math.sin on complex, imaginary part

Percentage Accurate: 53.9% → 99.9%

Time: 5.0s

Alternatives: 8

Speedup: 1.3×

• 50.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(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 = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.9% accurate, 1.0× speedup

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(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 = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (* (sinh (- im)) (cos re)))

C

double code(double re, double im) {
	return sinh(-im) * cos(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 = sinh(-im) * cos(re)
end function

Java

public static double code(double re, double im) {
	return Math.sinh(-im) * Math.cos(re);
}

Python

def code(re, im):
	return math.sinh(-im) * math.cos(re)

Julia

function code(re, im)
	return Float64(sinh(Float64(-im)) * cos(re))
end

MATLAB

function tmp = code(re, im)
	tmp = sinh(-im) * cos(re);
end

Wolfram

code[re_, im_] := N[(N[Sinh[(-im)], $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(e^{0 - im} - e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(e^{0 - im} - e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

   6. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{e^{0 - im} - e^{im}}{2}} \cdot \cos re \]

   7. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{0 - im} - e^{im}}}{2} \cdot \cos re \]

   8. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{0 - im}} - e^{im}}{2} \cdot \cos re \]

   9. lift-exp.f64 N/A

      \[\leadsto \frac{e^{0 - im} - \color{blue}{e^{im}}}{2} \cdot \cos re \]

   10. --rgt-identity N/A

       \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{im - 0}}}{2} \cdot \cos re \]

   11. sub-negate-rev N/A

       \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{\mathsf{neg}\left(\left(0 - im\right)\right)}}}{2} \cdot \cos re \]

   12. lift--.f64 N/A

       \[\leadsto \frac{e^{0 - im} - e^{\mathsf{neg}\left(\color{blue}{\left(0 - im\right)}\right)}}{2} \cdot \cos re \]

   13. sinh-def N/A

       \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

   14. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sinh \left(0 - im\right) \cdot \cos re} \]

   15. lower-sinh.f64 99.9%

       \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

   16. lift--.f64 N/A

       \[\leadsto \sinh \color{blue}{\left(0 - im\right)} \cdot \cos re \]

   17. sub0-neg N/A

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

   18. lower-neg.f64 99.9%

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\sinh \left(-im\right) \cdot \cos re} \]
4. Add Preprocessing

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(-\left|im\right|\right)\\ t_1 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - \left|im\right|} - e^{\left|im\right|}\right)\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(-\cos re\right) \cdot \left|im\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + -0.5 \cdot {re}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (sinh (- (fabs im))))
       (t_1
        (*
         (* 0.5 (cos re))
         (- (exp (- 0.0 (fabs im))) (exp (fabs im))))))
  (*
   (copysign 1.0 im)
   (if (<= t_1 -0.5)
     t_0
     (if (<= t_1 2e-7)
       (* (- (cos re)) (fabs im))
       (* t_0 (+ 1.0 (* -0.5 (pow re 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = sinh(-fabs(im));
	double t_1 = (0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)));
	double tmp;
	if (t_1 <= -0.5) {
		tmp = t_0;
	} else if (t_1 <= 2e-7) {
		tmp = -cos(re) * fabs(im);
	} else {
		tmp = t_0 * (1.0 + (-0.5 * pow(re, 2.0)));
	}
	return copysign(1.0, im) * tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.sinh(-Math.abs(im));
	double t_1 = (0.5 * Math.cos(re)) * (Math.exp((0.0 - Math.abs(im))) - Math.exp(Math.abs(im)));
	double tmp;
	if (t_1 <= -0.5) {
		tmp = t_0;
	} else if (t_1 <= 2e-7) {
		tmp = -Math.cos(re) * Math.abs(im);
	} else {
		tmp = t_0 * (1.0 + (-0.5 * Math.pow(re, 2.0)));
	}
	return Math.copySign(1.0, im) * tmp;
}

Python

def code(re, im):
	t_0 = math.sinh(-math.fabs(im))
	t_1 = (0.5 * math.cos(re)) * (math.exp((0.0 - math.fabs(im))) - math.exp(math.fabs(im)))
	tmp = 0
	if t_1 <= -0.5:
		tmp = t_0
	elif t_1 <= 2e-7:
		tmp = -math.cos(re) * math.fabs(im)
	else:
		tmp = t_0 * (1.0 + (-0.5 * math.pow(re, 2.0)))
	return math.copysign(1.0, im) * tmp

Julia

function code(re, im)
	t_0 = sinh(Float64(-abs(im)))
	t_1 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - abs(im))) - exp(abs(im))))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = t_0;
	elseif (t_1 <= 2e-7)
		tmp = Float64(Float64(-cos(re)) * abs(im));
	else
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 * (re ^ 2.0))));
	end
	return Float64(copysign(1.0, im) * tmp)
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sinh(-abs(im));
	t_1 = (0.5 * cos(re)) * (exp((0.0 - abs(im))) - exp(abs(im)));
	tmp = 0.0;
	if (t_1 <= -0.5)
		tmp = t_0;
	elseif (t_1 <= 2e-7)
		tmp = -cos(re) * abs(im);
	else
		tmp = t_0 * (1.0 + (-0.5 * (re ^ 2.0)));
	end
	tmp_2 = (sign(im) * abs(1.0)) * tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -0.5], t$95$0, If[LessEqual[t$95$1, 2e-7], N[((-N[Cos[re], $MachinePrecision]) * N[Abs[im], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 + N[(-0.5 * N[Power[re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.5

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 39.8%

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. fp-cancel-sub-sign N/A

         \[\leadsto \frac{e^{-im}}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{e^{-im}}{2} - e^{im} \cdot \frac{1}{\color{blue}{2}} \]

      9. mult-flip N/A

         \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{\color{blue}{2}} \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{2} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)}}{2} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      13. div-sub N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{\color{blue}{2}} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      15. sinh-def N/A

          \[\leadsto \sinh \left(-im\right) \]

      16. lift-sinh.f64 64.0%

          \[\leadsto \sinh \left(-im\right) \]

   6. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\sinh \left(-im\right)} \]

   if -0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 52.4%

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

   6. Applied rewrites 52.4%

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

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} - e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{0 - im} - e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} - e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{0 - im} - e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{0 - im} - e^{im}}{2}} \cdot \cos re \]

      7. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im} - e^{im}}}{2} \cdot \cos re \]

      8. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{0 - im}} - e^{im}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{e^{0 - im} - \color{blue}{e^{im}}}{2} \cdot \cos re \]

      10. --rgt-identity N/A

          \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{im - 0}}}{2} \cdot \cos re \]

      11. sub-negate-rev N/A

          \[\leadsto \frac{e^{0 - im} - e^{\color{blue}{\mathsf{neg}\left(\left(0 - im\right)\right)}}}{2} \cdot \cos re \]

      12. lift--.f64 N/A

          \[\leadsto \frac{e^{0 - im} - e^{\mathsf{neg}\left(\color{blue}{\left(0 - im\right)}\right)}}{2} \cdot \cos re \]

      13. sinh-def N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      14. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(0 - im\right) \cdot \cos re} \]

      15. lower-sinh.f64 99.9%

          \[\leadsto \color{blue}{\sinh \left(0 - im\right)} \cdot \cos re \]

      16. lift--.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(0 - im\right)} \cdot \cos re \]

      17. sub0-neg N/A

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

      18. lower-neg.f64 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 62.9%

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

   6. Applied rewrites 62.9%

      \[\leadsto \sinh \left(-im\right) \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - \left|im\right|} - e^{\left|im\right|}\right)\\ \mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\sinh \left(-\left|im\right|\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\left(-\cos re\right) \cdot \left|im\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \left|im\right|, 0.5 \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{{re}^{8}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0
        (*
         (* 0.5 (cos re))
         (- (exp (- 0.0 (fabs im))) (exp (fabs im))))))
  (*
   (copysign 1.0 im)
   (if (<= t_0 -0.5)
     (sinh (- (fabs im)))
     (if (<= t_0 2e-7)
       (* (- (cos re)) (fabs im))
       (fma
        -1.0
        (fabs im)
        (* 0.5 (* (fabs im) (sqrt (sqrt (pow re 8.0)))))))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = sinh(-fabs(im));
	} else if (t_0 <= 2e-7) {
		tmp = -cos(re) * fabs(im);
	} else {
		tmp = fma(-1.0, fabs(im), (0.5 * (fabs(im) * sqrt(sqrt(pow(re, 8.0))))));
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - abs(im))) - exp(abs(im))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = sinh(Float64(-abs(im)));
	elseif (t_0 <= 2e-7)
		tmp = Float64(Float64(-cos(re)) * abs(im));
	else
		tmp = fma(-1.0, abs(im), Float64(0.5 * Float64(abs(im) * sqrt(sqrt((re ^ 8.0))))));
	end
	return Float64(copysign(1.0, im) * tmp)
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, -0.5], N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision], If[LessEqual[t$95$0, 2e-7], N[((-N[Cos[re], $MachinePrecision]) * N[Abs[im], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Abs[im], $MachinePrecision] + N[(0.5 * N[(N[Abs[im], $MachinePrecision] * N[Sqrt[N[Sqrt[N[Power[re, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.5

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 39.8%

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. fp-cancel-sub-sign N/A

         \[\leadsto \frac{e^{-im}}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{e^{-im}}{2} - e^{im} \cdot \frac{1}{\color{blue}{2}} \]

      9. mult-flip N/A

         \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{\color{blue}{2}} \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{2} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)}}{2} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      13. div-sub N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{\color{blue}{2}} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      15. sinh-def N/A

          \[\leadsto \sinh \left(-im\right) \]

      16. lift-sinh.f64 64.0%

          \[\leadsto \sinh \left(-im\right) \]

   6. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\sinh \left(-im\right)} \]

   if -0.5 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1.9999999999999999e-7

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 52.4%

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

   6. Applied rewrites 52.4%

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

   if 1.9999999999999999e-7 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.7%

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

   7. Applied rewrites 36.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 37.3%

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.3%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.3%

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

   9. Applied rewrites 37.3%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-sqrt.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow-prod-down N/A

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

       7. pow-prod-up N/A

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

       8. lift-*.f64 N/A

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

       9. pow-prod-down N/A

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

       10. pow-prod-up N/A

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

       11. lower-pow.f64 N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(-1, im, \frac{1}{2} \cdot \left(im \cdot \sqrt{\sqrt{{re}^{\left(4 + \left(2 + 2\right)\right)}}}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(-1, im, \frac{1}{2} \cdot \left(im \cdot \sqrt{\sqrt{{re}^{\left(4 + 4\right)}}}\right)\right) \]

       14. metadata-eval 38.3%

           \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot \sqrt{\sqrt{{re}^{8}}}\right)\right) \]

   11. Applied rewrites 38.3%

       \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot \sqrt{\sqrt{{re}^{8}}}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, im\right) \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - \left|im\right|} - e^{\left|im\right|}\right) \leq 0:\\ \;\;\;\;\sinh \left(-\left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \left|im\right|, 0.5 \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{{re}^{8}}}\right)\right)\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<=
      (* (* 0.5 (cos re)) (- (exp (- 0.0 (fabs im))) (exp (fabs im))))
      0.0)
   (sinh (- (fabs im)))
   (fma
    -1.0
    (fabs im)
    (* 0.5 (* (fabs im) (sqrt (sqrt (pow re 8.0)))))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)))) <= 0.0) {
		tmp = sinh(-fabs(im));
	} else {
		tmp = fma(-1.0, fabs(im), (0.5 * (fabs(im) * sqrt(sqrt(pow(re, 8.0))))));
	}
	return copysign(1.0, im) * tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - abs(im))) - exp(abs(im)))) <= 0.0)
		tmp = sinh(Float64(-abs(im)));
	else
		tmp = fma(-1.0, abs(im), Float64(0.5 * Float64(abs(im) * sqrt(sqrt((re ^ 8.0))))));
	end
	return Float64(copysign(1.0, im) * tmp)
end

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision], N[(-1.0 * N[Abs[im], $MachinePrecision] + N[(0.5 * N[(N[Abs[im], $MachinePrecision] * N[Sqrt[N[Sqrt[N[Power[re, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 39.8%

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. fp-cancel-sub-sign N/A

         \[\leadsto \frac{e^{-im}}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{e^{-im}}{2} - e^{im} \cdot \frac{1}{\color{blue}{2}} \]

      9. mult-flip N/A

         \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{\color{blue}{2}} \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{2} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)}}{2} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      13. div-sub N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{\color{blue}{2}} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      15. sinh-def N/A

          \[\leadsto \sinh \left(-im\right) \]

      16. lift-sinh.f64 64.0%

          \[\leadsto \sinh \left(-im\right) \]

   6. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\sinh \left(-im\right)} \]

   if -0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.7%

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

   7. Applied rewrites 36.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 37.3%

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.3%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.3%

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

   9. Applied rewrites 37.3%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-sqrt.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow-prod-down N/A

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

       7. pow-prod-up N/A

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

       8. lift-*.f64 N/A

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

       9. pow-prod-down N/A

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

       10. pow-prod-up N/A

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

       11. lower-pow.f64 N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(-1, im, \frac{1}{2} \cdot \left(im \cdot \sqrt{\sqrt{{re}^{\left(4 + \left(2 + 2\right)\right)}}}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(-1, im, \frac{1}{2} \cdot \left(im \cdot \sqrt{\sqrt{{re}^{\left(4 + 4\right)}}}\right)\right) \]

       14. metadata-eval 38.3%

           \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot \sqrt{\sqrt{{re}^{8}}}\right)\right) \]

   11. Applied rewrites 38.3%

       \[\leadsto \mathsf{fma}\left(-1, im, 0.5 \cdot \left(im \cdot \sqrt{\sqrt{{re}^{8}}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
  :precision binary64
  (*
 (copysign 1.0 im)
 (if (<=
      (* (* 0.5 (cos re)) (- (exp (- 0.0 (fabs im))) (exp (fabs im))))
      0.0)
   (sinh (- (fabs im)))
   (fma
    -1.0
    (fabs im)
    (* 0.5 (* (fabs im) (sqrt (* (* re re) (* re re)))))))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp((0.0 - fabs(im))) - exp(fabs(im)))) <= 0.0) {
		tmp = sinh(-fabs(im));
	} else {
		tmp = fma(-1.0, fabs(im), (0.5 * (fabs(im) * sqrt(((re * re) * (re * re))))));
	}
	return copysign(1.0, im) * tmp;
}

Julia

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

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision], N[(-1.0 * N[Abs[im], $MachinePrecision] + N[(0.5 * N[(N[Abs[im], $MachinePrecision] * N[Sqrt[N[(N[(re * re), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 39.8%

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. fp-cancel-sub-sign N/A

         \[\leadsto \frac{e^{-im}}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{e^{-im}}{2} - e^{im} \cdot \frac{1}{\color{blue}{2}} \]

      9. mult-flip N/A

         \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{\color{blue}{2}} \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{2} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)}}{2} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      13. div-sub N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{\color{blue}{2}} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      15. sinh-def N/A

          \[\leadsto \sinh \left(-im\right) \]

      16. lift-sinh.f64 64.0%

          \[\leadsto \sinh \left(-im\right) \]

   6. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\sinh \left(-im\right)} \]

   if -0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.7%

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

   7. Applied rewrites 36.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 37.3%

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.3%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.3%

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

   9. Applied rewrites 37.3%

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

Alternative 6: 74.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(re, im):
	tmp = 0
	if ((0.5 * math.cos(re)) * (math.exp((0.0 - math.fabs(im))) - math.exp(math.fabs(im)))) <= 0.0:
		tmp = math.sinh(-math.fabs(im))
	else:
		tmp = (((re * re) * math.fabs(im)) * 0.5) - math.fabs(im)
	return math.copysign(1.0, im) * tmp

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Sinh[(-N[Abs[im], $MachinePrecision])], $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[Abs[im], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in re around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 39.8%

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

   4. Applied rewrites 39.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. mult-flip-rev N/A

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

      7. fp-cancel-sub-sign N/A

         \[\leadsto \frac{e^{-im}}{2} - \color{blue}{e^{im} \cdot \frac{1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{e^{-im}}{2} - e^{im} \cdot \frac{1}{\color{blue}{2}} \]

      9. mult-flip N/A

         \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{\color{blue}{2}} \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{im}}{2} \]

      11. remove-double-neg N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(im\right)\right)\right)}}{2} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{e^{-im}}{2} - \frac{e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      13. div-sub N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{\color{blue}{2}} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{e^{-im} - e^{\mathsf{neg}\left(\left(-im\right)\right)}}{2} \]

      15. sinh-def N/A

          \[\leadsto \sinh \left(-im\right) \]

      16. lift-sinh.f64 64.0%

          \[\leadsto \sinh \left(-im\right) \]

   6. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\sinh \left(-im\right)} \]

   if -0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.7%

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

   7. Applied rewrites 36.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 37.3%

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.3%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.3%

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

   9. Applied rewrites 37.3%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. sub-flip-reverse N/A

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

       5. lower--.f64 37.3%

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 37.3%

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. sqrt-unprod N/A

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

       13. rem-square-sqrt N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 36.7%

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

   11. Applied rewrites 36.7%

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

Alternative 7: 39.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - N[Abs[im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Exp[N[Abs[im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], (-N[Abs[im], $MachinePrecision]), N[(N[(N[(N[(re * re), $MachinePrecision] * N[Abs[im], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[Abs[im], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in re around 0

      \[\leadsto -1 \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 29.6%

      \[\leadsto -1 \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(im\right) \]

      3. lower-neg.f64 29.6%

         \[\leadsto -im \]

   9. Applied rewrites 29.6%

      \[\leadsto -im \]

   if -0.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (-.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 53.9%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 52.4%

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in re around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 36.7%

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

   7. Applied rewrites 36.7%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 37.3%

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.3%

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.3%

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

   9. Applied rewrites 37.3%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. sub-flip-reverse N/A

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

       5. lower--.f64 37.3%

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 37.3%

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. sqrt-unprod N/A

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

       13. rem-square-sqrt N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 36.7%

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

   11. Applied rewrites 36.7%

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

Alternative 8: 29.6% accurate, 33.9× speedup

Math

\[-im \]

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 Float64(-im)
end

MATLAB

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

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 53.9%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]

2. Taylor expanded in im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 52.4%

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

4. Applied rewrites 52.4%

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

5. Taylor expanded in re around 0

   \[\leadsto -1 \cdot im \]
6. Step-by-step derivation

7. Applied rewrites 29.6%

   \[\leadsto -1 \cdot im \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(im\right) \]

   3. lower-neg.f64 29.6%

      \[\leadsto -im \]

9. Applied rewrites 29.6%

   \[\leadsto -im \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025213 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))