math.sqrt on complex, real part

Percentage Accurate: 41.4% → 87.4%

Time: 4.2s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + 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 = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

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

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

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

Initial Program: 41.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	return 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + 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 = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

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

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

Wolfram

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

Alternative 1: 87.4% accurate, N/A× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -240:\\ \;\;\;\;0.5 \cdot e^{\mathsf{fma}\left(2, \log im\_m, \log \left(\frac{-1}{re}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(im\_m, re\right) + re\right) \cdot 2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -240.0)
   (* 0.5 (exp (* (fma 2.0 (log im_m) (log (/ -1.0 re))) 0.5)))
   (* (sqrt (* (+ (hypot im_m re) re) 2.0)) 0.5)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -240.0) {
		tmp = 0.5 * exp((fma(2.0, log(im_m), log((-1.0 / re))) * 0.5));
	} else {
		tmp = sqrt(((hypot(im_m, re) + re) * 2.0)) * 0.5;
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -240.0)
		tmp = Float64(0.5 * exp(Float64(fma(2.0, log(im_m), log(Float64(-1.0 / re))) * 0.5)));
	else
		tmp = Float64(sqrt(Float64(Float64(hypot(im_m, re) + re) * 2.0)) * 0.5);
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -240.0], N[(0.5 * N[Exp[N[(N[(2.0 * N[Log[im$95$m], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Sqrt[im$95$m ^ 2 + re ^ 2], $MachinePrecision] + re), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -240

   1. Initial program 9.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 46.5

         \[\leadsto 0.5 \cdot \sqrt{-\frac{im \cdot im}{re}} \]

   5. Applied rewrites 46.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-\frac{im \cdot im}{re}}} \]

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 44.0%

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

   8. Taylor expanded in re around -inf

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

      1. +-commutative N/A

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

      2. log-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 38.6

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

   10. Applied rewrites 38.6%

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

   if -240 < re

   1. Initial program 53.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 94.6%

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

Alternative 2: 73.8% accurate, N/A× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im\_m \cdot im\_m} + re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot e^{\mathsf{fma}\left(2, \log im\_m, \log \left(\frac{-1}{re}\right)\right) \cdot 0.5}\\ \mathbf{elif}\;t\_0 \leq 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im_m im_m))) re))))))
   (if (<= t_0 0.0)
     (* 0.5 (exp (* (fma 2.0 (log im_m) (log (/ -1.0 re))) 0.5)))
     (if (<= t_0 1e+76) t_0 (* 0.5 (sqrt (* 2.0 im_m)))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im_m * im_m))) + re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * exp((fma(2.0, log(im_m), log((-1.0 / re))) * 0.5));
	} else if (t_0 <= 1e+76) {
		tmp = t_0;
	} else {
		tmp = 0.5 * sqrt((2.0 * im_m));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im_m * im_m))) + re))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * exp(Float64(fma(2.0, log(im_m), log(Float64(-1.0 / re))) * 0.5)));
	elseif (t_0 <= 1e+76)
		tmp = t_0;
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[Exp[N[(N[(2.0 * N[Log[im$95$m], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+76], t$95$0, N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 0.0

   1. Initial program 4.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 59.0

         \[\leadsto 0.5 \cdot \sqrt{-\frac{im \cdot im}{re}} \]

   5. Applied rewrites 59.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-\frac{im \cdot im}{re}}} \]

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 55.4%

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

   8. Taylor expanded in re around -inf

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

      1. +-commutative N/A

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

      2. log-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 47.5

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

   10. Applied rewrites 47.5%

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

   if 0.0 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))) < 1e76

   1. Initial program 94.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   if 1e76 < (*.f64 #s(literal 1/2 binary64) (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))))

   1. Initial program 3.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 24.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{im}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 63.3% accurate, N/A× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -240:\\ \;\;\;\;0.5 \cdot e^{\mathsf{fma}\left(2, \log im\_m, \log \left(\frac{-1}{re}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im\_m}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -240.0)
   (* 0.5 (exp (* (fma 2.0 (log im_m) (log (/ -1.0 re))) 0.5)))
   (* 0.5 (sqrt (* 2.0 im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -240.0) {
		tmp = 0.5 * exp((fma(2.0, log(im_m), log((-1.0 / re))) * 0.5));
	} else {
		tmp = 0.5 * sqrt((2.0 * im_m));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -240.0)
		tmp = Float64(0.5 * exp(Float64(fma(2.0, log(im_m), log(Float64(-1.0 / re))) * 0.5)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -240.0], N[(0.5 * N[Exp[N[(N[(2.0 * N[Log[im$95$m], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -240

   1. Initial program 9.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 46.5

         \[\leadsto 0.5 \cdot \sqrt{-\frac{im \cdot im}{re}} \]

   5. Applied rewrites 46.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-\frac{im \cdot im}{re}}} \]

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 44.0%

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

   8. Taylor expanded in re around -inf

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

      1. +-commutative N/A

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

      2. log-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 38.6

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

   10. Applied rewrites 38.6%

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

   if -240 < re

   1. Initial program 53.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 28.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 60.2% accurate, N/A× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -240:\\ \;\;\;\;0.5 \cdot e^{\mathsf{fma}\left(2, \log im\_m, \log \left(\frac{-1}{re}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot e^{\log \left(im\_m \cdot 2\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -240.0)
   (* 0.5 (exp (* (fma 2.0 (log im_m) (log (/ -1.0 re))) 0.5)))
   (* 0.5 (exp (* (log (* im_m 2.0)) 0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -240.0) {
		tmp = 0.5 * exp((fma(2.0, log(im_m), log((-1.0 / re))) * 0.5));
	} else {
		tmp = 0.5 * exp((log((im_m * 2.0)) * 0.5));
	}
	return tmp;
}

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -240.0)
		tmp = Float64(0.5 * exp(Float64(fma(2.0, log(im_m), log(Float64(-1.0 / re))) * 0.5)));
	else
		tmp = Float64(0.5 * exp(Float64(log(Float64(im_m * 2.0)) * 0.5)));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -240.0], N[(0.5 * N[Exp[N[(N[(2.0 * N[Log[im$95$m], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Exp[N[(N[Log[N[(im$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -240

   1. Initial program 9.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 46.5

         \[\leadsto 0.5 \cdot \sqrt{-\frac{im \cdot im}{re}} \]

   5. Applied rewrites 46.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-\frac{im \cdot im}{re}}} \]

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 44.0%

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

   8. Taylor expanded in re around -inf

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

      1. +-commutative N/A

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

      2. log-pow N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 38.6

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

   10. Applied rewrites 38.6%

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

   if -240 < re

   1. Initial program 53.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 28.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{im}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{2 \cdot im}} \]

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 26.1

         \[\leadsto 0.5 \cdot e^{\color{blue}{\log \left(2 \cdot im\right)} \cdot 0.5} \]

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 26.1

         \[\leadsto 0.5 \cdot e^{\log \color{blue}{\left(im \cdot 2\right)} \cdot 0.5} \]

   7. Applied rewrites 26.1%

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

Alternative 5: 24.0% accurate, N/A× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot e^{\mathsf{fma}\left(2, \log im\_m, \log \left(\frac{-1}{re}\right)\right) \cdot 0.5} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (* 0.5 (exp (* (fma 2.0 (log im_m) (log (/ -1.0 re))) 0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5 * exp((fma(2.0, log(im_m), log((-1.0 / re))) * 0.5));
}

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(0.5 * exp(Float64(fma(2.0, log(im_m), log(Float64(-1.0 / re))) * 0.5)))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[Exp[N[(N[(2.0 * N[Log[im$95$m], $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.4%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Add Preprocessing

3. Taylor expanded in re around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 14.7

      \[\leadsto 0.5 \cdot \sqrt{-\frac{im \cdot im}{re}} \]

5. Applied rewrites 14.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-\frac{im \cdot im}{re}}} \]

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 14.0%

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

8. Taylor expanded in re around -inf

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

   1. +-commutative N/A

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

   2. log-pow N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-/.f64 11.5

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

10. Applied rewrites 11.5%

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

Alternative 6: 23.8% accurate, N/A× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \log im\_m \cdot 2\\ t_1 := \log \left(\frac{-1}{re}\right)\\ 0.5 \cdot e^{\frac{\mathsf{fma}\left({\log im\_m}^{3}, 8, {t\_1}^{3}\right)}{{t\_0}^{2} + \left({t\_1}^{2} - t\_0 \cdot t\_1\right)} \cdot 0.5} \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (* (log im_m) 2.0)) (t_1 (log (/ -1.0 re))))
   (*
    0.5
    (exp
     (*
      (/
       (fma (pow (log im_m) 3.0) 8.0 (pow t_1 3.0))
       (+ (pow t_0 2.0) (- (pow t_1 2.0) (* t_0 t_1))))
      0.5)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = log(im_m) * 2.0;
	double t_1 = log((-1.0 / re));
	return 0.5 * exp(((fma(pow(log(im_m), 3.0), 8.0, pow(t_1, 3.0)) / (pow(t_0, 2.0) + (pow(t_1, 2.0) - (t_0 * t_1)))) * 0.5));
}

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = Float64(log(im_m) * 2.0)
	t_1 = log(Float64(-1.0 / re))
	return Float64(0.5 * exp(Float64(Float64(fma((log(im_m) ^ 3.0), 8.0, (t_1 ^ 3.0)) / Float64((t_0 ^ 2.0) + Float64((t_1 ^ 2.0) - Float64(t_0 * t_1)))) * 0.5)))
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[(N[Log[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]}, N[(0.5 * N[Exp[N[(N[(N[(N[Power[N[Log[im$95$m], $MachinePrecision], 3.0], $MachinePrecision] * 8.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 42.4%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Add Preprocessing

3. Taylor expanded in re around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 14.7

      \[\leadsto 0.5 \cdot \sqrt{-\frac{im \cdot im}{re}} \]

5. Applied rewrites 14.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-\frac{im \cdot im}{re}}} \]

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 14.0%

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

8. Taylor expanded in re around -inf

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

   1. +-commutative N/A

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

   2. log-pow N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-/.f64 11.5

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

10. Applied rewrites 11.5%

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

12. Applied rewrites 11.3%

    \[\leadsto 0.5 \cdot e^{\frac{\mathsf{fma}\left({\log im}^{3}, 8, {\log \left(\frac{-1}{re}\right)}^{3}\right)}{\color{blue}{{\left(\log im \cdot 2\right)}^{2} + \left({\log \left(\frac{-1}{re}\right)}^{2} - \left(\log im \cdot 2\right) \cdot \log \left(\frac{-1}{re}\right)\right)}} \cdot 0.5} \]
13. Add Preprocessing

Alternative 7: 11.0% accurate, N/A× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ 0.5 \cdot e^{\frac{{\left(\log im\_m \cdot 2\right)}^{2} - {\log \left(\frac{-1}{re}\right)}^{2}}{\log \left(\frac{im\_m \cdot im\_m}{\frac{-1}{re}}\right)} \cdot 0.5} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (*
  0.5
  (exp
   (*
    (/
     (- (pow (* (log im_m) 2.0) 2.0) (pow (log (/ -1.0 re)) 2.0))
     (log (/ (* im_m im_m) (/ -1.0 re))))
    0.5))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return 0.5 * exp((((pow((log(im_m) * 2.0), 2.0) - pow(log((-1.0 / re)), 2.0)) / log(((im_m * im_m) / (-1.0 / re)))) * 0.5));
}

Fortran

im_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = 0.5d0 * exp((((((log(im_m) * 2.0d0) ** 2.0d0) - (log(((-1.0d0) / re)) ** 2.0d0)) / log(((im_m * im_m) / ((-1.0d0) / re)))) * 0.5d0))
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return 0.5 * Math.exp((((Math.pow((Math.log(im_m) * 2.0), 2.0) - Math.pow(Math.log((-1.0 / re)), 2.0)) / Math.log(((im_m * im_m) / (-1.0 / re)))) * 0.5));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return 0.5 * math.exp((((math.pow((math.log(im_m) * 2.0), 2.0) - math.pow(math.log((-1.0 / re)), 2.0)) / math.log(((im_m * im_m) / (-1.0 / re)))) * 0.5))

Julia

im_m = abs(im)
function code(re, im_m)
	return Float64(0.5 * exp(Float64(Float64(Float64((Float64(log(im_m) * 2.0) ^ 2.0) - (log(Float64(-1.0 / re)) ^ 2.0)) / log(Float64(Float64(im_m * im_m) / Float64(-1.0 / re)))) * 0.5)))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = 0.5 * exp((((((log(im_m) * 2.0) ^ 2.0) - (log((-1.0 / re)) ^ 2.0)) / log(((im_m * im_m) / (-1.0 / re)))) * 0.5));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[(0.5 * N[Exp[N[(N[(N[(N[Power[N[(N[Log[im$95$m], $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Log[N[(N[(im$95$m * im$95$m), $MachinePrecision] / N[(-1.0 / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.4%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Add Preprocessing

3. Taylor expanded in re around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 14.7

      \[\leadsto 0.5 \cdot \sqrt{-\frac{im \cdot im}{re}} \]

5. Applied rewrites 14.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-\frac{im \cdot im}{re}}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-\frac{im \cdot im}{re}}} \]

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 14.0%

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

8. Taylor expanded in re around -inf

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

   1. +-commutative N/A

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

   2. log-pow N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-/.f64 11.5

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

10. Applied rewrites 11.5%

    \[\leadsto 0.5 \cdot e^{\color{blue}{\mathsf{fma}\left(2, \log im, \log \left(\frac{-1}{re}\right)\right)} \cdot 0.5} \]
11. Step-by-step derivation

    1. lift-log.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-log.f64 N/A

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

    5. flip-+ N/A

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

    6. lower-/.f64 N/A

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

    7. lower--.f64 N/A

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

    8. pow2 N/A

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

    9. lower-pow.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. lift-log.f64 N/A

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

    13. pow2 N/A

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

    14. lower-pow.f64 N/A

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

    15. lift-log.f64 N/A

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

    16. lift-/.f64 N/A

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

    17. log-pow-rev N/A

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

12. Applied rewrites 5.5%

    \[\leadsto 0.5 \cdot e^{\frac{{\left(\log im \cdot 2\right)}^{2} - {\log \left(\frac{-1}{re}\right)}^{2}}{\color{blue}{\log \left(\frac{im \cdot im}{\frac{-1}{re}}\right)}} \cdot 0.5} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))