math.cos on complex, imaginary part

Percentage Accurate: 66.1% → 99.9%

Time: 5.0s

Alternatives: 14

Speedup: 0.9×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;im\_m \leq 0.00135:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ \end{array} \end{array} \]

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 (<= im_m 0.00135)
    (* (* (sin re) (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m)
    (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp 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 (im_m <= 0.00135) {
		tmp = (sin(re) * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im_m) - exp(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 (im_m <= 0.00135)
		tmp = Float64(Float64(sin(re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(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[im$95$m, 0.00135], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if im < 0.0013500000000000001

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   if 0.0013500000000000001 < im

   1. Initial program 66.1%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ t_1 := \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(1 - e^{im\_m}\right) \cdot 0.5\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;\left(\sin re \cdot t\_1\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re\right) \cdot t\_1\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))))
        (t_1 (fma (* -0.16666666666666666 im_m) im_m -1.0)))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* (- 1.0 (exp im_m)) 0.5) re)
      (if (<= t_0 1e-13)
        (* (* (sin re) t_1) im_m)
        (* (* (* (* (* re re) -0.16666666666666666) re) t_1) 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 = (0.5 * sin(re)) * (exp(-im_m) - exp(im_m));
	double t_1 = fma((-0.16666666666666666 * im_m), im_m, -1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((1.0 - exp(im_m)) * 0.5) * re;
	} else if (t_0 <= 1e-13) {
		tmp = (sin(re) * t_1) * im_m;
	} else {
		tmp = ((((re * re) * -0.16666666666666666) * re) * t_1) * 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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	t_1 = fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(1.0 - exp(im_m)) * 0.5) * re);
	elseif (t_0 <= 1e-13)
		tmp = Float64(Float64(sin(re) * t_1) * im_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * re) * -0.16666666666666666) * re) * t_1) * 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[(N[(N[Sin[re], $MachinePrecision] * t$95$1), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision] * t$95$1), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 51.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1e-13

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   if 1e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.7

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 25.2

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

   10. Applied rewrites 25.2%

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

Alternative 3: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(1 - e^{im\_m}\right) \cdot 0.5\right) \cdot re\\ \mathbf{elif}\;t\_0 \leq 10^{-13}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(re \cdot re\right) \cdot -0.16666666666666666\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \end{array} \]

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 (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 (- INFINITY))
      (* (* (- 1.0 (exp im_m)) 0.5) re)
      (if (<= t_0 1e-13)
        (* (- (sin re)) im_m)
        (*
         (*
          (* (* (* re re) -0.16666666666666666) re)
          (fma (* -0.16666666666666666 im_m) im_m -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 = (0.5 * sin(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((1.0 - exp(im_m)) * 0.5) * re;
	} else if (t_0 <= 1e-13) {
		tmp = -sin(re) * im_m;
	} else {
		tmp = ((((re * re) * -0.16666666666666666) * re) * fma((-0.16666666666666666 * im_m), im_m, -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 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(1.0 - exp(im_m)) * 0.5) * re);
	elseif (t_0 <= 1e-13)
		tmp = Float64(Float64(-sin(re)) * im_m);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(re * re) * -0.16666666666666666) * re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], If[LessEqual[t$95$0, 1e-13], N[((-N[Sin[re], $MachinePrecision]) * im$95$m), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 51.8%

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

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 1e-13

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   if 1e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.7

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 25.2

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

   10. Applied rewrites 25.2%

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

Alternative 4: 56.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(1 - e^{im\_m}\right) \cdot 0.5\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re, -0.16666666666666666 \cdot re, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) -1e-13)
    (* (* (- 1.0 (exp im_m)) 0.5) re)
    (*
     (*
      (* (fma re (* -0.16666666666666666 re) 1.0) re)
      (fma (* -0.16666666666666666 im_m) im_m -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 (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -1e-13) {
		tmp = ((1.0 - exp(im_m)) * 0.5) * re;
	} else {
		tmp = ((fma(re, (-0.16666666666666666 * re), 1.0) * re) * fma((-0.16666666666666666 * im_m), im_m, -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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -1e-13)
		tmp = Float64(Float64(Float64(1.0 - exp(im_m)) * 0.5) * re);
	else
		tmp = Float64(Float64(Float64(fma(re, Float64(-0.16666666666666666 * re), 1.0) * re) * fma(Float64(-0.16666666666666666 * im_m), im_m, -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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-13], N[(N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(re * N[(-0.16666666666666666 * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1e-13

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 51.8%

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

   if -1e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 50.7

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

   7. Applied rewrites 50.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(re \cdot re, -0.16666666666666666, 1\right) \cdot re\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\right) \cdot im \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 50.7

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

   9. Applied rewrites 50.7%

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

Alternative 5: 52.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(1 - e^{im\_m}\right) \cdot 0.5\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.16666666666666666 \cdot im\_m, -im\_m\right) \cdot re\\ \end{array} \end{array} \]

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 (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) -1e-13)
    (* (* (- 1.0 (exp im_m)) 0.5) re)
    (* (fma (* re re) (* 0.16666666666666666 im_m) (- im_m)) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -1e-13) {
		tmp = ((1.0 - exp(im_m)) * 0.5) * re;
	} else {
		tmp = fma((re * re), (0.16666666666666666 * im_m), -im_m) * re;
	}
	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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -1e-13)
		tmp = Float64(Float64(Float64(1.0 - exp(im_m)) * 0.5) * re);
	else
		tmp = Float64(fma(Float64(re * re), Float64(0.16666666666666666 * im_m), Float64(-im_m)) * re);
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-13], N[(N[(N[(1.0 - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * im$95$m), $MachinePrecision] + (-im$95$m)), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1e-13

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

   7. Applied rewrites 51.8%

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

   if -1e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{6}, \mathsf{neg}\left(im\right)\right) \cdot re \]

      11. lift-neg.f64 35.9

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

   7. Applied rewrites 35.9%

      \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, 0.16666666666666666, -im\right) \cdot \color{blue}{re} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{1}{6} \cdot im, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lift-neg.f64 35.9

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

   9. Applied rewrites 35.9%

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

Alternative 6: 52.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(re \cdot re, 0.16666666666666666 \cdot im\_m, -im\_m\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im\_m \cdot \left(-0.16666666666666666 \cdot im\_m\right) - 1\right) \cdot im\_m\right) \cdot re\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) -0.01)
    (* (fma (* re re) (* 0.16666666666666666 im_m) (- im_m)) re)
    (* (* (- (* im_m (* -0.16666666666666666 im_m)) 1.0) im_m) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma((re * re), (0.16666666666666666 * im_m), -im_m) * re;
	} else {
		tmp = (((im_m * (-0.16666666666666666 * im_m)) - 1.0) * im_m) * re;
	}
	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(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(re * re), Float64(0.16666666666666666 * im_m), Float64(-im_m)) * re);
	else
		tmp = Float64(Float64(Float64(Float64(im_m * Float64(-0.16666666666666666 * im_m)) - 1.0) * im_m) * re);
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * im$95$m), $MachinePrecision] + (-im$95$m)), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im$95$m * N[(-0.16666666666666666 * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{6}, \mathsf{neg}\left(im\right)\right) \cdot re \]

      11. lift-neg.f64 35.9

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

   7. Applied rewrites 35.9%

      \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, 0.16666666666666666, -im\right) \cdot \color{blue}{re} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(re \cdot re, \frac{1}{6} \cdot im, \mathsf{neg}\left(im\right)\right) \cdot re \]

      16. lift-neg.f64 35.9

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

   9. Applied rewrites 35.9%

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

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.3

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\right) \cdot re \]

   7. Applied rewrites 52.3%

      \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\right) \cdot re \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 52.3

         \[\leadsto \left(\left(im \cdot \left(-0.16666666666666666 \cdot im\right) - 1\right) \cdot im\right) \cdot re \]

   9. Applied rewrites 52.3%

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

Alternative 7: 49.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\_m\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im\_m \cdot \left(-0.16666666666666666 \cdot im\_m\right) - 1\right) \cdot im\_m\right) \cdot re\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) -0.01)
    (* (* (* (* re re) im_m) 0.16666666666666666) re)
    (* (* (- (* im_m (* -0.16666666666666666 im_m)) 1.0) im_m) re))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re;
	} else {
		tmp = (((im_m * (-0.16666666666666666 * im_m)) - 1.0) * im_m) * re;
	}
	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 ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = (((re * re) * im_m) * 0.16666666666666666d0) * re
    else
        tmp = (((im_m * ((-0.16666666666666666d0) * im_m)) - 1.0d0) * im_m) * re
    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 ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re;
	} else {
		tmp = (((im_m * (-0.16666666666666666 * im_m)) - 1.0) * im_m) * re;
	}
	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 (0.5 * math.sin(re)) <= -0.01:
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re
	else:
		tmp = (((im_m * (-0.16666666666666666 * im_m)) - 1.0) * im_m) * re
	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(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * re) * im_m) * 0.16666666666666666) * re);
	else
		tmp = Float64(Float64(Float64(Float64(im_m * Float64(-0.16666666666666666 * im_m)) - 1.0) * im_m) * re);
	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 ((0.5 * sin(re)) <= -0.01)
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re;
	else
		tmp = (((im_m * (-0.16666666666666666 * im_m)) - 1.0) * im_m) * re;
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im$95$m * N[(-0.16666666666666666 * im$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * im$95$m), $MachinePrecision] * re), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{6}, \mathsf{neg}\left(im\right)\right) \cdot re \]

      11. lift-neg.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 24.1

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re \]

   10. Applied rewrites 24.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.3

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\right) \cdot re \]

   7. Applied rewrites 52.3%

      \[\leadsto \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666 - 1\right) \cdot im\right) \cdot re \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 52.3

         \[\leadsto \left(\left(im \cdot \left(-0.16666666666666666 \cdot im\right) - 1\right) \cdot im\right) \cdot re \]

   9. Applied rewrites 52.3%

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

Alternative 8: 44.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot \left(-0.16666666666666666 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im\_m, im\_m, -1\right)\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\_m\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \end{array} \]

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 (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -1e-13)
      (* (* (* im_m im_m) im_m) (* -0.16666666666666666 re))
      (if (<= t_0 0.0)
        (* (* re (fma (* -0.16666666666666666 im_m) im_m -1.0)) im_m)
        (* (* (* (* re re) re) im_m) 0.16666666666666666))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -1e-13) {
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re);
	} else if (t_0 <= 0.0) {
		tmp = (re * fma((-0.16666666666666666 * im_m), im_m, -1.0)) * im_m;
	} else {
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666;
	}
	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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -1e-13)
		tmp = Float64(Float64(Float64(im_m * im_m) * im_m) * Float64(-0.16666666666666666 * re));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(re * fma(Float64(-0.16666666666666666 * im_m), im_m, -1.0)) * im_m);
	else
		tmp = Float64(Float64(Float64(Float64(re * re) * re) * im_m) * 0.16666666666666666);
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -1e-13], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(re * N[(N[(-0.16666666666666666 * im$95$m), $MachinePrecision] * im$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * im$95$m), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1e-13

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 41.6

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 \]

   10. Applied rewrites 41.6%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. pow3 N/A

          \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{3} \cdot \left(re \cdot \frac{-1}{6}\right) \]

       7. *-commutative N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       8. lower-*.f64 N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       9. pow3 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 41.6

           \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(-0.16666666666666666 \cdot re\right) \]

   12. Applied rewrites 41.6%

       \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(-0.16666666666666666 \cdot re\right) \]

   if -1e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 49.2%

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

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

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{6}, \mathsf{neg}\left(im\right)\right) \cdot re \]

      11. lift-neg.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      3. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.1

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\right) \cdot 0.16666666666666666 \]

   10. Applied rewrites 24.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\right) \cdot 0.16666666666666666 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 43.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right)\\ im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot \left(-0.16666666666666666 \cdot re\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-re\right) \cdot im\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\_m\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \end{array} \]

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 (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m)))))
   (*
    im_s
    (if (<= t_0 -1e-13)
      (* (* (* im_m im_m) im_m) (* -0.16666666666666666 re))
      (if (<= t_0 0.0)
        (* (- re) im_m)
        (* (* (* (* re re) re) im_m) 0.16666666666666666))))))

C

im\_m = fabs(im);
im\_s = copysign(1.0, im);
double code(double im_s, double re, double im_m) {
	double t_0 = (0.5 * sin(re)) * (exp(-im_m) - exp(im_m));
	double tmp;
	if (t_0 <= -1e-13) {
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re);
	} else if (t_0 <= 0.0) {
		tmp = -re * im_m;
	} else {
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * sin(re)) * (exp(-im_m) - exp(im_m))
    if (t_0 <= (-1d-13)) then
        tmp = ((im_m * im_m) * im_m) * ((-0.16666666666666666d0) * re)
    else if (t_0 <= 0.0d0) then
        tmp = -re * im_m
    else
        tmp = (((re * re) * re) * im_m) * 0.16666666666666666d0
    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 t_0 = (0.5 * Math.sin(re)) * (Math.exp(-im_m) - Math.exp(im_m));
	double tmp;
	if (t_0 <= -1e-13) {
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re);
	} else if (t_0 <= 0.0) {
		tmp = -re * im_m;
	} else {
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666;
	}
	return im_s * tmp;
}

Python

im\_m = math.fabs(im)
im\_s = math.copysign(1.0, im)
def code(im_s, re, im_m):
	t_0 = (0.5 * math.sin(re)) * (math.exp(-im_m) - math.exp(im_m))
	tmp = 0
	if t_0 <= -1e-13:
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re)
	elif t_0 <= 0.0:
		tmp = -re * im_m
	else:
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666
	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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m)))
	tmp = 0.0
	if (t_0 <= -1e-13)
		tmp = Float64(Float64(Float64(im_m * im_m) * im_m) * Float64(-0.16666666666666666 * re));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-re) * im_m);
	else
		tmp = Float64(Float64(Float64(Float64(re * re) * re) * im_m) * 0.16666666666666666);
	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)
	t_0 = (0.5 * sin(re)) * (exp(-im_m) - exp(im_m));
	tmp = 0.0;
	if (t_0 <= -1e-13)
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re);
	elseif (t_0 <= 0.0)
		tmp = -re * im_m;
	else
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666;
	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_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(im$95$s * If[LessEqual[t$95$0, -1e-13], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-re) * im$95$m), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1e-13

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 41.6

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 \]

   10. Applied rewrites 41.6%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. pow3 N/A

          \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{3} \cdot \left(re \cdot \frac{-1}{6}\right) \]

       7. *-commutative N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       8. lower-*.f64 N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       9. pow3 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 41.6

           \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(-0.16666666666666666 \cdot re\right) \]

   12. Applied rewrites 41.6%

       \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(-0.16666666666666666 \cdot re\right) \]

   if -1e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.0

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 32.1%

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

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

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{6}, \mathsf{neg}\left(im\right)\right) \cdot re \]

      11. lift-neg.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      3. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.1

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\right) \cdot 0.16666666666666666 \]

   10. Applied rewrites 24.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\right) \cdot 0.16666666666666666 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im\_m} - e^{im\_m}\right) \leq -1 \cdot 10^{-13}:\\ \;\;\;\;\left(\left(im\_m \cdot im\_m\right) \cdot im\_m\right) \cdot \left(-0.16666666666666666 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (* (* 0.5 (sin re)) (- (exp (- im_m)) (exp im_m))) -1e-13)
    (* (* (* im_m im_m) im_m) (* -0.16666666666666666 re))
    (* (* (- (* 0.16666666666666666 (* re re)) 1.0) 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 (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -1e-13) {
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re);
	} else {
		tmp = (((0.16666666666666666 * (re * re)) - 1.0) * 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 (((0.5d0 * sin(re)) * (exp(-im_m) - exp(im_m))) <= (-1d-13)) then
        tmp = ((im_m * im_m) * im_m) * ((-0.16666666666666666d0) * re)
    else
        tmp = (((0.16666666666666666d0 * (re * re)) - 1.0d0) * 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 (((0.5 * Math.sin(re)) * (Math.exp(-im_m) - Math.exp(im_m))) <= -1e-13) {
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re);
	} else {
		tmp = (((0.16666666666666666 * (re * re)) - 1.0) * 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 ((0.5 * math.sin(re)) * (math.exp(-im_m) - math.exp(im_m))) <= -1e-13:
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re)
	else:
		tmp = (((0.16666666666666666 * (re * re)) - 1.0) * 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(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im_m)) - exp(im_m))) <= -1e-13)
		tmp = Float64(Float64(Float64(im_m * im_m) * im_m) * Float64(-0.16666666666666666 * re));
	else
		tmp = Float64(Float64(Float64(Float64(0.16666666666666666 * Float64(re * re)) - 1.0) * 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 (((0.5 * sin(re)) * (exp(-im_m) - exp(im_m))) <= -1e-13)
		tmp = ((im_m * im_m) * im_m) * (-0.16666666666666666 * re);
	else
		tmp = (((0.16666666666666666 * (re * re)) - 1.0) * 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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im$95$m)], $MachinePrecision] - N[Exp[im$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-13], N[(N[(N[(im$95$m * im$95$m), $MachinePrecision] * im$95$m), $MachinePrecision] * N[(-0.16666666666666666 * re), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -1e-13

   1. Initial program 66.1%

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

   2. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lift--.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      3. lower-*.f64 N/A

         \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

      4. pow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 41.6

         \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 \]

   10. Applied rewrites 41.6%

       \[\leadsto \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot re\right) \cdot -0.16666666666666666 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. pow3 N/A

          \[\leadsto \left({im}^{3} \cdot re\right) \cdot \frac{-1}{6} \]

       6. associate-*l* N/A

          \[\leadsto {im}^{3} \cdot \left(re \cdot \frac{-1}{6}\right) \]

       7. *-commutative N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       8. lower-*.f64 N/A

          \[\leadsto {im}^{3} \cdot \left(\frac{-1}{6} \cdot re\right) \]

       9. pow3 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 41.6

           \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(-0.16666666666666666 \cdot re\right) \]

   12. Applied rewrites 41.6%

       \[\leadsto \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(-0.16666666666666666 \cdot re\right) \]

   if -1e-13 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (-.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 35.9

         \[\leadsto \left(\left(0.16666666666666666 \cdot \left(re \cdot re\right) - 1\right) \cdot re\right) \cdot im \]

   7. Applied rewrites 35.9%

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

Alternative 11: 34.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot im\_m\right) \cdot 0.16666666666666666\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) -0.01)
    (* (* (* (* re re) im_m) 0.16666666666666666) re)
    (* (- 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 ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re;
	} 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 ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = (((re * re) * im_m) * 0.16666666666666666d0) * re
    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 ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re;
	} 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 (0.5 * math.sin(re)) <= -0.01:
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re
	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(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * re) * im_m) * 0.16666666666666666) * re);
	else
		tmp = Float64(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 ((0.5 * sin(re)) <= -0.01)
		tmp = (((re * re) * im_m) * 0.16666666666666666) * re;
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{6}, \mathsf{neg}\left(im\right)\right) \cdot re \]

      11. lift-neg.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 24.1

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re \]

   10. Applied rewrites 24.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.16666666666666666\right) \cdot re \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 32.1%

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

Alternative 12: 34.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\_m\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\_m\\ \end{array} \end{array} \]

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 (<= (* 0.5 (sin re)) -0.01)
    (* (* (* (* re re) re) im_m) 0.16666666666666666)
    (* (- 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 ((0.5 * sin(re)) <= -0.01) {
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666;
	} 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 ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = (((re * re) * re) * im_m) * 0.16666666666666666d0
    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 ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666;
	} 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 (0.5 * math.sin(re)) <= -0.01:
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666
	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(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(re * re) * re) * im_m) * 0.16666666666666666);
	else
		tmp = Float64(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 ((0.5 * sin(re)) <= -0.01)
		tmp = (((re * re) * re) * im_m) * 0.16666666666666666;
	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[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * im$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[((-re) * im$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0100000000000000002

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot im, \frac{1}{6}, \mathsf{neg}\left(im\right)\right) \cdot re \]

      11. lift-neg.f64 35.9

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

   7. Applied rewrites 35.9%

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

   8. Taylor expanded in re around inf

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

      1. *-commutative N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(im \cdot {re}^{3}\right) \cdot \frac{1}{6} \]

      3. *-commutative N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      4. lower-*.f64 N/A

         \[\leadsto \left({re}^{3} \cdot im\right) \cdot \frac{1}{6} \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 24.1

         \[\leadsto \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\right) \cdot 0.16666666666666666 \]

   10. Applied rewrites 24.1%

       \[\leadsto \left(\left(\left(re \cdot re\right) \cdot re\right) \cdot im\right) \cdot 0.16666666666666666 \]

   if -0.0100000000000000002 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 66.1%

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

   2. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-sin.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \left(-re\right) \cdot im \]
   6. Step-by-step derivation

   7. Applied rewrites 32.1%

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

Alternative 13: 32.1% accurate, 12.7× speedup

Math

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

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(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 66.1%

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

2. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lift-sin.f64 51.0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in re around 0

   \[\leadsto \left(-re\right) \cdot im \]
6. Step-by-step derivation

7. Applied rewrites 32.1%

   \[\leadsto \left(-re\right) \cdot im \]
8. Add Preprocessing

Alternative 14: 14.8% accurate, 15.8× speedup

Math

\[\begin{array}{l} im\_m = \left|im\right| \\ im\_s = \mathsf{copysign}\left(1, im\right) \\ im\_s \cdot \left(0 \cdot im\_m\right) \end{array} \]

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 (* 0.0 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 * (0.0 * 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 * (0.0d0 * 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 * (0.0 * 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 * (0.0 * 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(0.0 * 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 * (0.0 * 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[(0.0 * im$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

2. Taylor expanded in im around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lift-sin.f64 51.0

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

4. Applied rewrites 51.0%

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

   1. lift-neg.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. sin-+PI-rev N/A

      \[\leadsto \sin \left(re + \mathsf{PI}\left(\right)\right) \cdot im \]

   4. sin-sum N/A

      \[\leadsto \left(\sin re \cdot \cos \mathsf{PI}\left(\right) + \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \mathsf{PI}\left(\right), \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   6. lift-sin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \mathsf{PI}\left(\right), \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   7. lower-cos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \mathsf{PI}\left(\right), \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   8. lower-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   10. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   11. lower-sin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \mathsf{PI}\left(\right)\right) \cdot im \]

   12. lower-PI.f64 29.2

       \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \pi\right) \cdot im \]

6. Applied rewrites 29.2%

   \[\leadsto \mathsf{fma}\left(\sin re, \cos \pi, \cos re \cdot \sin \pi\right) \cdot im \]

7. Taylor expanded in re around 0

   \[\leadsto \sin \mathsf{PI}\left(\right) \cdot im \]
8. Step-by-step derivation

   1. sin-PI 14.8

      \[\leadsto 0 \cdot im \]

9. Applied rewrites 14.8%

   \[\leadsto 0 \cdot im \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64
  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))