math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 17

Speedup: 1.0×

• 24.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-19} \lor \neg \left(t\_0 \leq 0.9999999999996004\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.02)
       (* (- re -1.0) (cos im))
       (if (or (<= t_0 1e-19) (not (<= t_0 0.9999999999996004)))
         (exp re)
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.02) {
		tmp = (re - -1.0) * cos(im);
	} else if ((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004)) {
		tmp = exp(re);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(re - -1.0) * cos(im));
	elseif ((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004))
		tmp = exp(re);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	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, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(re - -1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-19], N[Not[LessEqual[t$95$0, 0.9999999999996004]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.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

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 100.0

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 100.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.9999999999999998e-20 or 0.999999999999600431 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

   if 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999600431

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-19} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999999996004\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-19} \lor \neg \left(t\_0 \leq 0.9999999999996004\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.02)
       (* (- re -1.0) (cos im))
       (if (or (<= t_0 1e-19) (not (<= t_0 0.9999999999996004)))
         (exp re)
         (* (fma (fma 0.5 re 1.0) re 1.0) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.02) {
		tmp = (re - -1.0) * cos(im);
	} else if ((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004)) {
		tmp = exp(re);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(re - -1.0) * cos(im));
	elseif ((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004))
		tmp = exp(re);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	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, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(re - -1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-19], N[Not[LessEqual[t$95$0, 0.9999999999996004]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.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

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 100.0

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 100.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.9999999999999998e-20 or 0.999999999999600431 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

   if 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999600431

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-19} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999999996004\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 10^{-19} \lor \neg \left(t\_0 \leq 0.9999999999996004\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 1e-19) (not (<= t_0 0.9999999999996004)))))
       (* (- 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 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if ((t_0 <= -0.02) || !((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004))) {
		tmp = (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 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif ((t_0 <= -0.02) || !((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004)))
		tmp = Float64(Float64(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[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 1e-19], N[Not[LessEqual[t$95$0, 0.9999999999996004]], $MachinePrecision]]], $MachinePrecision]], N[(N[(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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999600431

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 98.4

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 98.4%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.9999999999999998e-20 or 0.999999999999600431 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 10^{-19} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999999996004\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-19} \lor \neg \left(t\_0 \leq 0.9999999999996004\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, re, \cos im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.02)
       (cos im)
       (if (or (<= t_0 1e-19) (not (<= t_0 0.9999999999996004)))
         (exp re)
         (fma 1.0 re (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if ((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004)) {
		tmp = exp(re);
	} else {
		tmp = fma(1.0, re, cos(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif ((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004))
		tmp = exp(re);
	else
		tmp = fma(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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-19], N[Not[LessEqual[t$95$0, 0.9999999999996004]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(1.0 * re + 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.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

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

      1. lift-cos.f64 99.8

         \[\leadsto \cos im \]

   5. Applied rewrites 99.8%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.9999999999999998e-20 or 0.999999999999600431 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

   if 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999600431

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\cos im, re, \cos im\right) \]

      5. lift-cos.f64 97.2

         \[\leadsto \mathsf{fma}\left(\cos im, re, \cos im\right) \]

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(1, re, \cos im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 10^{-19} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999999996004\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, re, \cos im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 10^{-19} \lor \neg \left(t\_0 \leq 0.9999999999996004\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 1e-19) (not (<= t_0 0.9999999999996004)))))
       (cos im)
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if ((t_0 <= -0.02) || !((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004))) {
		tmp = 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 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif ((t_0 <= -0.02) || !((t_0 <= 1e-19) || !(t_0 <= 0.9999999999996004)))
		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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 1e-19], N[Not[LessEqual[t$95$0, 0.9999999999996004]], $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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999600431

   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. lift-cos.f64 96.7

         \[\leadsto \cos im \]

   5. Applied rewrites 96.7%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 9.9999999999999998e-20 or 0.999999999999600431 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{e^{re}} \]
   4. Step-by-step derivation

      1. lift-exp.f64 100.0

         \[\leadsto e^{re} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 10^{-19} \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999999999996004\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.9998:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (* (* re re) 0.16666666666666666) re 1.0)
      (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.02)
       (cos im)
       (if (<= t_0 0.0)
         (* (- re -1.0) (* (* im im) -0.5))
         (if (<= t_0 0.9998)
           (cos im)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (- (* 0.041666666666666664 (* im im)) 0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = (re - -1.0) * ((im * im) * -0.5);
	} else if (t_0 <= 0.9998) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(re - -1.0) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.9998)
		tmp = cos(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 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, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(re - -1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9998], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   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. lift-cos.f64 94.7

         \[\leadsto \cos im \]

   5. Applied rewrites 94.7%

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

   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

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 2.3

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 2.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 2.0

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 29.2

          \[\leadsto \left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right) \]

   11. Applied rewrites 29.2%

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

   if 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, \color{blue}{{im}^{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {\color{blue}{im}}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot \color{blue}{im}, 1\right) \]

      9. lower-*.f64 91.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot \color{blue}{im}, 1\right) \]

   8. Applied rewrites 91.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0.9998:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* (- re -1.0) (* (* im im) -0.5))
     (if (<= t_0 0.9998)
       1.0
       (*
        (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
        (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (re - -1.0) * ((im * im) * -0.5);
	} else if (t_0 <= 0.9998) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(re - -1.0) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.9998)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 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, 0.0], N[(N[(re - -1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9998], 1.0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 30.3

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 18.2

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

   8. Applied rewrites 18.2%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 32.5

          \[\leadsto \left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right) \]

   11. Applied rewrites 32.5%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99980000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lift-cos.f64 90.2

         \[\leadsto \cos im \]

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 20.8%

      \[\leadsto 1 \]

   if 0.99980000000000002 < (*.f64 (exp.f64 re) (cos.f64 im))

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \left(\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, \color{blue}{{im}^{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {\color{blue}{im}}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, re, \frac{1}{2}\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot \color{blue}{im}, 1\right) \]

      9. lower-*.f64 91.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot \color{blue}{im}, 1\right) \]

   8. Applied rewrites 91.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 41.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 + re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.5, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 1e-19)
     (fma (* im im) -0.5 1.0)
     (if (<= t_0 2.0) (+ 1.0 re) (fma (* re re) 0.5 re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 1e-19) {
		tmp = fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + re;
	} else {
		tmp = fma((re * re), 0.5, re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 1e-19)
		tmp = fma(Float64(im * im), -0.5, 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 + re);
	else
		tmp = fma(Float64(re * re), 0.5, 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, 1e-19], N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 + re), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * 0.5 + re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 9.9999999999999998e-20

   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. lift-cos.f64 30.0

         \[\leadsto \cos im \]

   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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 14.9

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

   8. Applied rewrites 14.9%

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

   if 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 71.9

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

   8. Applied rewrites 71.9%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 + re \]
   10. Step-by-step derivation

       1. lower-+.f64 71.4

          \[\leadsto 1 + re \]

   11. Applied rewrites 71.4%

       \[\leadsto 1 + re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 62.2

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in re around inf

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

       1. distribute-lft-in N/A

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

       2. inv-pow N/A

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

       3. pow-prod-up N/A

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

       4. metadata-eval N/A

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

       5. unpow1 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 62.2

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

   11. Applied rewrites 62.2%

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

Alternative 10: 41.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 + re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 1e-19)
     (fma (* im im) -0.5 1.0)
     (if (<= t_0 2.0) (+ 1.0 re) (* (* re re) 0.5)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 1e-19) {
		tmp = fma((im * im), -0.5, 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 + re;
	} 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 <= 1e-19)
		tmp = fma(Float64(im * im), -0.5, 1.0);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 + re);
	else
		tmp = Float64(Float64(re * re) * 0.5);
	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, 1e-19], N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 + re), $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)) < 9.9999999999999998e-20

   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. lift-cos.f64 30.0

         \[\leadsto \cos im \]

   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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 14.9

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

   8. Applied rewrites 14.9%

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

   if 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 71.9

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

   8. Applied rewrites 71.9%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 + re \]
   10. Step-by-step derivation

       1. lower-+.f64 71.4

          \[\leadsto 1 + re \]

   11. Applied rewrites 71.4%

       \[\leadsto 1 + re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 62.2

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 62.2

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

   11. Applied rewrites 62.2%

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

Alternative 11: 52.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 10^{-19}:\\ \;\;\;\;\left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 1e-19)
   (* (- re -1.0) (* (* im im) -0.5))
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 1e-19) {
		tmp = (re - -1.0) * ((im * im) * -0.5);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 1e-19)
		tmp = Float64(Float64(re - -1.0) * Float64(Float64(im * im) * -0.5));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 1e-19], N[(N[(re - -1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 9.9999999999999998e-20

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 29.8

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.9

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

   8. Applied rewrites 17.9%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 32.0

          \[\leadsto \left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right) \]

   11. Applied rewrites 32.0%

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

   if 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im))

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 71.4%

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

Alternative 12: 49.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (- re -1.0) (* (* 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 = (re - -1.0) * ((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(re - -1.0) * 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[(re - -1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 30.3

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 18.2

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

   8. Applied rewrites 18.2%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 32.5

          \[\leadsto \left(re - -1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right) \]

   11. Applied rewrites 32.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 67.8

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

   8. Applied rewrites 67.8%

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

Alternative 13: 42.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* re (fma (* im im) -0.5 1.0))
   (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 = re * fma((im * im), -0.5, 1.0);
	} 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(re * fma(Float64(im * im), -0.5, 1.0));
	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[(re * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(re - -1 \cdot 1\right) \cdot \cos im \]

      5. metadata-eval N/A

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 30.3

         \[\leadsto \left(re - -1\right) \cdot \cos im \]

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 18.2

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

   8. Applied rewrites 18.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 17.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 67.8

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

   8. Applied rewrites 67.8%

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

Alternative 14: 41.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 1e-19)
   (fma (* im im) -0.5 1.0)
   (fma (fma 0.5 re 1.0) re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 1e-19) {
		tmp = fma((im * im), -0.5, 1.0);
	} 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)) <= 1e-19)
		tmp = fma(Float64(im * im), -0.5, 1.0);
	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], 1e-19], N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $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)) < 9.9999999999999998e-20

   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. lift-cos.f64 30.0

         \[\leadsto \cos im \]

   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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 14.9

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

   8. Applied rewrites 14.9%

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

   if 9.9999999999999998e-20 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 68.7

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

   8. Applied rewrites 68.7%

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

Alternative 15: 38.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 2.0) {
		tmp = 1.0 + re;
	} 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) :: tmp
    if ((exp(re) * cos(im)) <= 2.0d0) then
        tmp = 1.0d0 + re
    else
        tmp = (re * re) * 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 34.1

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

   8. Applied rewrites 34.1%

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

   9. Taylor expanded in re around 0

      \[\leadsto 1 + re \]
   10. Step-by-step derivation

       1. lower-+.f64 33.9

          \[\leadsto 1 + re \]

   11. Applied rewrites 33.9%

       \[\leadsto 1 + re \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 62.2

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 62.2

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

   11. Applied rewrites 62.2%

       \[\leadsto \left(re \cdot re\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 29.5% accurate, 51.5× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 re))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return 1.0 + re

Julia

function code(re, im)
	return Float64(1.0 + re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[(1.0 + re), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt1-in N/A

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

   6. +-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 64.7

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

5. Applied rewrites 64.7%

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

6. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 39.3

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

8. Applied rewrites 39.3%

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

9. Taylor expanded in re around 0

   \[\leadsto 1 + re \]
10. Step-by-step derivation

    1. lower-+.f64 28.6

       \[\leadsto 1 + re \]

11. Applied rewrites 28.6%

    \[\leadsto 1 + re \]
12. Add Preprocessing

Alternative 17: 29.0% 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. Taylor expanded in re around 0

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

   1. lift-cos.f64 50.0

      \[\leadsto \cos im \]

5. Applied rewrites 50.0%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 28.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025076 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))