math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 17

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return exp(re) * cos(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 = exp(re) * cos(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return exp(re) * cos(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 = exp(re) * cos(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return exp(re) * cos(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 = exp(re) * cos(im)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 0.999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, 1 + re\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (/ (* t_1 t_1) (- 1.0 re))
      (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
     (if (<= t_0 -0.02)
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (if (or (<= t_0 5e-8) (not (<= t_0 0.999)))
         (exp re)
         (* (fma (* re re) 0.5 (+ 1.0 re)) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * t_1) / (1.0 - re)) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else if ((t_0 <= 5e-8) || !(t_0 <= 0.999)) {
		tmp = exp(re);
	} else {
		tmp = fma((re * re), 0.5, (1.0 + re)) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * t_1) / Float64(1.0 - re)) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.02)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	elseif ((t_0 <= 5e-8) || !(t_0 <= 0.999))
		tmp = exp(re);
	else
		tmp = Float64(fma(Float64(re * re), 0.5, Float64(1.0 + re)) * cos(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-8], N[Not[LessEqual[t$95$0, 0.999]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5 + N[(1.0 + re), $MachinePrecision]), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8 or 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 99.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{e^{re}} \]

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   7. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(re \cdot re, \color{blue}{0.5}, 1 + re\right) \cdot \cos im \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right) \cdot \mathsf{fma}\left(-re, re, 1\right)}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8} \lor \neg \left(e^{re} \cdot \cos im \leq 0.999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, 1 + re\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 0.999\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (/ (* t_1 t_1) (- 1.0 re))
      (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-8) (not (<= t_0 0.999)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * t_1) / (1.0 - re)) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * t_1) / Float64(1.0 - re)) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-8], N[Not[LessEqual[t$95$0, 0.999]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8 or 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 99.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right) \cdot \mathsf{fma}\left(-re, re, 1\right)}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 5 \cdot 10^{-8} \lor \neg \left(e^{re} \cdot \cos im \leq 0.999\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 0.999\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (/ (* t_1 t_1) (- 1.0 re))
      (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-8) (not (<= t_0 0.999)))))
       (/ (cos im) (- 1.0 re))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * t_1) / (1.0 - re)) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999))) {
		tmp = cos(im) / (1.0 - re);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * t_1) / Float64(1.0 - re)) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999)))
		tmp = Float64(cos(im) / Float64(1.0 - re));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-8], N[Not[LessEqual[t$95$0, 0.999]], $MachinePrecision]]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{\cos im}{\color{blue}{1 + -1 \cdot re}} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

         \[\leadsto \frac{\cos im}{1 - \color{blue}{1} \cdot re} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\cos im}{1 - \color{blue}{re}} \]

      4. lower--.f64 97.5

         \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   7. Applied rewrites 97.5%

      \[\leadsto \frac{\cos im}{\color{blue}{1 - re}} \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8 or 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 99.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right) \cdot \mathsf{fma}\left(-re, re, 1\right)}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 5 \cdot 10^{-8} \lor \neg \left(e^{re} \cdot \cos im \leq 0.999\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 0.999\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (/ (* t_1 t_1) (- 1.0 re))
      (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-8) (not (<= t_0 0.999)))))
       (* (+ 1.0 re) (cos im))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * t_1) / (1.0 - re)) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999))) {
		tmp = (1.0 + re) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * t_1) / Float64(1.0 - re)) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999)))
		tmp = Float64(Float64(1.0 + re) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-8], N[Not[LessEqual[t$95$0, 0.999]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8 or 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 99.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right) \cdot \mathsf{fma}\left(-re, re, 1\right)}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 5 \cdot 10^{-8} \lor \neg \left(e^{re} \cdot \cos im \leq 0.999\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 0.999\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (/ (* t_1 t_1) (- 1.0 re))
      (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-8) (not (<= t_0 0.999)))))
       (cos im)
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * t_1) / (1.0 - re)) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else if ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999))) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * t_1) / Float64(1.0 - re)) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999)))
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-8], N[Not[LessEqual[t$95$0, 0.999]], $MachinePrecision]]], $MachinePrecision]], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8 or 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 99.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right) \cdot \mathsf{fma}\left(-re, re, 1\right)}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 5 \cdot 10^{-8} \lor \neg \left(e^{re} \cdot \cos im \leq 0.999\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0.999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (/ (* t_1 t_1) (- 1.0 re))
      (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
     (if (<= t_0 -0.02)
       (cos im)
       (if (<= t_0 0.0)
         (* (* im im) -0.5)
         (if (<= t_0 0.999)
           (cos im)
           (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * t_1) / (1.0 - re)) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 0.999) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * t_1) / Float64(1.0 - re)) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (t_0 <= 0.999)
		tmp = cos(im);
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.999], N[Cos[im], $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or -0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 22.7%

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

   if 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 98.7

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 78.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right) \cdot \mathsf{fma}\left(-re, re, 1\right)}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 0.999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 -0.02)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
     (if (or (<= t_0 5e-8) (not (<= t_0 0.999)))
       (exp re)
       (/
        (cos im)
        (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else if ((t_0 <= 5e-8) || !(t_0 <= 0.999)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	elseif ((t_0 <= 5e-8) || !(t_0 <= 0.999))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-8], N[Not[LessEqual[t$95$0, 0.999]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \cos im \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot re} + 1\right) \cdot \cos im \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), re, 1\right)} \cdot \cos im \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \left(\mathsf{neg}\left(re\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}, re, 1\right) \cdot \cos im \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}, re, 1\right) \cdot \cos im \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, re, 1\right) \cdot \cos im \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1, re, 1\right) \cdot \cos im \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right) \cdot re} + 1, re, 1\right) \cdot \cos im \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot re, re, 1\right)}, re, 1\right) \cdot \cos im \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, re, 1\right), re, 1\right) \cdot \cos im \]

      11. lower-fma.f64 97.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}, re, 1\right), re, 1\right) \cdot \cos im \]

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \cos im \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8 or 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 99.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{e^{re}} \]

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\cos im}{\color{blue}{re \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\cos im}{\color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1\right) \cdot re} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\cos im}{\color{blue}{\mathsf{fma}\left(re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1, re, 1\right)}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\cos im}{\mathsf{fma}\left(\color{blue}{re \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) - 1}, re, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos im}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) \cdot re} - 1, re, 1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\cos im}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot re\right) \cdot re} - 1, re, 1\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\cos im}{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{6} \cdot re + \frac{1}{2}\right)} \cdot re - 1, re, 1\right)} \]

      8. lower-fma.f64 98.6

         \[\leadsto \frac{\cos im}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right)} \cdot re - 1, re, 1\right)} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\cos im}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8} \lor \neg \left(e^{re} \cdot \cos im \leq 0.999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_0 \leq 0.999\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (or (<= t_0 -0.02) (not (or (<= t_0 5e-8) (not (<= t_0 0.999)))))
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
     (exp re))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999))) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if ((t_0 <= -0.02) || !((t_0 <= 5e-8) || !(t_0 <= 0.999)))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 5e-8], N[Not[LessEqual[t$95$0, 0.999]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.998999999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \cos im \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot re} + 1\right) \cdot \cos im \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), re, 1\right)} \cdot \cos im \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 - \left(\mathsf{neg}\left(re\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}, re, 1\right) \cdot \cos im \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)}, re, 1\right) \cdot \cos im \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, re, 1\right) \cdot \cos im \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{re} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1, re, 1\right) \cdot \cos im \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right) \cdot re} + 1, re, 1\right) \cdot \cos im \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot re, re, 1\right)}, re, 1\right) \cdot \cos im \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, re, 1\right), re, 1\right) \cdot \cos im \]

      11. lower-fma.f64 97.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}, re, 1\right), re, 1\right) \cdot \cos im \]

   5. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \cos im \]

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8 or 0.998999999999999999 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 99.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{e^{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 5 \cdot 10^{-8} \lor \neg \left(e^{re} \cdot \cos im \leq 0.999\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot t\_1}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (/ (* t_1 t_1) (- 1.0 re))
      (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))
     (if (<= t_0 5e-8)
       (* (* im im) -0.5)
       (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((t_1 * t_1) / (1.0 - re)) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	} else if (t_0 <= 5e-8) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * t_1) / Float64(1.0 - re)) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	elseif (t_0 <= 5e-8)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-8], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -inf.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 0.0

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

   8. Applied rewrites 0.0%

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

   10. Applied rewrites 0.0%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 16.2%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 78.9

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 78.9%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 63.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-re, re, 1\right) \cdot \mathsf{fma}\left(-re, re, 1\right)}{1 - re} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;re - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 5e-8)
     (* (* im im) -0.5)
     (if (<= t_0 2.0) (- re -1.0) (* (fma 0.5 re 1.0) re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 2.0) {
		tmp = re - -1.0;
	} else {
		tmp = fma(0.5, re, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 5e-8)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (t_0 <= 2.0)
		tmp = Float64(re - -1.0);
	else
		tmp = Float64(fma(0.5, re, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-8], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re - -1.0), $MachinePrecision], N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 13.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.3%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 68.0

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 68.0%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 68.0%

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

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 97.3

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 97.3%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 42.5%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 42.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;re - -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, re, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;re - -1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 5e-8)
     (* (* im im) -0.5)
     (if (<= t_0 2.0) (- re -1.0) (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 2.0) {
		tmp = re - -1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * cos(im)
    if (t_0 <= 5d-8) then
        tmp = (im * im) * (-0.5d0)
    else if (t_0 <= 2.0d0) then
        tmp = re - (-1.0d0)
    else
        tmp = (re * re) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= 5e-8) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 2.0) {
		tmp = re - -1.0;
	} else {
		tmp = (re * re) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= 5e-8:
		tmp = (im * im) * -0.5
	elif t_0 <= 2.0:
		tmp = re - -1.0
	else:
		tmp = (re * re) * 0.5
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 5e-8)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (t_0 <= 2.0)
		tmp = Float64(re - -1.0);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= 5e-8)
		tmp = (im * im) * -0.5;
	elseif (t_0 <= 2.0)
		tmp = re - -1.0;
	else
		tmp = (re * re) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-8], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(re - -1.0), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 13.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.3%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 68.0

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 68.0%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 68.0%

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

   if 2 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 97.3

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 97.3%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 42.5%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 42.5%

       \[\leadsto \left(re \cdot re\right) \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 2:\\ \;\;\;\;re - -1\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 5e-8)
   (* (* im im) -0.5)
   (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 5e-8) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 5e-8)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 13.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.3%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 78.9

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 78.9%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 63.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (* im im) -0.5)
   (fma (fma 0.5 re 1.0) re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 30.0

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

   5. Applied rewrites 30.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 13.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.5%

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

   if -0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 79.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 79.1%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 58.2%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;re - -1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 5e-8) (* (* im im) -0.5) (- re -1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 5e-8) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = re - -1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 5d-8) then
        tmp = (im * im) * (-0.5d0)
    else
        tmp = re - (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 5e-8) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = re - -1.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 5e-8:
		tmp = (im * im) * -0.5
	else:
		tmp = re - -1.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 5e-8)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = Float64(re - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= 5e-8)
		tmp = (im * im) * -0.5;
	else
		tmp = re - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(re - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 4.9999999999999998e-8

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 29.8

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 13.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 24.3%

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

   if 4.9999999999999998e-8 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 78.9

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 78.9%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 44.7%

       \[\leadsto re - \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;re - -1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 28.7% accurate, 51.5× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (- re -1.0))

C

double code(double re, double im) {
	return re - -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 = re - (-1.0d0)
end function

Java

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

Python

def code(re, im):
	return re - -1.0

Julia

function code(re, im)
	return Float64(re - -1.0)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(re - -1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. remove-double-neg N/A

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

   4. rec-exp N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. lift-cos.f64 N/A

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

   8. sin-PI/2 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. sin-PI/2 N/A

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

   12. *-commutative N/A

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

   13. *-lft-identity N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-neg.f64 100.0

       \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
6. Step-by-step derivation

   1. exp-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

   2. remove-double-div N/A

      \[\leadsto \color{blue}{e^{re}} \]

   3. lower-exp.f64 70.4

      \[\leadsto \color{blue}{e^{re}} \]

7. Applied rewrites 70.4%

   \[\leadsto \color{blue}{e^{re}} \]

8. Taylor expanded in re around 0

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

10. Applied rewrites 28.3%

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

11. Final simplification 28.3%

    \[\leadsto re - -1 \]
12. Add Preprocessing

Alternative 17: 28.2% accurate, 206.0× 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%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. remove-double-neg N/A

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

   4. rec-exp N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. lift-cos.f64 N/A

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

   8. sin-PI/2 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. sin-PI/2 N/A

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

   12. *-commutative N/A

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

   13. *-lft-identity N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-neg.f64 100.0

       \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
6. Step-by-step derivation

   1. exp-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

   2. remove-double-div N/A

      \[\leadsto \color{blue}{e^{re}} \]

   3. lower-exp.f64 70.4

      \[\leadsto \color{blue}{e^{re}} \]

7. Applied rewrites 70.4%

   \[\leadsto \color{blue}{e^{re}} \]

8. Taylor expanded in re around 0

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

10. Applied rewrites 27.9%

    \[\leadsto 1 \]

11. Final simplification 27.9%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))