math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 20

Speedup: 1.5×

• Could not converge on a ground truth

  1. re = 3.675550106996214e-103
  2. im = 31288.949553310547

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im}\\ \left(0.5 \cdot \sin re\right) \cdot \left(t\_0 + \frac{1}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (exp (- im)))) (* (* 0.5 (sin re)) (+ t_0 (/ 1.0 t_0)))))

C

double code(double re, double im) {
	double t_0 = exp(-im);
	return (0.5 * sin(re)) * (t_0 + (1.0 / t_0));
}

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) :: t_0
    t_0 = exp(-im)
    code = (0.5d0 * sin(re)) * (t_0 + (1.0d0 / t_0))
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im);
	return (0.5 * Math.sin(re)) * (t_0 + (1.0 / t_0));
}

Python

def code(re, im):
	t_0 = math.exp(-im)
	return (0.5 * math.sin(re)) * (t_0 + (1.0 / t_0))

Julia

function code(re, im)
	t_0 = exp(Float64(-im))
	return Float64(Float64(0.5 * sin(re)) * Float64(t_0 + Float64(1.0 / t_0)))
end

MATLAB

function tmp = code(re, im)
	t_0 = exp(-im);
	tmp = (0.5 * sin(re)) * (t_0 + (1.0 / t_0));
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Exp[(-im)], $MachinePrecision]}, N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

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

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

   3. flip-+ N/A

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

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

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

   5. sub0-neg N/A

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

   6. lift--.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cosh.f64 N/A

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

   12. lower-cosh.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sinh.f64 N/A

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

   15. lower-sinh.f64 55.8

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

   16. lift--.f64 N/A

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

   17. sub0-neg N/A

       \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

   18. lower-neg.f64 55.8

       \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

4. Applied rewrites 55.8%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]
5. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cosh.f64 N/A

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

   4. lift-cosh.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sinh.f64 N/A

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

   7. lift-sinh.f64 N/A

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

   8. sinh-cosh 100.0

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\color{blue}{1}}{e^{-im}}\right) \]

6. Applied rewrites 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\color{blue}{1}}{e^{-im}}\right) \]

7. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + \frac{1}{e^{-im}}\right) \]
8. Add Preprocessing

Alternative 2: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (*
      (* (* 2.0 (cosh im)) (* (fma -0.16666666666666666 (* re re) 1.0) re))
      0.5)
     (if (<= t_0 1.0)
       (* (fma (* 0.5 im) im 1.0) (sin re))
       (*
        (* 0.5 re)
        (fma
         (fma
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          (* im im)
          1.0)
         (* im im)
         2.0))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((2.0 * cosh(im)) * (fma(-0.16666666666666666, (re * re), 1.0) * re)) * 0.5;
	} else if (t_0 <= 1.0) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(2.0 * cosh(im)) * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)) * 0.5);
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \sin re\right)} \cdot 0.5 \]

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub0-neg N/A

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

      14. cosh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 67.2

         \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{re \cdot re}, 1\right) \cdot re\right)\right) \cdot 0.5 \]

   7. Applied rewrites 67.2%

      \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot 0.5 \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 77.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 77.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 67.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))))
        (t_1
         (fma
          (fma
           (fma 0.002777777777777778 (* im im) 0.08333333333333333)
           (* im im)
           1.0)
          (* im im)
          2.0)))
   (if (<= t_0 (- INFINITY))
     (* (* (fma (* re re) -0.08333333333333333 0.5) re) t_1)
     (if (<= t_0 1.0)
       (* (fma (* 0.5 im) im 1.0) (sin re))
       (* (* 0.5 re) t_1)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double t_1 = fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else {
		tmp = (0.5 * re) * t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * t_1);
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * t_1);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 85.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 85.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-*.f64 62.3

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 62.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 77.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 77.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 67.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))))
        (t_1
         (fma
          (fma
           (fma 0.002777777777777778 (* im im) 0.08333333333333333)
           (* im im)
           1.0)
          (* im im)
          2.0)))
   (if (<= t_0 (- INFINITY))
     (* (* (fma (* re re) -0.08333333333333333 0.5) re) t_1)
     (if (<= t_0 1.0) (* 1.0 (sin re)) (* (* 0.5 re) t_1)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double t_1 = fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * t_1;
	} else if (t_0 <= 1.0) {
		tmp = 1.0 * sin(re);
	} else {
		tmp = (0.5 * re) * t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	t_1 = fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * t_1);
	elseif (t_0 <= 1.0)
		tmp = Float64(1.0 * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * t_1);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(1.0 * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 85.3

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 85.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-*.f64 62.3

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 62.3%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in im around 0

      \[\leadsto 1 \cdot \sin \color{blue}{re} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.1%

      \[\leadsto 1 \cdot \sin \color{blue}{re} \]

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 77.4

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 77.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 67.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, -0.08333333333333333, 0.5 \cdot im\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (* (* (fma (* (* re im) re) -0.08333333333333333 (* 0.5 im)) re) im)
   (fma (* (* 0.5 im) im) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = (fma(((re * im) * re), -0.08333333333333333, (0.5 * im)) * re) * im;
	} else {
		tmp = fma(((0.5 * im) * im), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(fma(Float64(Float64(re * im) * re), -0.08333333333333333, Float64(0.5 * im)) * re) * im);
	else
		tmp = fma(Float64(Float64(0.5 * im) * im), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * -0.08333333333333333 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 20.8%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 20.4%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.4%

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

   10. Applied rewrites 64.4%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, -0.08333333333333333, 0.5 \cdot im\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (* (* (* (fma (* re re) -0.08333333333333333 0.5) im) im) re)
   (fma (* (* 0.5 im) im) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = ((fma((re * re), -0.08333333333333333, 0.5) * im) * im) * re;
	} else {
		tmp = fma(((0.5 * im) * im), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * im) * im) * re);
	else
		tmp = fma(Float64(Float64(0.5 * im) * im), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 30.0%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.4%

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

   10. Applied rewrites 64.4%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot im\right) \cdot im\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (*
    (* (* (fma (* im im) -0.08333333333333333 -0.16666666666666666) re) re)
    re)
   (fma (* (* 0.5 im) im) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = ((fma((im * im), -0.08333333333333333, -0.16666666666666666) * re) * re) * re;
	} else {
		tmp = fma(((0.5 * im) * im), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(Float64(fma(Float64(im * im), -0.08333333333333333, -0.16666666666666666) * re) * re) * re);
	else
		tmp = fma(Float64(Float64(0.5 * im) * im), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + -0.16666666666666666), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{12} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 18.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re\right) \cdot re\right) \cdot re \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.4%

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

   10. Applied rewrites 64.4%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\left(\left(im \cdot im\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (* (* (* (* (* im im) -0.08333333333333333) re) re) re)
   (fma (* (* 0.5 im) im) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = ((((im * im) * -0.08333333333333333) * re) * re) * re;
	} else {
		tmp = fma(((0.5 * im) * im), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(Float64(Float64(Float64(im * im) * -0.08333333333333333) * re) * re) * re);
	else
		tmp = fma(Float64(Float64(0.5 * im) * im), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{12} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 18.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re\right) \cdot re\right) \cdot re \]

   12. Taylor expanded in im around inf

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

   14. Applied rewrites 18.4%

       \[\leadsto \left(\left(\left(\left(im \cdot im\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot re\right) \cdot re \]

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.4%

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

   10. Applied rewrites 64.4%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(\left(\left(im \cdot im\right) \cdot -0.08333333333333333\right) \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (* (* (* -0.16666666666666666 re) re) re)
   (fma (* (* 0.5 im) im) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = ((-0.16666666666666666 * re) * re) * re;
	} else {
		tmp = fma(((0.5 * im) * im), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * re) * re) * re);
	else
		tmp = fma(Float64(Float64(0.5 * im) * im), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(-0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{12} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 18.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re\right) \cdot re\right) \cdot re \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 13.1%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.4%

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

   10. Applied rewrites 64.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 \cdot im\right) \cdot im, re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.1)
   (* (* (* -0.16666666666666666 re) re) re)
   (* (fma (* im im) 0.5 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.1) {
		tmp = ((-0.16666666666666666 * re) * re) * re;
	} else {
		tmp = fma((im * im), 0.5, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.1)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * re) * re) * re);
	else
		tmp = Float64(fma(Float64(im * im), 0.5, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(-0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 11.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{12} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 18.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re\right) \cdot re\right) \cdot re \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 13.1%

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

   if -0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 64.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.1:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 50.4%

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

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 38.0%

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

4. Final simplification 46.6%

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

Alternative 12: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(2 \cdot \cosh im\right) \cdot \sin re\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* (* 2.0 (cosh im)) (sin re)) 0.5))

C

double code(double re, double im) {
	return ((2.0 * cosh(im)) * sin(re)) * 0.5;
}

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 = ((2.0d0 * cosh(im)) * sin(re)) * 0.5d0
end function

Java

public static double code(double re, double im) {
	return ((2.0 * Math.cosh(im)) * Math.sin(re)) * 0.5;
}

Python

def code(re, im):
	return ((2.0 * math.cosh(im)) * math.sin(re)) * 0.5

Julia

function code(re, im)
	return Float64(Float64(Float64(2.0 * cosh(im)) * sin(re)) * 0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = ((2.0 * cosh(im)) * sin(re)) * 0.5;
end

Wolfram

code[re_, im_] := N[(N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 100.0

      \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \sin re\right)} \cdot 0.5 \]

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub0-neg N/A

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

   14. cosh-undef N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cosh.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \cosh im\right) \cdot \sin re\right) \cdot 0.5} \]
5. Add Preprocessing

Alternative 13: 57.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-5)
   (*
    (* (fma (* re re) -0.08333333333333333 0.5) re)
    (fma
     (fma
      (fma 0.002777777777777778 (* im im) 0.08333333333333333)
      (* im im)
      1.0)
     (* im im)
     2.0))
   (*
    (fma (* 0.5 im) im 1.0)
    (*
     (fma
      (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
      (* re re)
      1.0)
     re))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 1e-5) {
		tmp = (fma((re * re), -0.08333333333333333, 0.5) * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	} else {
		tmp = fma((0.5 * im), im, 1.0) * (fma(((0.008333333333333333 * (re * re)) - 0.16666666666666666), (re * re), 1.0) * re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 1e-5)
		tmp = Float64(Float64(fma(Float64(re * re), -0.08333333333333333, 0.5) * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	else
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * Float64(fma(Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), Float64(re * re), 1.0) * re));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-5], N[(N[(N[(N[(re * re), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[(N[(N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1.00000000000000008e-5

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 55.6

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 55.6

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 55.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 91.2

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 91.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{{re}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

      7. lower-*.f64 69.6

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{re \cdot re}, -0.08333333333333333, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 69.6%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   if 1.00000000000000008e-5 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.9%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re \cdot re, 1\right) \cdot \color{blue}{re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 85.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00132:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 7.6 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00132)
   (* (fma (* 0.5 im) im 1.0) (sin re))
   (if (<= im 7.6e+49)
     (*
      (* (* 2.0 (cosh im)) (* (fma -0.16666666666666666 (* re re) 1.0) re))
      0.5)
     (*
      (* 0.5 (sin re))
      (fma
       (fma
        (fma 0.002777777777777778 (* im im) 0.08333333333333333)
        (* im im)
        1.0)
       (* im im)
       2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00132) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else if (im <= 7.6e+49) {
		tmp = ((2.0 * cosh(im)) * (fma(-0.16666666666666666, (re * re), 1.0) * re)) * 0.5;
	} else {
		tmp = (0.5 * sin(re)) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00132)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	elseif (im <= 7.6e+49)
		tmp = Float64(Float64(Float64(2.0 * cosh(im)) * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)) * 0.5);
	else
		tmp = Float64(Float64(0.5 * sin(re)) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00132], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 7.6e+49], N[(N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00132

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 0.00132 < im < 7.5999999999999997e49

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \sin re\right)} \cdot 0.5 \]

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub0-neg N/A

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

      14. cosh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.2

         \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{re \cdot re}, 1\right) \cdot re\right)\right) \cdot 0.5 \]

   7. Applied rewrites 81.2%

      \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot 0.5 \]

   if 7.5999999999999997e49 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 98.1

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 98.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, -0.08333333333333333, 0.5 \cdot im\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.02)
   (* (* (fma (* (* re im) re) -0.08333333333333333 (* 0.5 im)) re) im)
   (*
    (* 0.5 re)
    (fma
     (fma
      (fma 0.002777777777777778 (* im im) 0.08333333333333333)
      (* im im)
      1.0)
     (* im im)
     2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.02) {
		tmp = (fma(((re * im) * re), -0.08333333333333333, (0.5 * im)) * re) * im;
	} else {
		tmp = (0.5 * re) * fma(fma(fma(0.002777777777777778, (im * im), 0.08333333333333333), (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.02)
		tmp = Float64(Float64(fma(Float64(Float64(re * im) * re), -0.08333333333333333, Float64(0.5 * im)) * re) * im);
	else
		tmp = Float64(Float64(0.5 * re) * fma(fma(fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * -0.08333333333333333 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(N[(N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 23.9%

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

   if -0.0200000000000000004 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 59.6

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 59.6

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 59.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right), {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 1}, {im}^{2}, 2\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1, {im}^{2}, 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), {im}^{2}, 1\right), {im}^{2}, 2\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      13. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      14. lower-*.f64 91.7

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 91.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right)} \]

   8. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, im \cdot im, \frac{1}{12}\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 72.2

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]

   10. Applied rewrites 72.2%

       \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), im \cdot im, 1\right), im \cdot im, 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 85.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 0.00132:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot im, im, 1\right) \cdot \sin re\\ \mathbf{elif}\;im \leq 2.75 \cdot 10^{+76}:\\ \;\;\;\;\left(\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), im \cdot im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 0.00132)
   (* (fma (* 0.5 im) im 1.0) (sin re))
   (if (<= im 2.75e+76)
     (*
      (* (* 2.0 (cosh im)) (* (fma -0.16666666666666666 (* re re) 1.0) re))
      0.5)
     (*
      (* 0.5 (sin re))
      (fma (fma 0.08333333333333333 (* im im) 1.0) (* im im) 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 0.00132) {
		tmp = fma((0.5 * im), im, 1.0) * sin(re);
	} else if (im <= 2.75e+76) {
		tmp = ((2.0 * cosh(im)) * (fma(-0.16666666666666666, (re * re), 1.0) * re)) * 0.5;
	} else {
		tmp = (0.5 * sin(re)) * fma(fma(0.08333333333333333, (im * im), 1.0), (im * im), 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 0.00132)
		tmp = Float64(fma(Float64(0.5 * im), im, 1.0) * sin(re));
	elseif (im <= 2.75e+76)
		tmp = Float64(Float64(Float64(2.0 * cosh(im)) * Float64(fma(-0.16666666666666666, Float64(re * re), 1.0) * re)) * 0.5);
	else
		tmp = Float64(Float64(0.5 * sin(re)) * fma(fma(0.08333333333333333, Float64(im * im), 1.0), Float64(im * im), 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[im, 0.00132], N[(N[(N[(0.5 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.75e+76], N[(N[(N[(2.0 * N[Cosh[im], $MachinePrecision]), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(0.08333333333333333 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * im), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 0.00132

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 86.3

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

   5. Applied rewrites 86.3%

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

   if 0.00132 < im < 2.75e76

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

         \[\leadsto \color{blue}{\left(\left(e^{0 - im} + e^{im}\right) \cdot \sin re\right)} \cdot 0.5 \]

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub0-neg N/A

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

      14. cosh-undef N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cosh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.2

         \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \color{blue}{re \cdot re}, 1\right) \cdot re\right)\right) \cdot 0.5 \]

   7. Applied rewrites 77.2%

      \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.16666666666666666, re \cdot re, 1\right) \cdot re\right)}\right) \cdot 0.5 \]

   if 2.75e76 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

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

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

      3. flip-+ N/A

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

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

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{\color{blue}{e^{\mathsf{neg}\left(im\right)}}}\right) \]

      5. sub0-neg N/A

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

      6. lift--.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cosh.f64 N/A

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

      12. lower-cosh.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sinh.f64 N/A

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

      15. lower-sinh.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \color{blue}{\sinh im}}{e^{0 - im}}\right) \]

      16. lift--.f64 N/A

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

      17. sub0-neg N/A

          \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{\mathsf{neg}\left(im\right)}}}\right) \]

      18. lower-neg.f64 0.0

          \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{\color{blue}{-im}}}\right) \]

   4. Applied rewrites 0.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + \color{blue}{\frac{\cosh im \cdot \cosh im - \sinh im \cdot \sinh im}{e^{-im}}}\right) \]

   5. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + \frac{1}{12} \cdot {im}^{2}\right) + 2\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{\left(1 + \frac{1}{12} \cdot {im}^{2}\right) \cdot {im}^{2}} + 2\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {im}^{2}, {im}^{2}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{12} \cdot {im}^{2} + 1}, {im}^{2}, 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12}, {im}^{2}, 1\right)}, {im}^{2}, 2\right) \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, \color{blue}{im \cdot im}, 1\right), {im}^{2}, 2\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

      9. lower-*.f64 95.7

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), \color{blue}{im \cdot im}, 2\right) \]

   7. Applied rewrites 95.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, im \cdot im, 1\right), im \cdot im, 2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 51.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, -0.08333333333333333, 0.5 \cdot im\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) -0.02)
   (* (* (fma (* (* re im) re) -0.08333333333333333 (* 0.5 im)) re) im)
   (* (fma (/ (* (* im im) im) im) 0.5 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= -0.02) {
		tmp = (fma(((re * im) * re), -0.08333333333333333, (0.5 * im)) * re) * im;
	} else {
		tmp = fma((((im * im) * im) / im), 0.5, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= -0.02)
		tmp = Float64(Float64(fma(Float64(Float64(re * im) * re), -0.08333333333333333, Float64(0.5 * im)) * re) * im);
	else
		tmp = Float64(fma(Float64(Float64(Float64(im * im) * im) / im), 0.5, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(N[(N[(re * im), $MachinePrecision] * re), $MachinePrecision] * -0.08333333333333333 + N[(0.5 * im), $MachinePrecision]), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] / im), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 24.4%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 23.9%

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

   if -0.0200000000000000004 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 62.7%

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

   10. Applied rewrites 65.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq -0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(re \cdot im\right) \cdot re, -0.08333333333333333, 0.5 \cdot im\right) \cdot re\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(im \cdot im\right) \cdot im}{im}, 0.5, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 17.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0.5 \cdot \sin re \leq 10^{-300}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\\ \mathbf{else}:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* 0.5 (sin re)) 1e-300)
   (* (* (* -0.16666666666666666 re) re) re)
   (* (* (* im im) 0.5) re)))

C

double code(double re, double im) {
	double tmp;
	if ((0.5 * sin(re)) <= 1e-300) {
		tmp = ((-0.16666666666666666 * re) * re) * re;
	} else {
		tmp = ((im * im) * 0.5) * re;
	}
	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)) <= 1d-300) then
        tmp = (((-0.16666666666666666d0) * re) * re) * re
    else
        tmp = ((im * im) * 0.5d0) * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((0.5 * Math.sin(re)) <= 1e-300) {
		tmp = ((-0.16666666666666666 * re) * re) * re;
	} else {
		tmp = ((im * im) * 0.5) * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (0.5 * math.sin(re)) <= 1e-300:
		tmp = ((-0.16666666666666666 * re) * re) * re
	else:
		tmp = ((im * im) * 0.5) * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(0.5 * sin(re)) <= 1e-300)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * re) * re) * re);
	else
		tmp = Float64(Float64(Float64(im * im) * 0.5) * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((0.5 * sin(re)) <= 1e-300)
		tmp = ((-0.16666666666666666 * re) * re) * re;
	else
		tmp = ((im * im) * 0.5) * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision], 1e-300], N[(N[(N[(-0.16666666666666666 * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision], N[(N[(N[(im * im), $MachinePrecision] * 0.5), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) < 1.00000000000000003e-300

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 36.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re, re, \mathsf{fma}\left(im \cdot im, 0.5, 1\right)\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{2} \cdot \left(\frac{-1}{12} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 14.7%

       \[\leadsto \left(\left(\mathsf{fma}\left(im \cdot im, -0.08333333333333333, -0.16666666666666666\right) \cdot re\right) \cdot re\right) \cdot re \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 11.0%

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

   if 1.00000000000000003e-300 < (*.f64 #s(literal 1/2 binary64) (sin.f64 re))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 54.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 25.0%

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

Alternative 19: 24.7% accurate, 19.8× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (* (* im im) 0.5) re))

C

double code(double re, double im) {
	return ((im * im) * 0.5) * 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 = ((im * im) * 0.5d0) * re
end function

Java

public static double code(double re, double im) {
	return ((im * im) * 0.5) * re;
}

Python

def code(re, im):
	return ((im * im) * 0.5) * re

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 76.3

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

5. Applied rewrites 76.3%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 52.2%

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

9. Taylor expanded in im around inf

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

11. Applied rewrites 23.8%

    \[\leadsto \left(\left(im \cdot im\right) \cdot 0.5\right) \cdot re \]
12. Add Preprocessing

Alternative 20: 19.2% accurate, 19.8× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (* (* (* re im) 0.5) im))

C

double code(double re, double im) {
	return ((re * im) * 0.5) * 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) * 0.5d0) * im
end function

Java

public static double code(double re, double im) {
	return ((re * im) * 0.5) * im;
}

Python

def code(re, im):
	return ((re * im) * 0.5) * im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 76.3

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

5. Applied rewrites 76.3%

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

6. Taylor expanded in im around inf

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

8. Applied rewrites 18.5%

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

9. Taylor expanded in re around 0

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

11. Applied rewrites 18.3%

    \[\leadsto \left(\left(re \cdot im\right) \cdot 0.5\right) \cdot im \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))