math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 98.5%

Time: 7.3s

Alternatives: 18

Speedup: 1.0×

• Could not converge on a ground truth

  1. re = 18912.78069246999
  2. im = 3.383981519535803e+211

• 24.4% 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: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right)\\ t_1 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\_m\\ \mathbf{elif}\;t\_1 \leq -0.04:\\ \;\;\;\;\frac{\sin im\_m}{\mathsf{fma}\left(-4, re, 4\right)} \cdot 4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-154} \lor \neg \left(t\_1 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) -0.16666666666666666 1.0))
        (t_1 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (fma (* (fma 0.5 re 1.0) t_0) re t_0) im_m)
      (if (<= t_1 -0.04)
        (* (/ (sin im_m) (fma -4.0 re 4.0)) 4.0)
        (if (or (<= t_1 2e-154) (not (<= t_1 1.0)))
          (* (exp re) im_m)
          (* (fma (fma 0.5 re 1.0) re 1.0) (sin im_m))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((im_m * im_m), -0.16666666666666666, 1.0);
	double t_1 = exp(re) * sin(im_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((fma(0.5, re, 1.0) * t_0), re, t_0) * im_m;
	} else if (t_1 <= -0.04) {
		tmp = (sin(im_m) / fma(-4.0, re, 4.0)) * 4.0;
	} else if ((t_1 <= 2e-154) || !(t_1 <= 1.0)) {
		tmp = exp(re) * im_m;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), -0.16666666666666666, 1.0)
	t_1 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(0.5, re, 1.0) * t_0), re, t_0) * im_m);
	elseif (t_1 <= -0.04)
		tmp = Float64(Float64(sin(im_m) / fma(-4.0, re, 4.0)) * 4.0);
	elseif ((t_1 <= 2e-154) || !(t_1 <= 1.0))
		tmp = Float64(exp(re) * im_m);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im$95$m), $MachinePrecision], If[LessEqual[t$95$1, -0.04], N[(N[(N[Sin[im$95$m], $MachinePrecision] / N[(-4.0 * re + 4.0), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision], If[Or[LessEqual[t$95$1, 2e-154], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 48.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\cosh re \cdot \cosh re - \sinh re \cdot \sinh re}{\cosh re - \sinh re}} \cdot \sin im \]

      5. sinh---cosh-rev N/A

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. sinh---cosh-rev N/A

         \[\leadsto \frac{\sin im}{\color{blue}{\cosh re - \sinh re}} \]

      10. cosh-def N/A

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

      11. sinh-def N/A

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

      12. frac-sub N/A

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

      13. associate-/r/ N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\sin im}{4 \cdot \cosh re - 4 \cdot \sinh re} \cdot 4} \]

   5. Taylor expanded in re around 0

      \[\leadsto \frac{\sin im}{\color{blue}{4 + -4 \cdot re}} \cdot 4 \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin im}{\color{blue}{-4 \cdot re + 4}} \cdot 4 \]

      2. lower-fma.f64 99.3

         \[\leadsto \frac{\sin im}{\color{blue}{\mathsf{fma}\left(-4, re, 4\right)}} \cdot 4 \]

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\sin im}{\color{blue}{\mathsf{fma}\left(-4, re, 4\right)}} \cdot 4 \]

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-154 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 1.9999999999999999e-154 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

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

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

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

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

      10. remove-double-neg N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \sin im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \sin im \]

   5. Applied rewrites 100.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.04:\\ \;\;\;\;\frac{\sin im}{\mathsf{fma}\left(-4, re, 4\right)} \cdot 4\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-154} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right)\\ t_1 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\_m\\ \mathbf{elif}\;t\_1 \leq -0.04:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\_m\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-154} \lor \neg \left(t\_1 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) -0.16666666666666666 1.0))
        (t_1 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (fma (* (fma 0.5 re 1.0) t_0) re t_0) im_m)
      (if (<= t_1 -0.04)
        (* (+ 1.0 re) (sin im_m))
        (if (or (<= t_1 2e-154) (not (<= t_1 1.0)))
          (* (exp re) im_m)
          (* (fma (fma 0.5 re 1.0) re 1.0) (sin im_m))))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((im_m * im_m), -0.16666666666666666, 1.0);
	double t_1 = exp(re) * sin(im_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((fma(0.5, re, 1.0) * t_0), re, t_0) * im_m;
	} else if (t_1 <= -0.04) {
		tmp = (1.0 + re) * sin(im_m);
	} else if ((t_1 <= 2e-154) || !(t_1 <= 1.0)) {
		tmp = exp(re) * im_m;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), -0.16666666666666666, 1.0)
	t_1 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(0.5, re, 1.0) * t_0), re, t_0) * im_m);
	elseif (t_1 <= -0.04)
		tmp = Float64(Float64(1.0 + re) * sin(im_m));
	elseif ((t_1 <= 2e-154) || !(t_1 <= 1.0))
		tmp = Float64(exp(re) * im_m);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im$95$m), $MachinePrecision], If[LessEqual[t$95$1, -0.04], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 2e-154], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 48.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-154 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 1.9999999999999999e-154 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

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

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

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

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

      10. remove-double-neg N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \sin im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \sin im \]

   5. Applied rewrites 100.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.04:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-154} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right)\\ t_1 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\_m\\ \mathbf{elif}\;t\_1 \leq -0.04 \lor \neg \left(t\_1 \leq 2 \cdot 10^{-154} \lor \neg \left(t\_1 \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\_m\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) -0.16666666666666666 1.0))
        (t_1 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (fma (* (fma 0.5 re 1.0) t_0) re t_0) im_m)
      (if (or (<= t_1 -0.04) (not (or (<= t_1 2e-154) (not (<= t_1 1.0)))))
        (* (+ 1.0 re) (sin im_m))
        (* (exp re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((im_m * im_m), -0.16666666666666666, 1.0);
	double t_1 = exp(re) * sin(im_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((fma(0.5, re, 1.0) * t_0), re, t_0) * im_m;
	} else if ((t_1 <= -0.04) || !((t_1 <= 2e-154) || !(t_1 <= 1.0))) {
		tmp = (1.0 + re) * sin(im_m);
	} else {
		tmp = exp(re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), -0.16666666666666666, 1.0)
	t_1 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(0.5, re, 1.0) * t_0), re, t_0) * im_m);
	elseif ((t_1 <= -0.04) || !((t_1 <= 2e-154) || !(t_1 <= 1.0)))
		tmp = Float64(Float64(1.0 + re) * sin(im_m));
	else
		tmp = Float64(exp(re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im$95$m), $MachinePrecision], If[Or[LessEqual[t$95$1, -0.04], N[Not[Or[LessEqual[t$95$1, 2e-154], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 48.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008 or 1.9999999999999999e-154 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-154 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.04 \lor \neg \left(e^{re} \cdot \sin im \leq 2 \cdot 10^{-154} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right)\\ t_1 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\_m\\ \mathbf{elif}\;t\_1 \leq -0.04 \lor \neg \left(t\_1 \leq 2 \cdot 10^{-154} \lor \neg \left(t\_1 \leq 1\right)\right):\\ \;\;\;\;\sin im\_m\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) -0.16666666666666666 1.0))
        (t_1 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (fma (* (fma 0.5 re 1.0) t_0) re t_0) im_m)
      (if (or (<= t_1 -0.04) (not (or (<= t_1 2e-154) (not (<= t_1 1.0)))))
        (sin im_m)
        (* (exp re) im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((im_m * im_m), -0.16666666666666666, 1.0);
	double t_1 = exp(re) * sin(im_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((fma(0.5, re, 1.0) * t_0), re, t_0) * im_m;
	} else if ((t_1 <= -0.04) || !((t_1 <= 2e-154) || !(t_1 <= 1.0))) {
		tmp = sin(im_m);
	} else {
		tmp = exp(re) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), -0.16666666666666666, 1.0)
	t_1 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(0.5, re, 1.0) * t_0), re, t_0) * im_m);
	elseif ((t_1 <= -0.04) || !((t_1 <= 2e-154) || !(t_1 <= 1.0)))
		tmp = sin(im_m);
	else
		tmp = Float64(exp(re) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im$95$m), $MachinePrecision], If[Or[LessEqual[t$95$1, -0.04], N[Not[Or[LessEqual[t$95$1, 2e-154], N[Not[LessEqual[t$95$1, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[Sin[im$95$m], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 48.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0400000000000000008 or 1.9999999999999999e-154 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-154 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.04 \lor \neg \left(e^{re} \cdot \sin im \leq 2 \cdot 10^{-154} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\_m\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-154} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_0 -0.04)
      (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im_m))
      (if (or (<= t_0 2e-154) (not (<= t_0 1.0)))
        (* (exp re) im_m)
        (* (fma (fma 0.5 re 1.0) re 1.0) (sin im_m)))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = exp(re) * sin(im_m);
	double tmp;
	if (t_0 <= -0.04) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im_m);
	} else if ((t_0 <= 2e-154) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im_m;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im_m);
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im_m));
	elseif ((t_0 <= 2e-154) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im_m);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im_m));
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -0.04], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 2e-154], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

   5. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \sin im \]

   if -0.0400000000000000008 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-154 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 1.9999999999999999e-154 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

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

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

      6. distribute-lft-in N/A

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

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

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

      8. *-rgt-identity N/A

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

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

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

      10. remove-double-neg N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + \color{blue}{\left(0 + 1\right)}\right) \cdot \sin im \]

      12. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(\left(\left(re + re \cdot \left(\frac{1}{2} \cdot re\right)\right) + 0\right) + 1\right)} \cdot \sin im \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)} \cdot \sin im \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-154} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right)\\ t_1 := e^{re} \cdot \sin im\_m\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\_m\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) -0.16666666666666666 1.0))
        (t_1 (* (exp re) (sin im_m))))
   (*
    im_s
    (if (<= t_1 (- INFINITY))
      (* (fma (* (fma 0.5 re 1.0) t_0) re t_0) im_m)
      (if (<= t_1 1.0)
        (sin im_m)
        (*
         (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
         im_m))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((im_m * im_m), -0.16666666666666666, 1.0);
	double t_1 = exp(re) * sin(im_m);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((fma(0.5, re, 1.0) * t_0), re, t_0) * im_m;
	} else if (t_1 <= 1.0) {
		tmp = sin(im_m);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), -0.16666666666666666, 1.0)
	t_1 = Float64(exp(re) * sin(im_m))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(fma(0.5, re, 1.0) * t_0), re, t_0) * im_m);
	elseif (t_1 <= 1.0)
		tmp = sin(im_m);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[im$95$m], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 48.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 46.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot t\_0, re, t\_0\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (let* ((t_0 (fma (* im_m im_m) -0.16666666666666666 1.0)))
   (*
    im_s
    (if (<= (* (exp re) (sin im_m)) 0.0)
      (* (fma (* (fma 0.5 re 1.0) t_0) re t_0) im_m)
      (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im_m)))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = fma((im_m * im_m), -0.16666666666666666, 1.0);
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = fma((fma(0.5, re, 1.0) * t_0), re, t_0) * im_m;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	t_0 = fma(Float64(im_m * im_m), -0.16666666666666666, 1.0)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = Float64(fma(Float64(fma(0.5, re, 1.0) * t_0), re, t_0) * im_m);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := Block[{t$95$0 = N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * re + t$95$0), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 61.5

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

   8. Applied rewrites 61.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 29.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right), re, \mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right)\right) \cdot im \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im\_m, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 4e-309)
    (* (fma (* im_m im_m) -0.16666666666666666 1.0) (fma re im_m im_m))
    (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 4e-309) {
		tmp = fma((im_m * im_m), -0.16666666666666666, 1.0) * fma(re, im_m, im_m);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 4e-309)
		tmp = Float64(fma(Float64(im_m * im_m), -0.16666666666666666, 1.0) * fma(re, im_m, im_m));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 4e-309], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(re * im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999977e-309

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 61.5

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

   8. Applied rewrites 61.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 25.1%

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

   if 3.9999999999999977e-309 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 45.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im\_m, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 4e-309)
    (* (fma (* im_m im_m) -0.16666666666666666 1.0) (fma re im_m im_m))
    (* (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 4e-309) {
		tmp = fma((im_m * im_m), -0.16666666666666666, 1.0) * fma(re, im_m, im_m);
	} else {
		tmp = fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 4e-309)
		tmp = Float64(fma(Float64(im_m * im_m), -0.16666666666666666, 1.0) * fma(re, im_m, im_m));
	else
		tmp = Float64(fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 4e-309], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(re * im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999977e-309

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 61.5

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

   8. Applied rewrites 61.5%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 25.1%

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

   if 3.9999999999999977e-309 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 52.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 2 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im\_m, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 2e-154)
    (* (fma (* im_m im_m) -0.16666666666666666 1.0) (fma re im_m im_m))
    (* (fma (* (* re re) 0.16666666666666666) re 1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 2e-154) {
		tmp = fma((im_m * im_m), -0.16666666666666666, 1.0) * fma(re, im_m, im_m);
	} else {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 2e-154)
		tmp = Float64(fma(Float64(im_m * im_m), -0.16666666666666666, 1.0) * fma(re, im_m, im_m));
	else
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 2e-154], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(re * im$95$m + im$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-154

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-exp.f64 65.0

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

   8. Applied rewrites 65.0%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 31.7%

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

   if 1.9999999999999999e-154 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 45.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 45.6%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot -0.16666666666666666, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.0)
    (fma (* im_m im_m) (* im_m -0.16666666666666666) im_m)
    (* (fma (* (* re re) 0.16666666666666666) re 1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.0) {
		tmp = fma((im_m * im_m), (im_m * -0.16666666666666666), im_m);
	} else {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.0)
		tmp = fma(Float64(im_m * im_m), Float64(im_m * -0.16666666666666666), im_m);
	else
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + im$95$m), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 25.3%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 25.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 53.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im \]

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 52.0%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 42.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot -0.16666666666666666, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 4e-309)
    (fma (* im_m im_m) (* im_m -0.16666666666666666) im_m)
    (* (fma (fma 0.5 re 1.0) re 1.0) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 4e-309) {
		tmp = fma((im_m * im_m), (im_m * -0.16666666666666666), im_m);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 4e-309)
		tmp = fma(Float64(im_m * im_m), Float64(im_m * -0.16666666666666666), im_m);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 4e-309], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + im$95$m), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 3.9999999999999977e-309

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 25.3%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 25.3%

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

   if 3.9999999999999977e-309 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.7%

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

Alternative 13: 42.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(im\_m \cdot im\_m, im\_m \cdot -0.16666666666666666, im\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 2e-8)
    (fma (* im_m im_m) (* im_m -0.16666666666666666) im_m)
    (* (* (* re re) 0.5) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 2e-8) {
		tmp = fma((im_m * im_m), (im_m * -0.16666666666666666), im_m);
	} else {
		tmp = ((re * re) * 0.5) * im_m;
	}
	return im_s * tmp;
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 2e-8)
		tmp = fma(Float64(im_m * im_m), Float64(im_m * -0.16666666666666666), im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im_m);
	end
	return Float64(im_s * tmp)
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 2e-8], N[(N[(im$95$m * im$95$m), $MachinePrecision] * N[(im$95$m * -0.16666666666666666), $MachinePrecision] + im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 37.1%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 37.1%

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

   if 2e-8 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 44.5

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

   5. Applied rewrites 44.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 28.4%

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

Alternative 14: 37.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.15:\\ \;\;\;\;1 \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (*
  im_s
  (if (<= (* (exp re) (sin im_m)) 0.15)
    (* 1.0 im_m)
    (* (* (* re re) 0.5) im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.15) {
		tmp = 1.0 * im_m;
	} else {
		tmp = ((re * re) * 0.5) * im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(re) * sin(im_m)) <= 0.15d0) then
        tmp = 1.0d0 * im_m
    else
        tmp = ((re * re) * 0.5d0) * im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im_m)) <= 0.15) {
		tmp = 1.0 * im_m;
	} else {
		tmp = ((re * re) * 0.5) * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(re) * math.sin(im_m)) <= 0.15:
		tmp = 1.0 * im_m
	else:
		tmp = ((re * re) * 0.5) * im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.15)
		tmp = Float64(1.0 * im_m);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(re) * sin(im_m)) <= 0.15)
		tmp = 1.0 * im_m;
	else
		tmp = ((re * re) * 0.5) * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.15], N[(1.0 * im$95$m), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.149999999999999994

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 33.9%

      \[\leadsto 1 \cdot im \]

   if 0.149999999999999994 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 24.5%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 30.0%

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

Alternative 15: 30.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im\_m \leq 0.92:\\ \;\;\;\;1 \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\_m\\ \end{array} \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m)
 :precision binary64
 (* im_s (if (<= (* (exp re) (sin im_m)) 0.92) (* 1.0 im_m) (* re im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((exp(re) * sin(im_m)) <= 0.92) {
		tmp = 1.0 * im_m;
	} else {
		tmp = re * im_m;
	}
	return im_s * tmp;
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if ((exp(re) * sin(im_m)) <= 0.92d0) then
        tmp = 1.0d0 * im_m
    else
        tmp = re * im_m
    end if
    code = im_s * tmp
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im_m)) <= 0.92) {
		tmp = 1.0 * im_m;
	} else {
		tmp = re * im_m;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	tmp = 0
	if (math.exp(re) * math.sin(im_m)) <= 0.92:
		tmp = 1.0 * im_m
	else:
		tmp = re * im_m
	return im_s * tmp

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	tmp = 0.0
	if (Float64(exp(re) * sin(im_m)) <= 0.92)
		tmp = Float64(1.0 * im_m);
	else
		tmp = Float64(re * im_m);
	end
	return Float64(im_s * tmp)
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp_2 = code(im_s, re, im_m)
	tmp = 0.0;
	if ((exp(re) * sin(im_m)) <= 0.92)
		tmp = 1.0 * im_m;
	else
		tmp = re * im_m;
	end
	tmp_2 = im_s * tmp;
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision], 0.92], N[(1.0 * im$95$m), $MachinePrecision], N[(re * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.92000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 30.2%

      \[\leadsto 1 \cdot im \]

   if 0.92000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 2.9%

      \[\leadsto 1 \cdot im \]

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 21.6%

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

   12. Taylor expanded in re around inf

       \[\leadsto im \cdot re \]
   13. Step-by-step derivation

   14. Applied rewrites 21.7%

       \[\leadsto re \cdot im \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(e^{re} \cdot \sin im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* (exp re) (sin im_m))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (exp(re) * sin(im_m));
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (exp(re) * sin(im_m))
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (Math.exp(re) * Math.sin(im_m));
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (math.exp(re) * math.sin(im_m))

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(exp(re) * sin(im_m)))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (exp(re) * sin(im_m));
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(N[Exp[re], $MachinePrecision] * N[Sin[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

Alternative 17: 30.2% accurate, 29.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \mathsf{fma}\left(im\_m, re, im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (fma im_m re im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * fma(im_m, re, im_m);
}

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * fma(im_m, re, im_m))
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(im$95$m * re + im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 67.6

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

5. Applied rewrites 67.6%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 30.0%

   \[\leadsto \mathsf{fma}\left(im, \color{blue}{re}, im\right) \]
9. Add Preprocessing

Alternative 18: 6.7% accurate, 34.3× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(re \cdot im\_m\right) \end{array} \]

FPCore

im\_m = (fabs.f64 im)
im\_s = (copysign.f64 #s(literal 1 binary64) im)
(FPCore (im_s re im_m) :precision binary64 (* im_s (* re im_m)))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	return im_s * (re * im_m);
}

Fortran

im\_m =     private
im\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(im_s, re, im_m)
use fmin_fmax_functions
    real(8), intent (in) :: im_s
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    code = im_s * (re * im_m)
end function

Java

im\_m = Math.abs(im);
im\_s = Math.copySign(1.0, im);
public static double code(double im_s, double re, double im_m) {
	return im_s * (re * im_m);
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	return im_s * (re * im_m)

Julia

im\_m = abs(im)
im\_s = copysign(1.0, im)
function code(im_s, re, im_m)
	return Float64(im_s * Float64(re * im_m))
end

MATLAB

im\_m = abs(im);
im\_s = sign(im) * abs(1.0);
function tmp = code(im_s, re, im_m)
	tmp = im_s * (re * im_m);
end

Wolfram

im\_m = N[Abs[im], $MachinePrecision]
im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[im$95$s_, re_, im$95$m_] := N[(im$95$s * N[(re * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 67.6

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

5. Applied rewrites 67.6%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 24.6%

   \[\leadsto 1 \cdot im \]

9. Taylor expanded in re around 0

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

11. Applied rewrites 30.0%

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

12. Taylor expanded in re around inf

    \[\leadsto im \cdot re \]
13. Step-by-step derivation

14. Applied rewrites 9.1%

    \[\leadsto re \cdot im \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025010 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))