math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Alternatives: 9

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00275:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.0017:\\ \;\;\;\;\sin im \cdot \left(re - -1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00275)
   (* (exp re) im)
   (if (<= re 0.0017)
     (* (sin im) (- re -1.0))
     (* (exp re) (fma (* im im) (* -0.16666666666666666 im) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00275) {
		tmp = exp(re) * im;
	} else if (re <= 0.0017) {
		tmp = sin(im) * (re - -1.0);
	} else {
		tmp = exp(re) * fma((im * im), (-0.16666666666666666 * im), im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00275)
		tmp = Float64(exp(re) * im);
	elseif (re <= 0.0017)
		tmp = Float64(sin(im) * Float64(re - -1.0));
	else
		tmp = Float64(exp(re) * fma(Float64(im * im), Float64(-0.16666666666666666 * im), im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00275], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 0.0017], N[(N[Sin[im], $MachinePrecision] * N[(re - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 * im), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0027499999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.7%

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

   if -0.0027499999999999998 < re < 0.00169999999999999991

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.1

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 51.1

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

   6. Applied rewrites 51.1%

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

   if 0.00169999999999999991 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.3

         \[\leadsto e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 60.3%

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

Alternative 3: 92.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.25 \cdot 10^{-7}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.25e-7)
   (* (exp re) im)
   (if (<= re 5.9e-6)
     (sin im)
     (* (exp re) (fma (* im im) (* -0.16666666666666666 im) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.25e-7) {
		tmp = exp(re) * im;
	} else if (re <= 5.9e-6) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * fma((im * im), (-0.16666666666666666 * im), im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.25e-7)
		tmp = Float64(exp(re) * im);
	elseif (re <= 5.9e-6)
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * fma(Float64(im * im), Float64(-0.16666666666666666 * im), im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.25e-7], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 5.9e-6], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 * im), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.25000000000000007e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.7%

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

   if -4.25000000000000007e-7 < re < 5.90000000000000026e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 50.5

         \[\leadsto \sin im \]

   4. Applied rewrites 50.5%

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

   if 5.90000000000000026e-6 < re

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.3

         \[\leadsto e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 60.3%

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

Alternative 4: 68.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;e^{re} \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (exp re) (fma (* im im) (* -0.16666666666666666 im) im))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = exp(re) * fma((im * im), (-0.16666666666666666 * im), im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(exp(re) * fma(Float64(im * im), Float64(-0.16666666666666666 * im), im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 * im), $MachinePrecision] + im), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 60.3

         \[\leadsto e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 60.3%

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.7%

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

Alternative 5: 61.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666 \cdot im, im\right) \cdot \left(re - -1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (fma (* im im) (* -0.16666666666666666 im) im) (- re -1.0))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = fma((im * im), (-0.16666666666666666 * im), im) * (re - -1.0);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(fma(Float64(im * im), Float64(-0.16666666666666666 * im), im) * Float64(re - -1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 * im), $MachinePrecision] + im), $MachinePrecision] * N[(re - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 30.9

         \[\leadsto \left(1 + re\right) \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   7. Applied rewrites 30.9%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 30.9

         \[\leadsto \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right) \cdot \left(1 + re\right)} \]

   9. Applied rewrites 30.9%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.7%

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

Alternative 6: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* im (fma (* im im) -0.16666666666666666 1.0))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = im * fma((im * im), -0.16666666666666666, 1.0);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(im * fma(Float64(im * im), -0.16666666666666666, 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 50.5

         \[\leadsto \sin im \]

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 29.8

         \[\leadsto im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right) \]

   7. Applied rewrites 29.8%

      \[\leadsto im \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 29.8

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

   9. Applied rewrites 29.8%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.7%

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

Alternative 7: 33.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{1}{1 + -1 \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* im (fma (* im im) -0.16666666666666666 1.0))
   (* im (/ 1.0 (+ 1.0 (* -1.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = im * fma((im * im), -0.16666666666666666, 1.0);
	} else {
		tmp = im * (1.0 / (1.0 + (-1.0 * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(im * fma(Float64(im * im), -0.16666666666666666, 1.0));
	else
		tmp = Float64(im * Float64(1.0 / Float64(1.0 + Float64(-1.0 * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(1.0 / N[(1.0 + N[(-1.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 50.5

         \[\leadsto \sin im \]

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 29.8

         \[\leadsto im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right) \]

   7. Applied rewrites 29.8%

      \[\leadsto im \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 29.8

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

   9. Applied rewrites 29.8%

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

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

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. exp-lft-sqr-rev N/A

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

      9. lower-exp.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2 N/A

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

      12. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 69.7

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

   6. Applied rewrites 69.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-sqrt-rev N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

         \[\leadsto im \cdot e^{1 \cdot re} \]

      7. *-lft-identity N/A

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

      8. remove-double-neg N/A

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

      9. rec-exp N/A

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

      10. lower-/.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 69.7

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in re around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 32.1

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

   11. Applied rewrites 32.1%

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

Alternative 8: 29.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ im \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* im (fma (* im im) -0.16666666666666666 1.0)))

C

double code(double re, double im) {
	return im * fma((im * im), -0.16666666666666666, 1.0);
}

Julia

function code(re, im)
	return Float64(im * fma(Float64(im * im), -0.16666666666666666, 1.0))
end

Wolfram

code[re_, im_] := N[(im * N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in re around 0

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

   1. lower-sin.f64 50.5

      \[\leadsto \sin im \]

4. Applied rewrites 50.5%

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

5. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 29.8

      \[\leadsto im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right) \]

7. Applied rewrites 29.8%

   \[\leadsto im \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 29.8

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

9. Applied rewrites 29.8%

   \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \]
10. Add Preprocessing

Alternative 9: 26.5% accurate, 45.8× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. exp-fabs N/A

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

   3. lift-exp.f64 N/A

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

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

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

   5. lower-sqrt.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. exp-lft-sqr-rev N/A

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

   9. lower-exp.f64 N/A

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

   10. *-commutative N/A

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

   11. count-2 N/A

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

   12. lower-+.f64 99.9

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 69.7

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

6. Applied rewrites 69.7%

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

7. Taylor expanded in re around 0

   \[\leadsto im \]
8. Step-by-step derivation

9. Applied rewrites 26.5%

   \[\leadsto im \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))