math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 7

Speedup: 1.2×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot \left(\cosh im \cdot 2\right)\\ \mathbf{elif}\;t\_0 \leq 0.9945036965538625:\\ \;\;\;\;\cos re\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* re -0.25) re 0.5) (* (cosh im) 2.0))
     (if (<= t_0 0.9945036965538625) (cos re) (* (* 2.0 (cosh im)) 0.5)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((re * -0.25), re, 0.5) * (cosh(im) * 2.0);
	} else if (t_0 <= 0.9945036965538625) {
		tmp = cos(re);
	} else {
		tmp = (2.0 * cosh(im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(re * -0.25), re, 0.5) * Float64(cosh(im) * 2.0));
	elseif (t_0 <= 0.9945036965538625)
		tmp = cos(re);
	else
		tmp = Float64(Float64(2.0 * cosh(im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(re * -0.25), $MachinePrecision] * re + 0.5), $MachinePrecision] * N[(N[Cosh[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9945036965538625], N[Cos[re], $MachinePrecision], N[(N[(2.0 * N[Cosh[im], $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

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

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied rewrites 63.6%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.9945036965538625

   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

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

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied rewrites 63.6%

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

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 50.4%

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

   if 0.9945036965538625 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.5%

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

Alternative 3: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(re \cdot -0.25, re, 0.5\right) \cdot \left(\cosh im \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.02)
   (* (fma (* re -0.25) re 0.5) (* (cosh im) 2.0))
   (* (* 2.0 (cosh im)) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma((re * -0.25), re, 0.5) * (cosh(im) * 2.0);
	} else {
		tmp = (2.0 * cosh(im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = Float64(fma(Float64(re * -0.25), re, 0.5) * Float64(cosh(im) * 2.0));
	else
		tmp = Float64(Float64(2.0 * cosh(im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(re * -0.25), $MachinePrecision] * re + 0.5), $MachinePrecision] * N[(N[Cosh[im], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[im], $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))) < -0.0200000000000000004

   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

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

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied rewrites 63.6%

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

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.5%

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

Alternative 4: 72.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(re * -0.25), $MachinePrecision] * re + 0.5), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied rewrites 63.6%

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

   8. Applied rewrites 12.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

   6. Applied rewrites 65.5%

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

Alternative 5: 35.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(re * -0.25), $MachinePrecision] * re + 0.5), $MachinePrecision] * 1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) < -0.0100000000000000002

   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

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

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied rewrites 63.6%

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

   8. Applied rewrites 12.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (cos.f64 re))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

   4. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied rewrites 63.6%

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

   8. Applied rewrites 32.7%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{1} \]

   11. Applied rewrites 28.6%

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

Alternative 6: 28.6% accurate, 62.8× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

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

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

6. Applied rewrites 63.6%

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

8. Applied rewrites 32.7%

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

9. Taylor expanded in re around 0

   \[\leadsto \color{blue}{1} \]

11. Applied rewrites 28.6%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Alternative 7: 8.6% accurate, 62.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.5)

C

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

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
end function

Java

public static double code(double re, double im) {
	return 0.5;
}

Python

def code(re, im):
	return 0.5

Julia

function code(re, im)
	return 0.5
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5;
end

Wolfram

code[re_, im_] := 0.5

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

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

4. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot \left(e^{-im} + e^{im}\right) \]

6. Applied rewrites 63.6%

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

8. Applied rewrites 12.7%

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

9. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]

11. Applied rewrites 8.6%

    \[\leadsto \color{blue}{0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))