math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 13

Speedup: 0.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. lift-neg.f64 N/A

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

   14. lift-exp.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. lift-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-flip N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 99.5

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

6. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\cos re \cdot 0.5, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(im, 0.5, -1\right), im, 1\right)}, \left(\cos re \cdot 0.5\right) \cdot e^{im}\right) \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im\_m} + e^{im\_m}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\cosh im\_m \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.999999999999999:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im\_m, im\_m, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im_m)) (exp im_m)))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* re re) -0.25 0.5) (* (cosh im_m) 2.0))
     (if (<= t_1 0.999999999999999)
       (* t_0 (fma im_m im_m 2.0))
       (* (* 2.0 (cosh im_m)) 0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im_m) + exp(im_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((re * re), -0.25, 0.5) * (cosh(im_m) * 2.0);
	} else if (t_1 <= 0.999999999999999) {
		tmp = t_0 * fma(im_m, im_m, 2.0);
	} else {
		tmp = (2.0 * cosh(im_m)) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(cosh(im_m) * 2.0));
	elseif (t_1 <= 0.999999999999999)
		tmp = Float64(t_0 * fma(im_m, im_m, 2.0));
	else
		tmp = Float64(Float64(2.0 * cosh(im_m)) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[Cosh[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999999999999999], N[(t$95$0 * N[(im$95$m * im$95$m + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \]

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(e^{-im} + e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. +-commutative N/A

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

      6. cosh-undef-rev N/A

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

      7. lift-cosh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.8

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

   6. Applied rewrites 63.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999999999001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.0

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

   4. Applied rewrites 76.0%

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

   if 0.999999999999999001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im\_m} + e^{im\_m}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(\cosh im\_m \cdot 2\right)\\ \mathbf{elif}\;t\_1 \leq 0.999999999999999:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im_m)) (exp im_m)))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* re re) -0.25 0.5) (* (cosh im_m) 2.0))
     (if (<= t_1 0.999999999999999) (* t_0 2.0) (* (* 2.0 (cosh im_m)) 0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im_m) + exp(im_m));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((re * re), -0.25, 0.5) * (cosh(im_m) * 2.0);
	} else if (t_1 <= 0.999999999999999) {
		tmp = t_0 * 2.0;
	} else {
		tmp = (2.0 * cosh(im_m)) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im_m)) + exp(im_m)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(re * re), -0.25, 0.5) * Float64(cosh(im_m) * 2.0));
	elseif (t_1 <= 0.999999999999999)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(Float64(2.0 * cosh(im_m)) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(re * re), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision] * N[(N[Cosh[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999999999999999], N[(t$95$0 * 2.0), $MachinePrecision], N[(N[(2.0 * N[Cosh[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \]

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(e^{-im} + e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. +-commutative N/A

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

      6. cosh-undef-rev N/A

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

      7. lift-cosh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.8

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

   6. Applied rewrites 63.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.999999999999999001

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   if 0.999999999999999001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

Alternative 5: 78.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(re \cdot re, -0.25, 0.5\right) \cdot \left(e^{-im} + e^{im}\right) \]

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re \cdot re, -0.25, 0.5\right)} \cdot \left(e^{-im} + e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. +-commutative N/A

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

      6. cosh-undef-rev N/A

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

      7. lift-cosh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 63.8

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

   6. Applied rewrites 63.8%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

Alternative 6: 75.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.9

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

   7. Applied rewrites 32.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. fabs-pow2-rev N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 35.7

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

   9. Applied rewrites 35.7%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

Alternative 7: 72.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.9

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

   7. Applied rewrites 32.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 8.2

         \[\leadsto \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot -0.5\right)\right) \cdot 2 \]

   10. Applied rewrites 8.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

Alternative 8: 63.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.9

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

   7. Applied rewrites 32.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 8.2

         \[\leadsto \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot -0.5\right)\right) \cdot 2 \]

   10. Applied rewrites 8.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 57.0

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

   9. Applied rewrites 57.0%

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

Alternative 9: 63.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.9

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

   7. Applied rewrites 32.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 8.2

         \[\leadsto \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot -0.5\right)\right) \cdot 2 \]

   10. Applied rewrites 8.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 56.8

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

   10. Applied rewrites 56.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 56.8

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 56.8

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

   12. Applied rewrites 56.8%

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

Alternative 10: 63.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * cos(re)) * (exp(-im_m) + exp(im_m))
    if (t_0 <= (-0.05d0)) then
        tmp = (0.5d0 * ((re * re) * (-0.5d0))) * 2.0d0
    else if (t_0 <= 2.0d0) then
        tmp = (0.5d0 * 1.0d0) * 2.0d0
    else
        tmp = (im_m * im_m) * ((im_m * im_m) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

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

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = (0.5 * math.cos(re)) * (math.exp(-im_m) + math.exp(im_m))
	tmp = 0
	if t_0 <= -0.05:
		tmp = (0.5 * ((re * re) * -0.5)) * 2.0
	elif t_0 <= 2.0:
		tmp = (0.5 * 1.0) * 2.0
	else:
		tmp = (im_m * im_m) * ((im_m * im_m) * 0.041666666666666664)
	return tmp

Julia

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

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = (0.5 * cos(re)) * (exp(-im_m) + exp(im_m));
	tmp = 0.0;
	if (t_0 <= -0.05)
		tmp = (0.5 * ((re * re) * -0.5)) * 2.0;
	elseif (t_0 <= 2.0)
		tmp = (0.5 * 1.0) * 2.0;
	else
		tmp = (im_m * im_m) * ((im_m * im_m) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.9

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

   7. Applied rewrites 32.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 8.2

         \[\leadsto \left(0.5 \cdot \left(\left(re \cdot re\right) \cdot -0.5\right)\right) \cdot 2 \]

   10. Applied rewrites 8.2%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 28.8%

      \[\leadsto \left(0.5 \cdot \color{blue}{1}\right) \cdot 2 \]

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 31.0

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

   10. Applied rewrites 31.0%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. unpow-prod-down N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 31.0

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

   12. Applied rewrites 31.0%

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

Alternative 11: 56.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (((0.5d0 * cos(re)) * (exp(-im_m) + exp(im_m))) <= 2.0d0) then
        tmp = (0.5d0 * 1.0d0) * 2.0d0
    else
        tmp = (im_m * im_m) * ((im_m * im_m) * 0.041666666666666664d0)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (((0.5 * Math.cos(re)) * (Math.exp(-im_m) + Math.exp(im_m))) <= 2.0) {
		tmp = (0.5 * 1.0) * 2.0;
	} else {
		tmp = (im_m * im_m) * ((im_m * im_m) * 0.041666666666666664);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp(-im_m) + math.exp(im_m))) <= 2.0:
		tmp = (0.5 * 1.0) * 2.0
	else:
		tmp = (im_m * im_m) * ((im_m * im_m) * 0.041666666666666664)
	return tmp

Julia

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

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (((0.5 * cos(re)) * (exp(-im_m) + exp(im_m))) <= 2.0)
		tmp = (0.5 * 1.0) * 2.0;
	else
		tmp = (im_m * im_m) * ((im_m * im_m) * 0.041666666666666664);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] + N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(0.5 * 1.0), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(N[(im$95$m * im$95$m), $MachinePrecision] * 0.041666666666666664), $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. Taylor expanded in im around 0

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

   4. Applied rewrites 50.7%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 28.8%

      \[\leadsto \left(0.5 \cdot \color{blue}{1}\right) \cdot 2 \]

   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. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cosh-undef N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cosh.f64 65.5

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.0

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 31.0

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

   10. Applied rewrites 31.0%

       \[\leadsto \left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. unpow-prod-down N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 31.0

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

   12. Applied rewrites 31.0%

       \[\leadsto \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.7% accurate, 7.0× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (fma im_m im_m 2.0) 0.5))

C

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

Julia

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(im$95$m * im$95$m + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cosh-undef N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cosh.f64 65.5

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

4. Applied rewrites 65.5%

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

5. Taylor expanded in im around 0

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

   1. cosh-undef-rev N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 47.7

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

7. Applied rewrites 47.7%

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

Alternative 13: 28.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \left(0.5 \cdot 1\right) \cdot 2 \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (* (* 0.5 1.0) 2.0))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return (0.5 * 1.0) * 2.0;
}

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = (0.5d0 * 1.0d0) * 2.0d0
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return (0.5 * 1.0) * 2.0;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return (0.5 * 1.0) * 2.0

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(Float64(0.5 * 1.0) * 2.0)
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = (0.5 * 1.0) * 2.0;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(N[(0.5 * 1.0), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in re around 0

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

7. Applied rewrites 28.8%

   \[\leadsto \left(0.5 \cdot \color{blue}{1}\right) \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))