}^{2} + \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left({im}^{2} \cdot \frac{1}{4} + \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.9

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

   7. Applied rewrites 46.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 56.5

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

   10. Applied rewrites 56.5%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 56.5

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

   12. Applied rewrites 56.5%

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

Alternative 9: 62.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot {re}^{2} + 1 \]

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.2

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

   10. Applied rewrites 31.2%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{1}{4} \cdot {im}^{2} + \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left({im}^{2} \cdot \frac{1}{4} + \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.9

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

   7. Applied rewrites 46.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 56.5

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

   10. Applied rewrites 56.5%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-fma.f64 N/A

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

       9. lift-*.f64 56.5

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

   12. Applied rewrites 56.5%

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

Alternative 10: 62.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	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, im, 2.0);
	} else {
		tmp = ((im * im) * (im * im)) * 0.041666666666666664;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $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. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot {re}^{2} + 1 \]

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.2

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

   10. Applied rewrites 31.2%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-undef N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 76.1

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

   6. Applied rewrites 76.1%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 46.9%

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

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{1}{4} \cdot {im}^{2} + \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left({im}^{2} \cdot \frac{1}{4} + \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.9

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

   7. Applied rewrites 46.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 56.5

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

   10. Applied rewrites 56.5%

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

   11. Taylor expanded in im around inf

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

       1. *-commutative N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} \]

       2. lower-*.f64 N/A

          \[\leadsto {im}^{4} \cdot \frac{1}{24} \]

       3. metadata-eval N/A

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

       4. pow-prod-up N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \frac{1}{24} \]

       9. lift-*.f64 31.4

          \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664 \]

   13. Applied rewrites 31.4%

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

Alternative 11: 62.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.005], N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot {re}^{2} + 1 \]

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.2

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

   10. Applied rewrites 31.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. cosh-undef N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cosh.f64 65.1

         \[\leadsto 2 \cdot \left(\cosh im \cdot 0.5\right) \]

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{1}{4} \cdot {im}^{2} + \frac{1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left({im}^{2} \cdot \frac{1}{4} + \frac{1}{2}\right) \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 46.9

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

   7. Applied rewrites 46.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 56.5

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

   10. Applied rewrites 56.5%

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

   11. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 56.2

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

   13. Applied rewrites 56.2%

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

Alternative 12: 53.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot {re}^{2} + 1 \]

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.2

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

   10. Applied rewrites 31.2%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 100.0

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

      6. lift-+.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-undef N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cosh.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

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

      1. cosh-undef-rev N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 76.1

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

   6. Applied rewrites 76.1%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 46.9%

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

Alternative 13: 34.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot {re}^{2} + 1 \]

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.2

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

   10. Applied rewrites 31.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around 0

      \[\leadsto 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 27.8%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 34.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((0.5d0 * cos(re)) <= (-0.005d0)) then
        tmp = (-0.5d0) * (re * re)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.cos(re)) <= -0.005) {
		tmp = -0.5 * (re * re);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.cos(re)) <= -0.005:
		tmp = -0.5 * (re * re)
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * cos(re)) <= -0.005)
		tmp = -0.5 * (re * re);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot {re}^{2} \]

      2. pow2 N/A

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

      3. lift-*.f64 7.6

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

   10. Applied rewrites 7.6%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-*.f64 N/A

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

      5. cosh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cosh.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 31.2

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

   7. Applied rewrites 31.2%

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

   8. Taylor expanded in re around 0

      \[\leadsto 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 27.8%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 27.8% accurate, 62.8× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. lower-*.f64 N/A

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

   5. cosh-undef N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cosh.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 61.9

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

4. Applied rewrites 61.9%

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

5. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 31.2

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

7. Applied rewrites 31.2%

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

8. Taylor expanded in re around 0

   \[\leadsto 1 \]
9. Step-by-step derivation

10. Applied rewrites 27.8%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))