math.cos on complex, imaginary part

Percentage Accurate: 65.5% → 99.9%

Time: 8.8s

Alternatives: 19

Speedup: 2.5×

• 49.8% 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: 65.5% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. *-commutative N/A

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

   11. lift-sinh.f64 N/A

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

   12. sinh-undef-rev N/A

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

   13. sinh-def N/A

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

   14. lift-sinh.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 58.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (fma
      (- (* -0.016666666666666666 (* im im)) 0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (*
     (fma
      (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
      (* re re)
      0.5)
     re)
    (*
     (fma
      (fma
       (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
       (* im im)
       -0.3333333333333333)
      (* im im)
      -2.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma(((-0.016666666666666666 * (im * im)) - 0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * (fma(fma(((-0.0003968253968253968 * (im * im)) - 0.016666666666666666), (im * im), -0.3333333333333333), (im * im), -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(fma(fma(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666), Float64(im * im), -0.3333333333333333), Float64(im * im), -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $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 55.1%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 85.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

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

   1. Initial program 67.3%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 83.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.2%

       \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right) \cdot im, im, -2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (*
     (fma
      (- (* -0.016666666666666666 (* im im)) 0.3333333333333333)
      (* im im)
      -2.0)
     im))
   (*
    (*
     re
     (*
      (fma
       (*
        (fma
         (fma -0.0003968253968253968 (* im im) -0.016666666666666666)
         (* im im)
         -0.3333333333333333)
        im)
       im
       -2.0)
      im))
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma(((-0.016666666666666666 * (im * im)) - 0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = (re * (fma((fma(fma(-0.0003968253968253968, (im * im), -0.016666666666666666), (im * im), -0.3333333333333333) * im), im, -2.0) * im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(Float64(Float64(-0.016666666666666666 * Float64(im * im)) - 0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(re * Float64(fma(Float64(fma(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666), Float64(im * im), -0.3333333333333333) * im), im, -2.0) * im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 85.8%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot \left(im \cdot im\right) - \frac{1}{3}, im \cdot im, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right) \]

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

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \left(re \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right) \cdot im, im, -2\right) \cdot im\right)}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right) - 0.3333333333333333, im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right) \cdot im, im, -2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right) \cdot im, im, -2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (fma (* im im) -0.3333333333333333 -2.0) im))
   (*
    (*
     re
     (*
      (fma
       (*
        (fma
         (fma -0.0003968253968253968 (* im im) -0.016666666666666666)
         (* im im)
         -0.3333333333333333)
        im)
       im
       -2.0)
      im))
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma((im * im), -0.3333333333333333, -2.0) * im);
	} else {
		tmp = (re * (fma((fma(fma(-0.0003968253968253968, (im * im), -0.016666666666666666), (im * im), -0.3333333333333333) * im), im, -2.0) * im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(Float64(im * im), -0.3333333333333333, -2.0) * im));
	else
		tmp = Float64(Float64(re * Float64(fma(Float64(fma(fma(-0.0003968253968253968, Float64(im * im), -0.016666666666666666), Float64(im * im), -0.3333333333333333) * im), im, -2.0) * im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision] + -0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * im), $MachinePrecision] * im + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 78.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

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

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \left(re \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right) \cdot im, im, -2\right) \cdot im\right)}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968, im \cdot im, -0.016666666666666666\right), im \cdot im, -0.3333333333333333\right) \cdot im, im, -2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-34}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im, im, -im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-34)
   (* (sinh (- im)) re)
   (*
    (fma
     (*
      (*
       (-
        (*
         (* (- (* -0.0001984126984126984 (* im im)) 0.008333333333333333) im)
         im)
        0.16666666666666666)
       im)
      im)
     im
     (- im))
    (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-34) {
		tmp = sinh(-im) * re;
	} else {
		tmp = fma((((((((-0.0001984126984126984 * (im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im), im, -im) * sin(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-34)
		tmp = Float64(sinh(Float64(-im)) * re);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.0001984126984126984 * Float64(im * im)) - 0.008333333333333333) * im) * im) - 0.16666666666666666) * im) * im), im, Float64(-im)) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-34], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.008333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im + (-im)), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.99999999999999928e-35

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 68.0%

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

   if 9.99999999999999928e-35 < re

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \cdot \sin re \]
   8. Step-by-step derivation

   9. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666, im \cdot im, -1\right) \cdot im\right)} \cdot \sin re \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im, \color{blue}{im}, -1 \cdot im\right) \cdot \sin re \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 10^{-34}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\left(-0.0001984126984126984 \cdot \left(im \cdot im\right) - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot im\right) \cdot im, im, -im\right) \cdot \sin re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-34}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), -1\right) \cdot im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-34)
   (* (sinh (- im)) re)
   (*
    (*
     (fma
      (* im im)
      (fma
       (fma (* im im) -0.0001984126984126984 -0.008333333333333333)
       (* im im)
       -0.16666666666666666)
      -1.0)
     im)
    (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-34) {
		tmp = sinh(-im) * re;
	} else {
		tmp = (fma((im * im), fma(fma((im * im), -0.0001984126984126984, -0.008333333333333333), (im * im), -0.16666666666666666), -1.0) * im) * sin(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-34)
		tmp = Float64(sinh(Float64(-im)) * re);
	else
		tmp = Float64(Float64(fma(Float64(im * im), fma(fma(Float64(im * im), -0.0001984126984126984, -0.008333333333333333), Float64(im * im), -0.16666666666666666), -1.0) * im) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-34], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.0001984126984126984 + -0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.99999999999999928e-35

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 68.0%

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

   if 9.99999999999999928e-35 < re

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \cdot \sin re \]
   8. Step-by-step derivation

   9. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666, im \cdot im, -1\right) \cdot im\right)} \cdot \sin re \]

   10. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \cdot \sin re \]

   12. Applied rewrites 94.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.0001984126984126984, -0.008333333333333333\right), im \cdot im, -0.16666666666666666\right), -1\right) \cdot im\right)} \cdot \sin re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (fma (* im im) -0.3333333333333333 -2.0) im))
   (*
    (* re (* (fma (* -0.016666666666666666 (* im im)) (* im im) -2.0) im))
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma((im * im), -0.3333333333333333, -2.0) * im);
	} else {
		tmp = (re * (fma((-0.016666666666666666 * (im * im)), (im * im), -2.0) * im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(Float64(im * im), -0.3333333333333333, -2.0) * im));
	else
		tmp = Float64(Float64(re * Float64(fma(Float64(-0.016666666666666666 * Float64(im * im)), Float64(im * im), -2.0) * im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 78.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 25.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

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

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.5%

       \[\leadsto \left(re \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)}\right) \cdot 0.5 \]

   11. Taylor expanded in im around inf

       \[\leadsto \left(re \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot {im}^{2}, im \cdot im, -2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.5%

       \[\leadsto \left(re \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (fma (* (* im re) re) 0.16666666666666666 (- im)) re)
   (*
    (* re (* (fma (* -0.016666666666666666 (* im im)) (* im im) -2.0) im))
    0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((im * re) * re), 0.16666666666666666, -im) * re;
	} else {
		tmp = (re * (fma((-0.016666666666666666 * (im * im)), (im * im), -2.0) * im)) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(im * re) * re), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(re * Float64(fma(Float64(-0.016666666666666666 * Float64(im * im)), Float64(im * im), -2.0) * im)) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(re * N[(N[(N[(-0.016666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.8%

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

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

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in im around 0

      \[\leadsto \left(re \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{60} \cdot {im}^{2} - \frac{1}{3}\right) - 2\right)\right)}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.5%

       \[\leadsto \left(re \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)}\right) \cdot 0.5 \]

   11. Taylor expanded in im around inf

       \[\leadsto \left(re \cdot \left(\mathsf{fma}\left(\frac{-1}{60} \cdot {im}^{2}, im \cdot im, -2\right) \cdot im\right)\right) \cdot \frac{1}{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.5%

       \[\leadsto \left(re \cdot \left(\mathsf{fma}\left(-0.016666666666666666 \cdot \left(im \cdot im\right), im \cdot im, -2\right) \cdot im\right)\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 95.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+103} \lor \neg \left(im \leq -0.235 \lor \neg \left(im \leq 0.075 \lor \neg \left(im \leq 1.05 \cdot 10^{+103}\right)\right)\right):\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -3.2e+103)
         (not
          (or (<= im -0.235)
              (not (or (<= im 0.075) (not (<= im 1.05e+103)))))))
   (* (* (fma (* im im) -0.16666666666666666 -1.0) im) (sin re))
   (* (sinh (- im)) re)))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -3.2e+103) || !((im <= -0.235) || !((im <= 0.075) || !(im <= 1.05e+103)))) {
		tmp = (fma((im * im), -0.16666666666666666, -1.0) * im) * sin(re);
	} else {
		tmp = sinh(-im) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -3.2e+103) || !((im <= -0.235) || !((im <= 0.075) || !(im <= 1.05e+103))))
		tmp = Float64(Float64(fma(Float64(im * im), -0.16666666666666666, -1.0) * im) * sin(re));
	else
		tmp = Float64(sinh(Float64(-im)) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -3.2e+103], N[Not[Or[LessEqual[im, -0.235], N[Not[Or[LessEqual[im, 0.075], N[Not[LessEqual[im, 1.05e+103]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + -1.0), $MachinePrecision] * im), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -3.19999999999999993e103 or -0.23499999999999999 < im < 0.0749999999999999972 or 1.0500000000000001e103 < im

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in im around 0

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

   9. Applied rewrites 99.6%

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

   if -3.19999999999999993e103 < im < -0.23499999999999999 or 0.0749999999999999972 < im < 1.0500000000000001e103

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 78.7%

      \[\leadsto \sinh \left(-im\right) \cdot \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3.2 \cdot 10^{+103} \lor \neg \left(im \leq -0.235 \lor \neg \left(im \leq 0.075 \lor \neg \left(im \leq 1.05 \cdot 10^{+103}\right)\right)\right):\\ \;\;\;\;\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, -1\right) \cdot im\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-34}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right) \cdot \sin re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-34)
   (* (sinh (- im)) re)
   (*
    (*
     (fma
      (fma (* im im) -0.008333333333333333 -0.16666666666666666)
      (* im im)
      -1.0)
     im)
    (sin re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-34) {
		tmp = sinh(-im) * re;
	} else {
		tmp = (fma(fma((im * im), -0.008333333333333333, -0.16666666666666666), (im * im), -1.0) * im) * sin(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-34)
		tmp = Float64(sinh(Float64(-im)) * re);
	else
		tmp = Float64(Float64(fma(fma(Float64(im * im), -0.008333333333333333, -0.16666666666666666), Float64(im * im), -1.0) * im) * sin(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-34], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -1.0), $MachinePrecision] * im), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.99999999999999928e-35

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 68.0%

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

   if 9.99999999999999928e-35 < re

   1. Initial program 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{5040} \cdot {im}^{2} - \frac{1}{120}\right) - \frac{1}{6}\right) - 1\right)\right)} \cdot \sin re \]
   8. Step-by-step derivation

   9. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\left(\left(im \cdot im\right) \cdot -0.0001984126984126984 - 0.008333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666, im \cdot im, -1\right) \cdot im\right)} \cdot \sin re \]

   10. Taylor expanded in im around 0

       \[\leadsto \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left(\frac{-1}{120} \cdot {im}^{2} - \frac{1}{6}\right) - 1\right)\right)} \cdot \sin re \]
   11. Step-by-step derivation

   12. Applied rewrites 90.6%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.008333333333333333, -0.16666666666666666\right), im \cdot im, -1\right) \cdot im\right)} \cdot \sin re \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-34}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-34)
   (* (sinh (- im)) re)
   (*
    (*
     (sin re)
     (fma
      (* im im)
      (fma -0.008333333333333333 (* im im) -0.16666666666666666)
      -1.0))
    im)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-34) {
		tmp = sinh(-im) * re;
	} else {
		tmp = (sin(re) * fma((im * im), fma(-0.008333333333333333, (im * im), -0.16666666666666666), -1.0)) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-34)
		tmp = Float64(sinh(Float64(-im)) * re);
	else
		tmp = Float64(Float64(sin(re) * fma(Float64(im * im), fma(-0.008333333333333333, Float64(im * im), -0.16666666666666666), -1.0)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-34], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(-0.008333333333333333 * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.99999999999999928e-35

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 68.0%

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

   if 9.99999999999999928e-35 < re

   1. Initial program 55.2%

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

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(-1 \cdot \sin re + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \sin re + \frac{-1}{120} \cdot \left({im}^{2} \cdot \sin re\right)\right)\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\left(\sin re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(-0.008333333333333333, im \cdot im, -0.16666666666666666\right), -1\right)\right) \cdot im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 52.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (fma (* (* im re) re) 0.16666666666666666 (- im)) re)
   (* (* 0.5 re) (* (fma (* im im) -0.3333333333333333 -2.0) im))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((im * re) * re), 0.16666666666666666, -im) * re;
	} else {
		tmp = (0.5 * re) * (fma((im * im), -0.3333333333333333, -2.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(im * re) * re), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(fma(Float64(im * im), -0.3333333333333333, -2.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im), $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 55.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.8%

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

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

   1. Initial program 67.3%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 83.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 60.4%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot re\right) \cdot re, 0.16666666666666666, -im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (fma (* (* im re) re) 0.16666666666666666 (- im)) re)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = fma(((im * re) * re), 0.16666666666666666, -im) * re;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(fma(Float64(Float64(im * re) * re), 0.16666666666666666, Float64(-im)) * re);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(im * re), $MachinePrecision] * re), $MachinePrecision] * 0.16666666666666666 + (-im)), $MachinePrecision] * re), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.8%

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

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

   1. Initial program 67.3%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.9%

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

Alternative 14: 34.6% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (fma 0.16666666666666666 (* re re) -1.0) re) im)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = (fma(0.16666666666666666, (re * re), -1.0) * re) * im;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(re * re), -1.0) * re) * im);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision] + -1.0), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.8%

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

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

   1. Initial program 67.3%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.9%

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

Alternative 15: 34.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.01:\\ \;\;\;\;\left(\left(0.16666666666666666 \cdot \left(re \cdot re\right)\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(-re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.01)
   (* (* (* 0.16666666666666666 (* re re)) re) im)
   (* (- re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.01) {
		tmp = ((0.16666666666666666 * (re * re)) * re) * im;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((0.5d0 * sin(re)) <= (-0.01d0)) then
        tmp = ((0.16666666666666666d0 * (re * re)) * re) * im
    else
        tmp = -re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= -0.01) {
		tmp = ((0.16666666666666666 * (re * re)) * re) * im;
	} else {
		tmp = -re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= -0.01:
		tmp = ((0.16666666666666666 * (re * re)) * re) * im
	else:
		tmp = -re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.01)
		tmp = Float64(Float64(Float64(0.16666666666666666 * Float64(re * re)) * re) * im);
	else
		tmp = Float64(Float64(-re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= -0.01)
		tmp = ((0.16666666666666666 * (re * re)) * re) * im;
	else
		tmp = -re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[((-re) * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 23.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 23.1%

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

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

   1. Initial program 67.3%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.9%

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

Alternative 16: 76.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.3 \cdot 10^{-34}:\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\sin re \cdot im\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.3e-34)
   (* (sinh (- im)) re)
   (* (* (sin re) im) (fma (* -0.16666666666666666 im) im -1.0))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.3e-34) {
		tmp = sinh(-im) * re;
	} else {
		tmp = (sin(re) * im) * fma((-0.16666666666666666 * im), im, -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.3e-34)
		tmp = Float64(sinh(Float64(-im)) * re);
	else
		tmp = Float64(Float64(sin(re) * im) * fma(Float64(-0.16666666666666666 * im), im, -1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.3e-34], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], N[(N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision] * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.30000000000000011e-34

   1. Initial program 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 68.0%

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

   if 2.30000000000000011e-34 < re

   1. Initial program 55.2%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 82.5%

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

Alternative 17: 86.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -0.235 \lor \neg \left(im \leq 0.00042\right):\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1e+105)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (fma (* im im) -0.3333333333333333 -2.0) im))
   (if (or (<= im -0.235) (not (<= im 0.00042)))
     (* (sinh (- im)) re)
     (* (- (sin re)) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1e+105) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma((im * im), -0.3333333333333333, -2.0) * im);
	} else if ((im <= -0.235) || !(im <= 0.00042)) {
		tmp = sinh(-im) * re;
	} else {
		tmp = -sin(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1e+105)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(Float64(im * im), -0.3333333333333333, -2.0) * im));
	elseif ((im <= -0.235) || !(im <= 0.00042))
		tmp = Float64(sinh(Float64(-im)) * re);
	else
		tmp = Float64(Float64(-sin(re)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1e+105], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -0.235], N[Not[LessEqual[im, 0.00042]], $MachinePrecision]], N[(N[Sinh[(-im)], $MachinePrecision] * re), $MachinePrecision], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.9999999999999994e104

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

   if -9.9999999999999994e104 < im < -0.23499999999999999 or 4.2000000000000002e-4 < im

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\sin re \cdot \left(2 \cdot \sinh \left(-im\right)\right)\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-sinh.f64 N/A

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

      12. sinh-undef-rev N/A

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

      13. sinh-def N/A

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

      14. lift-sinh.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 78.1%

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

   if -0.23499999999999999 < im < 4.2000000000000002e-4

   1. Initial program 26.4%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(-\sin re\right) \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -0.235 \lor \neg \left(im \leq 0.00042\right):\\ \;\;\;\;\sinh \left(-im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 82.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -500 \lor \neg \left(im \leq 700\right):\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1e+105)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (* (fma (* im im) -0.3333333333333333 -2.0) im))
   (if (or (<= im -500.0) (not (<= im 700.0)))
     (*
      (*
       (fma
        (- (* 0.004166666666666667 (* re re)) 0.08333333333333333)
        (* re re)
        0.5)
       re)
      (*
       (fma
        (fma
         (- (* -0.0003968253968253968 (* im im)) 0.016666666666666666)
         (* im im)
         -0.3333333333333333)
        (* im im)
        -2.0)
       im))
     (* (- (sin re)) im))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1e+105) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * (fma((im * im), -0.3333333333333333, -2.0) * im);
	} else if ((im <= -500.0) || !(im <= 700.0)) {
		tmp = (fma(((0.004166666666666667 * (re * re)) - 0.08333333333333333), (re * re), 0.5) * re) * (fma(fma(((-0.0003968253968253968 * (im * im)) - 0.016666666666666666), (im * im), -0.3333333333333333), (im * im), -2.0) * im);
	} else {
		tmp = -sin(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1e+105)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * Float64(fma(Float64(im * im), -0.3333333333333333, -2.0) * im));
	elseif ((im <= -500.0) || !(im <= 700.0))
		tmp = Float64(Float64(fma(Float64(Float64(0.004166666666666667 * Float64(re * re)) - 0.08333333333333333), Float64(re * re), 0.5) * re) * Float64(fma(fma(Float64(Float64(-0.0003968253968253968 * Float64(im * im)) - 0.016666666666666666), Float64(im * im), -0.3333333333333333), Float64(im * im), -2.0) * im));
	else
		tmp = Float64(Float64(-sin(re)) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1e+105], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.3333333333333333 + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -500.0], N[Not[LessEqual[im, 700.0]], $MachinePrecision]], N[(N[(N[(N[(N[(0.004166666666666667 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0003968253968253968 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.016666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + -2.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[((-N[Sin[re], $MachinePrecision]) * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < -9.9999999999999994e104

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

   if -9.9999999999999994e104 < im < -500 or 700 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 54.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{240} \cdot {re}^{2} - \frac{1}{12}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{3}, -2\right) \cdot im\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{240} \cdot \left(re \cdot re\right) - \frac{1}{12}, re \cdot re, \frac{1}{2}\right) \cdot re\right) \cdot \color{blue}{\left(im \cdot \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{2520} \cdot {im}^{2} - \frac{1}{60}\right) - \frac{1}{3}\right) - 2\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.1%

       \[\leadsto \left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)} \]

   if -500 < im < 700

   1. Initial program 26.4%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(-\sin re\right) \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.3333333333333333, -2\right) \cdot im\right)\\ \mathbf{elif}\;im \leq -500 \lor \neg \left(im \leq 700\right):\\ \;\;\;\;\left(\mathsf{fma}\left(0.004166666666666667 \cdot \left(re \cdot re\right) - 0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0003968253968253968 \cdot \left(im \cdot im\right) - 0.016666666666666666, im \cdot im, -0.3333333333333333\right), im \cdot im, -2\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 33.1% accurate, 39.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(re, im):
	return -re * im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in im around 0

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

5. Applied rewrites 50.6%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 29.2%

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

Developer Target 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(sin(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Sin[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025019 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs im) 1) (- (* (sin re) (+ im (* 1/6 im im im) (* 1/120 im im im im im)))) (* (* 1/2 (sin re)) (- (exp (- im)) (exp im)))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))