math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 14

Speedup: 1.0×

• 24.5% 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

Alternative 2: 92.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos im \cdot \left(re - -1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9988622371733984:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right) \cdot im, im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (cos im) (- re -1.0))) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_1 -0.02)
       t_0
       (if (<= t_1 0.0)
         (* (exp re) (fma im (* im -0.5) 1.0))
         (if (<= t_1 0.9988622371733984)
           t_0
           (*
            (exp re)
            (fma
             (* (- (* (* im im) 0.041666666666666664) 0.5) im)
             im
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = cos(im) * (re - -1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	} else if (t_1 <= 0.9988622371733984) {
		tmp = t_0;
	} else {
		tmp = exp(re) * fma(((((im * im) * 0.041666666666666664) - 0.5) * im), im, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(cos(im) * Float64(re - -1.0))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_1 <= 0.9988622371733984)
		tmp = t_0;
	else
		tmp = Float64(exp(re) * fma(Float64(Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5) * im), im, 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Cos[im], $MachinePrecision] * N[(re - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9988622371733984], t$95$0, N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 26.2

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

   7. Applied rewrites 26.2%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99886223717339839

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 62.5

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

   6. Applied rewrites 62.5%

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

   if 0.99886223717339839 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 3: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9988622371733984:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right) \cdot im, im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (<= t_0 -0.02)
       (cos im)
       (if (<= t_0 0.0)
         (* (exp re) (fma im (* im -0.5) 1.0))
         (if (<= t_0 0.9988622371733984)
           (cos im)
           (*
            (exp re)
            (fma
             (* (- (* (* im im) 0.041666666666666664) 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 = exp(re) * ((im * im) * -0.5);
	} else if (t_0 <= -0.02) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 0.9988622371733984) {
		tmp = cos(im);
	} else {
		tmp = exp(re) * fma(((((im * im) * 0.041666666666666664) - 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(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= -0.02)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= 0.9988622371733984)
		tmp = cos(im);
	else
		tmp = Float64(exp(re) * fma(Float64(Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5) * 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[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9988622371733984], N[Cos[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 26.2

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

   7. Applied rewrites 26.2%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99886223717339839

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. lift-cos.f64 50.7

         \[\leadsto \cos im \]

   4. Applied rewrites 50.7%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 62.5

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

   6. Applied rewrites 62.5%

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

   if 0.99886223717339839 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 4: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq 0.9988622371733984:\\ \;\;\;\;re + 1\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(\left(\left(im \cdot im\right) \cdot 0.041666666666666664 - 0.5\right) \cdot im, im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* (exp re) (fma im (* im -0.5) 1.0))
     (if (<= t_0 0.9988622371733984)
       (+ re 1.0)
       (*
        (exp re)
        (fma (* (- (* (* im im) 0.041666666666666664) 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 = exp(re) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= 0.9988622371733984) {
		tmp = re + 1.0;
	} else {
		tmp = exp(re) * fma(((((im * im) * 0.041666666666666664) - 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(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= 0.9988622371733984)
		tmp = Float64(re + 1.0);
	else
		tmp = Float64(exp(re) * fma(Float64(Float64(Float64(Float64(im * im) * 0.041666666666666664) - 0.5) * 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[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9988622371733984], N[(re + 1.0), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 62.5

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

   6. Applied rewrites 62.5%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99886223717339839

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. *-lft-identity N/A

          \[\leadsto re + 1 \]

       3. lower-+.f64 28.8

          \[\leadsto re + 1 \]

   12. Applied rewrites 28.8%

       \[\leadsto re + 1 \]

   if 0.99886223717339839 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 59.2

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

   6. Applied rewrites 59.2%

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

Alternative 5: 67.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos im \leq -0.02:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;\cos im \leq 0.9988622371733984:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (cos im) -0.02)
   (* (exp re) (* (* im im) -0.5))
   (if (<= (cos im) 0.9988622371733984)
     (* (exp re) (* (* (* im im) (* im im)) 0.041666666666666664))
     (* (exp re) (fma im (* im -0.5) 1.0)))))

C

double code(double re, double im) {
	double tmp;
	if (cos(im) <= -0.02) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if (cos(im) <= 0.9988622371733984) {
		tmp = exp(re) * (((im * im) * (im * im)) * 0.041666666666666664);
	} else {
		tmp = exp(re) * fma(im, (im * -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (cos(im) <= -0.02)
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif (cos(im) <= 0.9988622371733984)
		tmp = Float64(exp(re) * Float64(Float64(Float64(im * im) * Float64(im * im)) * 0.041666666666666664));
	else
		tmp = Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Cos[im], $MachinePrecision], -0.02], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[im], $MachinePrecision], 0.9988622371733984], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 im) < -0.0200000000000000004

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 26.2

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

   7. Applied rewrites 26.2%

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

   if -0.0200000000000000004 < (cos.f64 im) < 0.99886223717339839

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in im around inf

      \[\leadsto e^{re} \cdot \left(\frac{1}{24} \cdot \color{blue}{{im}^{4}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{4} \cdot \frac{1}{24}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{4} \cdot \frac{1}{24}\right) \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 26.1

         \[\leadsto e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\right) \]

   7. Applied rewrites 26.1%

      \[\leadsto e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.041666666666666664}\right) \]

   if 0.99886223717339839 < (cos.f64 im)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 62.5

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

   6. Applied rewrites 62.5%

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

Alternative 6: 62.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (fma im (* im -0.5) 1.0)))

C

double code(double re, double im) {
	return exp(re) * fma(im, (im * -0.5), 1.0);
}

Julia

function code(re, im)
	return Float64(exp(re) * fma(im, Float64(im * -0.5), 1.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in im around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 62.5

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

4. Applied rewrites 62.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 62.5

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

6. Applied rewrites 62.5%

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

Alternative 7: 61.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 0.0d0) then
        tmp = exp(re) * ((im * im) * (-0.5d0))
    else
        tmp = ((re * re) - 1.0d0) / (re - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 0.0) {
		tmp = Math.exp(re) * ((im * im) * -0.5);
	} else {
		tmp = ((re * re) - 1.0) / (re - 1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 0.0:
		tmp = math.exp(re) * ((im * im) * -0.5)
	else:
		tmp = ((re * 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(exp(re) * Float64(Float64(im * im) * -0.5));
	else
		tmp = Float64(Float64(Float64(re * re) - 1.0) / Float64(re - 1.0));
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $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. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 26.2

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

   7. Applied rewrites 26.2%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. flip-+ N/A

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

       3. *-lft-identity N/A

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

       4. *-lft-identity N/A

          \[\leadsto \frac{re \cdot re - 1 \cdot 1}{1 \cdot re - 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       8. unpow2 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       13. *-lft-identity N/A

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

       14. lower--.f64 37.2

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

   12. Applied rewrites 37.2%

       \[\leadsto \frac{re \cdot re - 1}{re - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 37.7

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

   7. Applied rewrites 37.7%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 14.1

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

   10. Applied rewrites 14.1%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in im around inf

      \[\leadsto e^{re} \cdot \left(\frac{1}{24} \cdot \color{blue}{{im}^{4}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{4} \cdot \frac{1}{24}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{4} \cdot \frac{1}{24}\right) \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 26.1

         \[\leadsto e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\right) \]

   7. Applied rewrites 26.1%

      \[\leadsto e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot \color{blue}{0.041666666666666664}\right) \]

   8. Taylor expanded in re around 0

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

      1. lower-+.f64 16.6

         \[\leadsto \left(1 + \color{blue}{re}\right) \cdot \left(\left(\left(im \cdot im\right) \cdot \left(im \cdot im\right)\right) \cdot 0.041666666666666664\right) \]

   10. Applied rewrites 16.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. unpow-prod-down N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

   12. Applied rewrites 16.6%

       \[\leadsto \color{blue}{\left(1 + re\right) \cdot \left(\left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot 0.041666666666666664\right)\right)} \]

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. flip-+ N/A

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

       3. *-lft-identity N/A

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

       4. *-lft-identity N/A

          \[\leadsto \frac{re \cdot re - 1 \cdot 1}{1 \cdot re - 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       8. unpow2 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       13. *-lft-identity N/A

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

       14. lower--.f64 37.2

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

   12. Applied rewrites 37.2%

       \[\leadsto \frac{re \cdot re - 1}{re - 1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 43.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02)
   (* (* (* re re) 0.5) (fma (* im im) -0.5 1.0))
   (/ (- (* re re) 1.0) (- re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = ((re * re) * 0.5) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = ((re * re) - 1.0) / (re - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(Float64(Float64(re * re) * 0.5) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = Float64(Float64(Float64(re * re) - 1.0) / Float64(re - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 37.7

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

   7. Applied rewrites 37.7%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 14.1

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

   10. Applied rewrites 14.1%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. flip-+ N/A

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

       3. *-lft-identity N/A

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

       4. *-lft-identity N/A

          \[\leadsto \frac{re \cdot re - 1 \cdot 1}{1 \cdot re - 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       8. unpow2 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       13. *-lft-identity N/A

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

       14. lower--.f64 37.2

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

   12. Applied rewrites 37.2%

       \[\leadsto \frac{re \cdot re - 1}{re - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 42.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02)
   (* (+ 1.0 re) (fma (* im im) -0.5 1.0))
   (/ (- (* re re) 1.0) (- re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = (1.0 + re) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = ((re * re) - 1.0) / (re - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = Float64(Float64(Float64(re * re) - 1.0) / Float64(re - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 31.2

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

   7. Applied rewrites 31.2%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. flip-+ N/A

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

       3. *-lft-identity N/A

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

       4. *-lft-identity N/A

          \[\leadsto \frac{re \cdot re - 1 \cdot 1}{1 \cdot re - 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       8. unpow2 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       13. *-lft-identity N/A

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

       14. lower--.f64 37.2

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

   12. Applied rewrites 37.2%

       \[\leadsto \frac{re \cdot re - 1}{re - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 42.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\frac{re \cdot re - 1}{re - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02)
   (* (fma (* im im) -0.5 1.0) re)
   (/ (- (* re re) 1.0) (- re 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = fma((im * im), -0.5, 1.0) * re;
	} else {
		tmp = ((re * re) - 1.0) / (re - 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(fma(Float64(im * im), -0.5, 1.0) * re);
	else
		tmp = Float64(Float64(Float64(re * re) - 1.0) / Float64(re - 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] - 1.0), $MachinePrecision] / N[(re - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. inv-pow N/A

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

      16. pow-plus N/A

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

      17. metadata-eval N/A

          \[\leadsto \left(-\left(\left(im \cdot im\right) \cdot re\right) \cdot \left(\frac{\frac{1}{2}}{re} + \frac{1}{2}\right)\right) + \left(1 \cdot re + {re}^{0}\right) \]

      18. metadata-eval N/A

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

   10. Applied rewrites 31.2%

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

   11. Taylor expanded in re around inf

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

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

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. +-commutative N/A

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

       9. lift-fma.f64 7.8

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

   13. Applied rewrites 7.8%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. flip-+ N/A

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

       3. *-lft-identity N/A

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

       4. *-lft-identity N/A

          \[\leadsto \frac{re \cdot re - 1 \cdot 1}{1 \cdot re - 1} \]

       5. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       6. metadata-eval N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{re \cdot re - -1 \cdot -1}{1 \cdot re - 1} \]

       8. unpow2 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. unpow2 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{re \cdot re - 1}{1 \cdot re - 1} \]

       13. *-lft-identity N/A

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

       14. lower--.f64 37.2

           \[\leadsto \frac{re \cdot re - 1}{re - 1} \]

   12. Applied rewrites 37.2%

       \[\leadsto \frac{re \cdot re - 1}{re - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 33.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.5, 1\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;re + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.02)
   (* (fma (* im im) -0.5 1.0) re)
   (+ re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.02) {
		tmp = fma((im * im), -0.5, 1.0) * re;
	} else {
		tmp = re + 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.02)
		tmp = Float64(fma(Float64(im * im), -0.5, 1.0) * re);
	else
		tmp = Float64(re + 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision], N[(re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. inv-pow N/A

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

      16. pow-plus N/A

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

      17. metadata-eval N/A

          \[\leadsto \left(-\left(\left(im \cdot im\right) \cdot re\right) \cdot \left(\frac{\frac{1}{2}}{re} + \frac{1}{2}\right)\right) + \left(1 \cdot re + {re}^{0}\right) \]

      18. metadata-eval N/A

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

   10. Applied rewrites 31.2%

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

   11. Taylor expanded in re around inf

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

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

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. +-commutative N/A

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

       9. lift-fma.f64 7.8

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

   13. Applied rewrites 7.8%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. *-lft-identity N/A

          \[\leadsto re + 1 \]

       3. lower-+.f64 28.8

          \[\leadsto re + 1 \]

   12. Applied rewrites 28.8%

       \[\leadsto re + 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 32.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[(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. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-rgt-in N/A

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

      15. inv-pow N/A

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

      16. pow-plus N/A

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

      17. metadata-eval N/A

          \[\leadsto \left(-\left(\left(im \cdot im\right) \cdot re\right) \cdot \left(\frac{\frac{1}{2}}{re} + \frac{1}{2}\right)\right) + \left(1 \cdot re + {re}^{0}\right) \]

      18. metadata-eval N/A

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

   10. Applied rewrites 31.2%

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

   11. Taylor expanded in re around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. +-commutative N/A

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

       7. lift-fma.f64 29.1

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

   13. Applied rewrites 29.1%

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

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

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]

   2. Taylor expanded in re around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 51.6

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in im around 0

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

      1. distribute-rgt-in N/A

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

      2. inv-pow N/A

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

      3. pow-plus N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 \cdot re + {re}^{0} \]

      5. metadata-eval N/A

         \[\leadsto 1 \cdot re + 1 \]

      6. lower-fma.f64 28.8

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

   10. Applied rewrites 28.8%

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

       1. lift-fma.f64 N/A

          \[\leadsto 1 \cdot re + 1 \]

       2. *-lft-identity N/A

          \[\leadsto re + 1 \]

       3. lower-+.f64 28.8

          \[\leadsto re + 1 \]

   12. Applied rewrites 28.8%

       \[\leadsto re + 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 28.8% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]

2. Taylor expanded in re around 0

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

   1. distribute-rgt1-in N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

       \[\leadsto \cos im \cdot \left(re - -1\right) \]

   11. metadata-eval N/A

       \[\leadsto \cos im \cdot \left(re - \left(\mathsf{neg}\left(1\right)\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto \cos im \cdot \left(re - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \]

   13. metadata-eval 51.6

       \[\leadsto \cos im \cdot \left(re - -1\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\cos im \cdot \left(re - -1\right)} \]

5. Taylor expanded in re around inf

   \[\leadsto re \cdot \color{blue}{\left(\cos im + \frac{\cos im}{re}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos im + \frac{\cos im}{re}\right) \cdot re \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos im + \frac{\cos im}{re}\right) \cdot re \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{\cos im}{re} + \cos im\right) \cdot re \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\frac{\cos im}{re} + \cos im\right) \cdot re \]

   5. lower-/.f64 N/A

      \[\leadsto \left(\frac{\cos im}{re} + \cos im\right) \cdot re \]

   6. lift-cos.f64 N/A

      \[\leadsto \left(\frac{\cos im}{re} + \cos im\right) \cdot re \]

   7. lift-cos.f64 51.5

      \[\leadsto \left(\frac{\cos im}{re} + \cos im\right) \cdot re \]

7. Applied rewrites 51.5%

   \[\leadsto \left(\frac{\cos im}{re} + \cos im\right) \cdot \color{blue}{re} \]

8. Taylor expanded in im around 0

   \[\leadsto re \cdot \left(1 + \color{blue}{\frac{1}{re}}\right) \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto 1 \cdot re + \frac{1}{re} \cdot re \]

   2. inv-pow N/A

      \[\leadsto 1 \cdot re + {re}^{-1} \cdot re \]

   3. pow-plus N/A

      \[\leadsto 1 \cdot re + {re}^{\left(-1 + 1\right)} \]

   4. metadata-eval N/A

      \[\leadsto 1 \cdot re + {re}^{0} \]

   5. metadata-eval N/A

      \[\leadsto 1 \cdot re + 1 \]

   6. lower-fma.f64 28.8

      \[\leadsto \mathsf{fma}\left(1, re, 1\right) \]

10. Applied rewrites 28.8%

    \[\leadsto \mathsf{fma}\left(1, re, 1\right) \]
11. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto 1 \cdot re + 1 \]

    2. *-lft-identity N/A

       \[\leadsto re + 1 \]

    3. lower-+.f64 28.8

       \[\leadsto re + 1 \]

12. Applied rewrites 28.8%

    \[\leadsto re + 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))