math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 10

Speedup: 1.0×

• 75.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999244:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (fma 0.5 re 1.0) re 1.0) (cos im)))
        (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (fma (* im im) -0.5 1.0))
     (if (<= t_1 -0.04)
       t_0
       (if (<= t_1 1e-8)
         (exp re)
         (if (<= t_1 0.9999999999999244) t_0 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else if (t_1 <= -0.04) {
		tmp = t_0;
	} else if (t_1 <= 1e-8) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999999999244) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im))
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_1 <= -0.04)
		tmp = t_0;
	elseif (t_1 <= 1e-8)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999999999244)
		tmp = t_0;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $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 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.04], t$95$0, If[LessEqual[t$95$1, 1e-8], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999244], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. 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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.4%

         \[\leadsto e^{re} \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 63.4%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.4%

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

   6. Applied rewrites 63.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 or 1e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999924394

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-fma.f64 63.4%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 63.4%

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

   8. Applied rewrites 63.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 63.4%

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

   10. Applied rewrites 63.4%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im)) < 1e-8 or 0.999999999999924394 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      3. lower-*.f64 71.7%

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

   9. Applied rewrites 71.7%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       3. exp-sqrt-rev N/A

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

          \[\leadsto e^{\frac{re + re}{2}} \]

       6. exp-sqrt-rev N/A

          \[\leadsto \sqrt{e^{re + re}} \]

       7. exp-sum N/A

          \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

          \[\leadsto \left|e^{re}\right| \]

       9. exp-fabs N/A

          \[\leadsto e^{re} \]

       10. lower-exp.f64 71.7%

           \[\leadsto e^{re} \]

   11. Applied rewrites 71.7%

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

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \cos im \cdot \left(re - -1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_1 \leq 0.9999999999999244:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

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) (fma (* im im) -0.5 1.0))
     (if (<= t_1 -0.04)
       t_0
       (if (<= t_1 1e-8)
         (exp re)
         (if (<= t_1 0.9999999999999244) t_0 (exp re)))))))

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) * fma((im * im), -0.5, 1.0);
	} else if (t_1 <= -0.04) {
		tmp = t_0;
	} else if (t_1 <= 1e-8) {
		tmp = exp(re);
	} else if (t_1 <= 0.9999999999999244) {
		tmp = t_0;
	} else {
		tmp = exp(re);
	}
	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) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_1 <= -0.04)
		tmp = t_0;
	elseif (t_1 <= 1e-8)
		tmp = exp(re);
	elseif (t_1 <= 0.9999999999999244)
		tmp = t_0;
	else
		tmp = exp(re);
	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 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.04], t$95$0, If[LessEqual[t$95$1, 1e-8], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999999244], t$95$0, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. 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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.4%

         \[\leadsto e^{re} \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 63.4%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.4%

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

   6. Applied rewrites 63.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 or 1e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999924394

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 51.4%

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

   4. Applied rewrites 51.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 51.4%

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

   6. Applied rewrites 51.4%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im)) < 1e-8 or 0.999999999999924394 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      3. lower-*.f64 71.7%

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

   9. Applied rewrites 71.7%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       3. exp-sqrt-rev N/A

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

          \[\leadsto e^{\frac{re + re}{2}} \]

       6. exp-sqrt-rev N/A

          \[\leadsto \sqrt{e^{re + re}} \]

       7. exp-sum N/A

          \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

          \[\leadsto \left|e^{re}\right| \]

       9. exp-fabs N/A

          \[\leadsto e^{re} \]

       10. lower-exp.f64 71.7%

           \[\leadsto e^{re} \]

   11. Applied rewrites 71.7%

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

Alternative 3: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.04:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 10^{-8}:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.04)
       (cos im)
       (if (<= t_0 1e-8) (exp re) (if (<= t_0 0.9995) (cos im) (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.04) {
		tmp = cos(im);
	} else if (t_0 <= 1e-8) {
		tmp = exp(re);
	} else if (t_0 <= 0.9995) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.04)
		tmp = cos(im);
	elseif (t_0 <= 1e-8)
		tmp = exp(re);
	elseif (t_0 <= 0.9995)
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.04], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. 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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.4%

         \[\leadsto e^{re} \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 63.4%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.4%

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

   6. Applied rewrites 63.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008 or 1e-8 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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. lower-cos.f64 50.5%

         \[\leadsto \cos im \]

   4. Applied rewrites 50.5%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im)) < 1e-8 or 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      3. lower-*.f64 71.7%

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

   9. Applied rewrites 71.7%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       3. exp-sqrt-rev N/A

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

          \[\leadsto e^{\frac{re + re}{2}} \]

       6. exp-sqrt-rev N/A

          \[\leadsto \sqrt{e^{re + re}} \]

       7. exp-sum N/A

          \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

          \[\leadsto \left|e^{re}\right| \]

       9. exp-fabs N/A

          \[\leadsto e^{re} \]

       10. lower-exp.f64 71.7%

           \[\leadsto e^{re} \]

   11. Applied rewrites 71.7%

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

Alternative 4: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.04)
   (* (exp re) (fma (* im im) -0.5 1.0))
   (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.04) {
		tmp = exp(re) * fma((im * im), -0.5, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.04)
		tmp = Float64(exp(re) * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.4%

         \[\leadsto e^{re} \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 63.4%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.4%

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

   6. Applied rewrites 63.4%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      3. lower-*.f64 71.7%

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

   9. Applied rewrites 71.7%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       3. exp-sqrt-rev N/A

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

          \[\leadsto e^{\frac{re + re}{2}} \]

       6. exp-sqrt-rev N/A

          \[\leadsto \sqrt{e^{re + re}} \]

       7. exp-sum N/A

          \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

          \[\leadsto \left|e^{re}\right| \]

       9. exp-fabs N/A

          \[\leadsto e^{re} \]

       10. lower-exp.f64 71.7%

           \[\leadsto e^{re} \]

   11. Applied rewrites 71.7%

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

Alternative 5: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\sqrt{\left(im \cdot im\right) \cdot \left(im \cdot im\right)}, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.04)
   (* 1.0 (fma (sqrt (* (* im im) (* im im))) -0.5 1.0))
   (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.04) {
		tmp = 1.0 * fma(sqrt(((im * im) * (im * im))), -0.5, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.04)
		tmp = Float64(1.0 * fma(sqrt(Float64(Float64(im * im) * Float64(im * im))), -0.5, 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.04], N[(1.0 * N[(N[Sqrt[N[(N[(im * im), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.4%

         \[\leadsto e^{re} \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 63.4%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.4%

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

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 29.8%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 31.2%

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

   11. Applied rewrites 31.2%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      3. lower-*.f64 71.7%

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

   9. Applied rewrites 71.7%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       3. exp-sqrt-rev N/A

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

          \[\leadsto e^{\frac{re + re}{2}} \]

       6. exp-sqrt-rev N/A

          \[\leadsto \sqrt{e^{re + re}} \]

       7. exp-sum N/A

          \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

          \[\leadsto \left|e^{re}\right| \]

       9. exp-fabs N/A

          \[\leadsto e^{re} \]

       10. lower-exp.f64 71.7%

           \[\leadsto e^{re} \]

   11. Applied rewrites 71.7%

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

Alternative 6: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;\left(re - -1\right) \cdot \mathsf{fma}\left(-0.5, im \cdot im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.04)
   (* (- re -1.0) (fma -0.5 (* im im) 1.0))
   (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.04) {
		tmp = (re - -1.0) * fma(-0.5, (im * im), 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.04)
		tmp = Float64(Float64(re - -1.0) * fma(-0.5, Float64(im * im), 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.04], N[(N[(re - -1.0), $MachinePrecision] * N[(-0.5 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 51.4%

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 31.8%

         \[\leadsto \left(1 + re\right) \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   7. Applied rewrites 31.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-flip N/A

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

      5. lift--.f64 31.8%

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(1 + -0.5 \cdot {im}^{2}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + \frac{-1}{2} \cdot {im}^{2}\right)\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(\frac{-1}{2} \cdot {im}^{2} + 1\right)\right) \]

      8. lift--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\frac{-1}{2} \cdot {im}^{2}\right)\right) + 1\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite=>}\left(lift-pow.f64, \left({im}^{2}\right)\right) + 1\right) \]

      10. lift--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite<=}\left(pow2, \left(im \cdot im\right)\right) + 1\right) \]

      11. lift--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \left(\frac{-1}{2} \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(im \cdot im\right)\right) + 1\right) \]

      12. lift--.f64 N/A

          \[\leadsto \left(re - \color{blue}{-1}\right) \cdot \mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{-1}{2}, im \cdot im, 1\right)\right)\right) \]

   9. Applied rewrites 31.8%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      3. lower-*.f64 71.7%

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

   9. Applied rewrites 71.7%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       3. exp-sqrt-rev N/A

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

          \[\leadsto e^{\frac{re + re}{2}} \]

       6. exp-sqrt-rev N/A

          \[\leadsto \sqrt{e^{re + re}} \]

       7. exp-sum N/A

          \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

          \[\leadsto \left|e^{re}\right| \]

       9. exp-fabs N/A

          \[\leadsto e^{re} \]

       10. lower-exp.f64 71.7%

           \[\leadsto e^{re} \]

   11. Applied rewrites 71.7%

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

Alternative 7: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.04:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) -0.04)
   (* 1.0 (fma (* im im) -0.5 1.0))
   (exp re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= -0.04) {
		tmp = 1.0 * fma((im * im), -0.5, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= -0.04)
		tmp = Float64(1.0 * fma(Float64(im * im), -0.5, 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], -0.04], N[(1.0 * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0400000000000000008

   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. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 63.4%

         \[\leadsto e^{re} \cdot \left(1 + -0.5 \cdot {im}^{\color{blue}{2}}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto e^{re} \cdot \color{blue}{\left(1 + -0.5 \cdot {im}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 63.4%

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 63.4%

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

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 29.8%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 63.4%

         \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

   6. Applied rewrites 63.4%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      2. lower-exp.f64 N/A

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

      3. lower-*.f64 71.7%

         \[\leadsto \sqrt{e^{2 \cdot re}} \]

   9. Applied rewrites 71.7%

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

       1. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       2. lift-exp.f64 N/A

          \[\leadsto \sqrt{e^{2 \cdot re}} \]

       3. exp-sqrt-rev N/A

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

       4. lift-*.f64 N/A

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

       5. count-2-rev N/A

          \[\leadsto e^{\frac{re + re}{2}} \]

       6. exp-sqrt-rev N/A

          \[\leadsto \sqrt{e^{re + re}} \]

       7. exp-sum N/A

          \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

          \[\leadsto \left|e^{re}\right| \]

       9. exp-fabs N/A

          \[\leadsto e^{re} \]

       10. lower-exp.f64 71.7%

           \[\leadsto e^{re} \]

   11. Applied rewrites 71.7%

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

Alternative 8: 71.7% accurate, 4.1× speedup

Math

\[e^{re} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(re, im)
	return exp(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Exp[re], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f64 N/A

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

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

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

   5. lower-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. prod-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-+.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in re around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 63.4%

      \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

6. Applied rewrites 63.4%

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

7. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{e^{2 \cdot re}} \]

   2. lower-exp.f64 N/A

      \[\leadsto \sqrt{e^{2 \cdot re}} \]

   3. lower-*.f64 71.7%

      \[\leadsto \sqrt{e^{2 \cdot re}} \]

9. Applied rewrites 71.7%

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

    1. lift-sqrt.f64 N/A

       \[\leadsto \sqrt{e^{2 \cdot re}} \]

    2. lift-exp.f64 N/A

       \[\leadsto \sqrt{e^{2 \cdot re}} \]

    3. exp-sqrt-rev N/A

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

    4. lift-*.f64 N/A

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

    5. count-2-rev N/A

       \[\leadsto e^{\frac{re + re}{2}} \]

    6. exp-sqrt-rev N/A

       \[\leadsto \sqrt{e^{re + re}} \]

    7. exp-sum N/A

       \[\leadsto \sqrt{e^{re} \cdot e^{re}} \]

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

       \[\leadsto \left|e^{re}\right| \]

    9. exp-fabs N/A

       \[\leadsto e^{re} \]

    10. lower-exp.f64 71.7%

        \[\leadsto e^{re} \]

11. Applied rewrites 71.7%

    \[\leadsto \color{blue}{e^{re}} \]
12. Add Preprocessing

Alternative 9: 28.6% accurate, 5.4× speedup

Math

\[\sqrt{1 + 2 \cdot re} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt (+ 1.0 (* 2.0 re))))

C

double code(double re, double im) {
	return sqrt((1.0 + (2.0 * re)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((1.0d0 + (2.0d0 * re)))
end function

Java

public static double code(double re, double im) {
	return Math.sqrt((1.0 + (2.0 * re)));
}

Python

def code(re, im):
	return math.sqrt((1.0 + (2.0 * re)))

Julia

function code(re, im)
	return sqrt(Float64(1.0 + Float64(2.0 * re)))
end

MATLAB

function tmp = code(re, im)
	tmp = sqrt((1.0 + (2.0 * re)));
end

Wolfram

code[re_, im_] := N[Sqrt[N[(1.0 + N[(2.0 * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f64 N/A

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

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

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

   5. lower-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. prod-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-+.f64 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in re around 0

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

   1. lower-+.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 63.4%

      \[\leadsto \cos im + re \cdot \left(\cos im + 0.5 \cdot \left(re \cdot \cos im\right)\right) \]

6. Applied rewrites 63.4%

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

7. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{e^{2 \cdot re}} \]

   2. lower-exp.f64 N/A

      \[\leadsto \sqrt{e^{2 \cdot re}} \]

   3. lower-*.f64 71.7%

      \[\leadsto \sqrt{e^{2 \cdot re}} \]

9. Applied rewrites 71.7%

   \[\leadsto \color{blue}{\sqrt{e^{2 \cdot re}}} \]

10. Taylor expanded in re around 0

    \[\leadsto \sqrt{1 + 2 \cdot re} \]
11. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \sqrt{1 + 2 \cdot re} \]

    2. lower-*.f64 28.6%

       \[\leadsto \sqrt{1 + 2 \cdot re} \]

12. Applied rewrites 28.6%

    \[\leadsto \sqrt{1 + 2 \cdot re} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))