math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 18

Speedup: 1.5×

• 49.6% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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 = cosh(im) * cos(re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. cosh-undef N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999999600385:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (cosh im) (fma -0.5 (* re re) 1.0))
     (if (<= t_0 0.9999999999600385)
       (*
        (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0)
        (cos re))
       (* (cosh im) 1.0)))))

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 = cosh(im) * fma(-0.5, (re * re), 1.0);
	} else if (t_0 <= 0.9999999999600385) {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * cos(re);
	} else {
		tmp = cosh(im) * 1.0;
	}
	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(cosh(im) * fma(-0.5, Float64(re * re), 1.0));
	elseif (t_0 <= 0.9999999999600385)
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * cos(re));
	else
		tmp = Float64(cosh(im) * 1.0);
	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, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999600385], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. 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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999960038521

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 0.999999999960038521 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999600385:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999600385:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (cosh im) (fma -0.5 (* re re) 1.0))
     (if (<= t_1 0.9999999999600385)
       (* t_0 (fma im im 2.0))
       (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(im) * fma(-0.5, (re * re), 1.0);
	} else if (t_1 <= 0.9999999999600385) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(-0.5, Float64(re * re), 1.0));
	elseif (t_1 <= 0.9999999999600385)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999600385], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. 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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999960038521

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.999999999960038521 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 100.0%

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

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999600385:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (* (cosh im) (fma -0.5 (* re re) 1.0))
     (if (<= t_1 0.9999999999600385) (* t_0 2.0) (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(im) * fma(-0.5, (re * re), 1.0);
	} else if (t_1 <= 0.9999999999600385) {
		tmp = t_0 * 2.0;
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(-0.5, Float64(re * re), 1.0));
	elseif (t_1 <= 0.9999999999600385)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999600385], N[(t$95$0 * 2.0), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. 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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999960038521

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 98.3%

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

   if 0.999999999960038521 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 100.0%

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

Alternative 5: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999600385:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma (* im im) 0.001388888888888889 0.041666666666666664)
        (* im im)
        0.5)
       (* im im)
       1.0)
      (fma -0.5 (* re re) 1.0))
     (if (<= t_1 0.9999999999600385) (* t_0 2.0) (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma(-0.5, (re * re), 1.0);
	} else if (t_1 <= 0.9999999999600385) {
		tmp = t_0 * 2.0;
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	elseif (t_1 <= 0.9999999999600385)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999600385], N[(t$95$0 * 2.0), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. 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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 96.9

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

   10. Applied rewrites 96.9%

       \[\leadsto \left(1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 96.9

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

   12. Applied rewrites 96.9%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999960038521

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 98.3%

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

   if 0.999999999960038521 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{elif}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999600385:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.995:\\ \;\;\;\;0.5 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 -0.002)
     (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
     (if (<= t_0 0.995)
       (* 0.5 2.0)
       (*
        (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5)
        (fma im im 2.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -0.002) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else if (t_0 <= 0.995) {
		tmp = 0.5 * 2.0;
	} else {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * fma(im, im, 2.0);
	}
	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 <= -0.002)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	elseif (t_0 <= 0.995)
		tmp = Float64(0.5 * 2.0);
	else
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im, im, 2.0));
	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, -0.002], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.995], N[(0.5 * 2.0), $MachinePrecision], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]]

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))) < -2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if -2e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.994999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 20.9%

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

   if 0.994999999999999996 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 78.1

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

   8. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999600385:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) 0.9999999999600385)
   (*
    (fma
     (fma
      (fma 0.001388888888888889 (* im im) 0.041666666666666664)
      (* im im)
      0.5)
     (* im im)
     1.0)
    (cos re))
   (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= 0.9999999999600385) {
		tmp = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * cos(re);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.9999999999600385)
		tmp = Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * cos(re));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.9999999999600385], N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999960038521

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 96.1

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

   7. Applied rewrites 96.1%

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

   if 0.999999999960038521 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.9999999999600385:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \cos re\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.002], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -2e-3

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Add Preprocessing
   3. 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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 45.9

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

   10. Applied rewrites 45.9%

       \[\leadsto \left(1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 45.9

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

   12. Applied rewrites 45.9%

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

   if -2e-3 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 86.9%

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

4. Final simplification 75.8%

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

Alternative 9: 47.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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))) <= 2.0d0) then
        tmp = 0.5d0 * 2.0d0
    else
        tmp = 0.5d0 * (im * im)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 50.7%

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

Alternative 10: 71.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;t\_0 \leq -0.0004 \lor \neg \left(t\_0 \leq 0.4998\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (or (<= t_0 -0.0004) (not (<= t_0 0.4998)))
     (*
      (fma
       (fma
        (fma (* im im) 0.001388888888888889 0.041666666666666664)
        (* im im)
        0.5)
       (* im im)
       1.0)
      (fma -0.5 (* re re) 1.0))
     (*
      (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5)
      (fma im im 2.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((t_0 <= -0.0004) || !(t_0 <= 0.4998)) {
		tmp = fma(fma(fma((im * im), 0.001388888888888889, 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if ((t_0 <= -0.0004) || !(t_0 <= 0.4998))
		tmp = Float64(fma(fma(fma(Float64(im * im), 0.001388888888888889, 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.0004], N[Not[LessEqual[t$95$0, 0.4998]], $MachinePrecision]], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -4.00000000000000019e-4 or 0.49980000000000002 < (*.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 73.4

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

   10. Applied rewrites 73.4%

       \[\leadsto \left(1 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)}\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-lft-identity 73.4

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

   12. Applied rewrites 73.4%

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

   if -4.00000000000000019e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < 0.49980000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.6

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

   8. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0004 \lor \neg \left(0.5 \cdot \cos re \leq 0.4998\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.001388888888888889, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;t\_0 \leq -0.0004 \lor \neg \left(t\_0 \leq 0.4998\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (or (<= t_0 -0.0004) (not (<= t_0 0.4998)))
     (*
      (fma (fma (* im im) 0.041666666666666664 0.5) (* im im) 1.0)
      (fma -0.5 (* re re) 1.0))
     (*
      (fma (fma 0.020833333333333332 (* re re) -0.25) (* re re) 0.5)
      (fma im im 2.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if ((t_0 <= -0.0004) || !(t_0 <= 0.4998)) {
		tmp = fma(fma((im * im), 0.041666666666666664, 0.5), (im * im), 1.0) * fma(-0.5, (re * re), 1.0);
	} else {
		tmp = fma(fma(0.020833333333333332, (re * re), -0.25), (re * re), 0.5) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if ((t_0 <= -0.0004) || !(t_0 <= 0.4998))
		tmp = Float64(fma(fma(Float64(im * im), 0.041666666666666664, 0.5), Float64(im * im), 1.0) * fma(-0.5, Float64(re * re), 1.0));
	else
		tmp = Float64(fma(fma(0.020833333333333332, Float64(re * re), -0.25), Float64(re * re), 0.5) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.0004], N[Not[LessEqual[t$95$0, 0.4998]], $MachinePrecision]], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-0.5 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision] + -0.25), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -4.00000000000000019e-4 or 0.49980000000000002 < (*.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cosh-undef N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.4

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

   10. Applied rewrites 70.4%

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

   if -4.00000000000000019e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < 0.49980000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.6

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

   8. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0004 \lor \neg \left(0.5 \cdot \cos re \leq 0.4998\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right), im \cdot im, 1\right) \cdot \mathsf{fma}\left(-0.5, re \cdot re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;t\_0 \leq -0.0004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.476:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.020833333333333332, -0.25\right) \cdot re, re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= t_0 -0.0004)
     (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
     (if (<= t_0 0.476)
       (* (fma (* (fma (* re re) 0.020833333333333332 -0.25) re) re 0.5) 2.0)
       (* 0.5 (fma im im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else if (t_0 <= 0.476) {
		tmp = fma((fma((re * re), 0.020833333333333332, -0.25) * re), re, 0.5) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (t_0 <= -0.0004)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	elseif (t_0 <= 0.476)
		tmp = Float64(fma(Float64(fma(Float64(re * re), 0.020833333333333332, -0.25) * re), re, 0.5) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0004], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.476], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] * re), $MachinePrecision] * re + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -4.00000000000000019e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if -4.00000000000000019e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < 0.47599999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.0

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

   8. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, re \cdot re, -0.25\right), re \cdot re, 0.5\right)} \cdot 2 \]
   9. Step-by-step derivation

   10. Applied rewrites 49.0%

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

   if 0.47599999999999998 < (*.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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 71.2%

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

Alternative 13: 59.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ \mathbf{if}\;t\_0 \leq -0.0004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.476:\\ \;\;\;\;\mathsf{fma}\left(0.020833333333333332 \cdot \left(re \cdot re\right), re \cdot re, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))))
   (if (<= t_0 -0.0004)
     (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
     (if (<= t_0 0.476)
       (* (fma (* 0.020833333333333332 (* re re)) (* re re) 0.5) 2.0)
       (* 0.5 (fma im im 2.0))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double tmp;
	if (t_0 <= -0.0004) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else if (t_0 <= 0.476) {
		tmp = fma((0.020833333333333332 * (re * re)), (re * re), 0.5) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	tmp = 0.0
	if (t_0 <= -0.0004)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	elseif (t_0 <= 0.476)
		tmp = Float64(fma(Float64(0.020833333333333332 * Float64(re * re)), Float64(re * re), 0.5) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0004], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.476], N[(N[(N[(0.020833333333333332 * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -4.00000000000000019e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if -4.00000000000000019e-4 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < 0.47599999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.0

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

   8. Applied rewrites 49.0%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 49.0%

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

   if 0.47599999999999998 < (*.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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 71.2%

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

Alternative 14: 59.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0004)
   (* (fma (* re re) -0.25 0.5) (fma im im 2.0))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0004) {
		tmp = fma((re * re), -0.25, 0.5) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0004)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0004], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -4.00000000000000019e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if -4.00000000000000019e-4 < (*.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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.1%

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

Alternative 15: 59.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0004:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot -0.25\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0004)
   (* (* (* re re) -0.25) (fma im im 2.0))
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0004) {
		tmp = ((re * re) * -0.25) * fma(im, im, 2.0);
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0004)
		tmp = Float64(Float64(Float64(re * re) * -0.25) * fma(im, im, 2.0));
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0004], N[(N[(N[(re * re), $MachinePrecision] * -0.25), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -4.00000000000000019e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 39.3%

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

   if -4.00000000000000019e-4 < (*.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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.1%

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

Alternative 16: 54.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \cos re \leq -0.0004:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (cos re)) -0.0004)
   (* (fma (* re re) -0.25 0.5) 2.0)
   (* 0.5 (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * cos(re)) <= -0.0004) {
		tmp = fma((re * re), -0.25, 0.5) * 2.0;
	} else {
		tmp = 0.5 * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * cos(re)) <= -0.0004)
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * 2.0);
	else
		tmp = Float64(0.5 * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.0004], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(0.5 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -4.00000000000000019e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 28.0

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

   8. Applied rewrites 28.0%

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

   if -4.00000000000000019e-4 < (*.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. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.1%

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

Alternative 17: 47.5% accurate, 26.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 75.0

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

5. Applied rewrites 75.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 45.6%

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

Alternative 18: 28.9% accurate, 52.7× speedup

Math

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

FPCore

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

C

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

Fortran

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 * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 52.1%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 28.0%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))