math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 19

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \sin im \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-276} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* (* im im) -0.16666666666666666) im))
     (if (or (<= t_0 -0.01) (not (or (<= t_0 5e-276) (not (<= t_0 1.0)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * (((im * im) * -0.16666666666666666) * im);
	} else if ((t_0 <= -0.01) || !((t_0 <= 5e-276) || !(t_0 <= 1.0))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	elseif ((t_0 <= -0.01) || !((t_0 <= 5e-276) || !(t_0 <= 1.0)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.01], N[Not[Or[LessEqual[t$95$0, 5e-276], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 29.6

         \[\leadsto e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right) \]

   8. Applied rewrites 29.6%

      \[\leadsto e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 4.99999999999999967e-276 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999967e-276 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-276} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-276} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma
        (fma
         (fma 0.001388888888888889 (* re re) 0.041666666666666664)
         (* re re)
         0.5)
        (* re re)
        1.0))
      (*
       (fma
        (-
         (*
          (* im im)
          (fma (* -0.0001984126984126984 im) im 0.008333333333333333))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (or (<= t_0 -0.01) (not (or (<= t_0 5e-276) (not (<= t_0 1.0)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0)) * (fma((((im * im) * fma((-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if ((t_0 <= -0.01) || !((t_0 <= 5e-276) || !(t_0 <= 1.0))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)) * Float64(fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif ((t_0 <= -0.01) || !((t_0 <= 5e-276) || !(t_0 <= 1.0)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.0001984126984126984 * im), $MachinePrecision] * im + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.01], N[Not[Or[LessEqual[t$95$0, 5e-276], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 78.6

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 78.6%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{im}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       7. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]

       8. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \color{blue}{im}\right) \]

   13. Applied rewrites 67.1%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 4.99999999999999967e-276 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999967e-276 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-276} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-18} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma
        (fma
         (fma 0.001388888888888889 (* re re) 0.041666666666666664)
         (* re re)
         0.5)
        (* re re)
        1.0))
      (*
       (fma
        (-
         (*
          (* im im)
          (fma (* -0.0001984126984126984 im) im 0.008333333333333333))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (or (<= t_0 -0.01) (not (or (<= t_0 2e-18) (not (<= t_0 1.0)))))
       (* (- re -1.0) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0)) * (fma((((im * im) * fma((-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if ((t_0 <= -0.01) || !((t_0 <= 2e-18) || !(t_0 <= 1.0))) {
		tmp = (re - -1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)) * Float64(fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif ((t_0 <= -0.01) || !((t_0 <= 2e-18) || !(t_0 <= 1.0)))
		tmp = Float64(Float64(re - -1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.0001984126984126984 * im), $MachinePrecision] * im + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.01], N[Not[Or[LessEqual[t$95$0, 2e-18], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 78.6

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 78.6%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{im}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       7. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]

       8. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \color{blue}{im}\right) \]

   13. Applied rewrites 67.1%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 2.0000000000000001e-18 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 97.6

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

   5. Applied rewrites 97.6%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000001e-18 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 2 \cdot 10^{-18} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01 \lor \neg \left(t\_0 \leq 10^{-8} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma
        (fma
         (fma 0.001388888888888889 (* re re) 0.041666666666666664)
         (* re re)
         0.5)
        (* re re)
        1.0))
      (*
       (fma
        (-
         (*
          (* im im)
          (fma (* -0.0001984126984126984 im) im 0.008333333333333333))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (or (<= t_0 -0.01) (not (or (<= t_0 1e-8) (not (<= t_0 1.0)))))
       (sin im)
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0)) * (fma((((im * im) * fma((-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if ((t_0 <= -0.01) || !((t_0 <= 1e-8) || !(t_0 <= 1.0))) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)) * Float64(fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif ((t_0 <= -0.01) || !((t_0 <= 1e-8) || !(t_0 <= 1.0)))
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.0001984126984126984 * im), $MachinePrecision] * im + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.01], N[Not[Or[LessEqual[t$95$0, 1e-8], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[Sin[im], $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 78.6

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 78.6%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{im}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       7. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]

       8. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \color{blue}{im}\right) \]

   13. Applied rewrites 67.1%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 1e-8 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lift-sin.f64 96.5

         \[\leadsto \sin im \]

   5. Applied rewrites 96.5%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-8 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 10^{-8} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma
        (fma
         (fma 0.001388888888888889 (* re re) 0.041666666666666664)
         (* re re)
         0.5)
        (* re re)
        1.0))
      (*
       (fma
        (-
         (*
          (* im im)
          (fma (* -0.0001984126984126984 im) im 0.008333333333333333))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (<= t_0 1.0) (sin im) (* (fma (fma 0.5 re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0)) * (fma((((im * im) * fma((-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)) * Float64(fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.0001984126984126984 * im), $MachinePrecision] * im + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 78.6

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 78.6%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{im}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       7. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]

       8. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \color{blue}{im}\right) \]

   13. Applied rewrites 67.1%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-sin.f64 66.7

         \[\leadsto \sin im \]

   5. Applied rewrites 66.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 46.0

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

   8. Applied rewrites 46.0%

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

Alternative 7: 25.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (*
    (*
     (* im im)
     (- (* (* (fma 0.5 re 1.0) re) -0.16666666666666666) 0.16666666666666666))
    im)
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = ((im * im) * (((fma(0.5, re, 1.0) * re) * -0.16666666666666666) - 0.16666666666666666)) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 25.1%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift--.f64 9.6

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

   11. Applied rewrites 9.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 43.2

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

   8. Applied rewrites 43.2%

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

Alternative 8: 37.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, im \cdot im, 0.5\right) \cdot \left(re \cdot re\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.01)
   (* (* (fma -0.08333333333333333 (* im im) 0.5) (* re re)) im)
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.01) {
		tmp = (fma(-0.08333333333333333, (im * im), 0.5) * (re * re)) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.01)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(im * im), 0.5) * Float64(re * re)) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(-0.08333333333333333 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 9.7%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {im}^{2}\right) \cdot {re}^{2}\right) \cdot im \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {im}^{2}\right) \cdot {re}^{2}\right) \cdot im \]

       3. +-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right) \cdot {re}^{2}\right) \cdot im \]

       4. lower-fma.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 23.3

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

   11. Applied rewrites 23.3%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 39.4

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

   8. Applied rewrites 39.4%

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

Alternative 9: 33.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (fma -0.16666666666666666 (* im im) 1.0) im)
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = fma(-0.16666666666666666, (im * im), 1.0) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-sin.f64 44.9

         \[\leadsto \sin im \]

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 25.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 43.2

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

   8. Applied rewrites 43.2%

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

Alternative 10: 30.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (fma -0.16666666666666666 (* im im) 1.0) im)
   (* (- re -1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = fma(-0.16666666666666666, (im * im), 1.0) * im;
	} else {
		tmp = (re - -1.0) * im;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(re - -1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-sin.f64 44.9

         \[\leadsto \sin im \]

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in im around 0

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

   11. Applied rewrites 25.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 34.6%

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

Alternative 11: 41.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \leq 0.06:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin im) 0.06)
   (*
    (+
     (* (fma (* re re) 0.16666666666666666 1.0) re)
     (fma
      (fma
       (fma 0.001388888888888889 (* re re) 0.041666666666666664)
       (* re re)
       0.5)
      (* re re)
      1.0))
    (*
     (fma
      (-
       (*
        (* im im)
        (fma (* -0.0001984126984126984 im) im 0.008333333333333333))
       0.16666666666666666)
      (* im im)
      1.0)
     im))
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(im) <= 0.06) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0)) * (fma((((im * im) * fma((-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(im) <= 0.06)
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)) * Float64(fma(Float64(Float64(Float64(im * im) * fma(Float64(-0.0001984126984126984 * im), im, 0.008333333333333333)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[im], $MachinePrecision], 0.06], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(im * im), $MachinePrecision] * N[(N[(-0.0001984126984126984 * im), $MachinePrecision] * im + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 0.059999999999999998

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 70.1

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 70.1%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{im}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       3. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       4. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{1}{120} + \frac{-1}{5040} \cdot \left(im \cdot im\right)\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       5. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot {im}^{2} - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot {im}^{2}\right) \cdot im\right) \]

       7. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right)\right) \cdot im\right) \]

       8. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot im\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\left(\left(\frac{-1}{5040} \cdot \left(im \cdot im\right) + \frac{1}{120}\right) \cdot \left(im \cdot im\right) - \frac{1}{6}\right) \cdot \left(im \cdot im\right) + 1\right) \cdot \color{blue}{im}\right) \]

   13. Applied rewrites 49.0%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(im \cdot im\right) \cdot \mathsf{fma}\left(-0.0001984126984126984 \cdot im, im, 0.008333333333333333\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)} \]

   if 0.059999999999999998 < (sin.f64 im)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 13.7

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

   8. Applied rewrites 13.7%

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

Alternative 12: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\\ \mathbf{if}\;re \leq -0.82:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.74:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, 0.16666666666666666\right), re \cdot re, 1\right) \cdot re + t\_0\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{+51}:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + t\_0\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma 0.001388888888888889 (* re re) 0.041666666666666664)
           (* re re)
           0.5)
          (* re re)
          1.0)))
   (if (<= re -0.82)
     (* (exp re) im)
     (if (<= re 0.74)
       (*
        (+
         (*
          (fma
           (fma (* re re) 0.008333333333333333 0.16666666666666666)
           (* re re)
           1.0)
          re)
         t_0)
        (sin im))
       (if (<= re 7.2e+51)
         (* (exp re) (* (fma (* im im) -0.16666666666666666 1.0) im))
         (*
          (+ (* (fma (* re re) 0.16666666666666666 1.0) re) t_0)
          (sin im)))))))

C

double code(double re, double im) {
	double t_0 = fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0);
	double tmp;
	if (re <= -0.82) {
		tmp = exp(re) * im;
	} else if (re <= 0.74) {
		tmp = ((fma(fma((re * re), 0.008333333333333333, 0.16666666666666666), (re * re), 1.0) * re) + t_0) * sin(im);
	} else if (re <= 7.2e+51) {
		tmp = exp(re) * (fma((im * im), -0.16666666666666666, 1.0) * im);
	} else {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + t_0) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)
	tmp = 0.0
	if (re <= -0.82)
		tmp = Float64(exp(re) * im);
	elseif (re <= 0.74)
		tmp = Float64(Float64(Float64(fma(fma(Float64(re * re), 0.008333333333333333, 0.16666666666666666), Float64(re * re), 1.0) * re) + t_0) * sin(im));
	elseif (re <= 7.2e+51)
		tmp = Float64(exp(re) * Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im));
	else
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + t_0) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[re, -0.82], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[LessEqual[re, 0.74], N[(N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision] + t$95$0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.2e+51], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + t$95$0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -0.819999999999999951

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 98.6%

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

   if -0.819999999999999951 < re < 0.73999999999999999

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 99.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 99.8%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   11. Taylor expanded in re around 0

       \[\leadsto \left(\color{blue}{re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {re}^{2}\right)\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {re}^{2}\right)\right) \cdot \color{blue}{re} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(1 + {re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {re}^{2}\right)\right) \cdot \color{blue}{re} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       3. +-commutative N/A

          \[\leadsto \left(\left({re}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {re}^{2}\right) + 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       4. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {re}^{2}\right) \cdot {re}^{2} + 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {re}^{2}, {re}^{2}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       6. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{1}{120} \cdot {re}^{2} + \frac{1}{6}, {re}^{2}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       7. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left({re}^{2} \cdot \frac{1}{120} + \frac{1}{6}, {re}^{2}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left({re}^{2}, \frac{1}{120}, \frac{1}{6}\right), {re}^{2}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       9. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{1}{6}\right), {re}^{2}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       10. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{1}{6}\right), {re}^{2}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       11. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, \frac{1}{120}, \frac{1}{6}\right), re \cdot re, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \sin im \]

       12. lift-*.f64 99.9

           \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, 0.16666666666666666\right), re \cdot re, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \sin im \]

   13. Applied rewrites 99.9%

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(re \cdot re, 0.008333333333333333, 0.16666666666666666\right), re \cdot re, 1\right) \cdot re} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \sin im \]

   if 0.73999999999999999 < re < 7.20000000000000022e51

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if 7.20000000000000022e51 < re

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 100.0

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 100.0%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 41.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \leq 0.06:\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sin im) 0.06)
   (*
    (+
     (* (fma (* re re) 0.16666666666666666 1.0) re)
     (fma
      (fma
       (fma 0.001388888888888889 (* re re) 0.041666666666666664)
       (* re re)
       0.5)
      (* re re)
      1.0))
    (* (fma (* im im) -0.16666666666666666 1.0) im))
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if (sin(im) <= 0.06) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0)) * (fma((im * im), -0.16666666666666666, 1.0) * im);
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (sin(im) <= 0.06)
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)) * Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im));
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sin[im], $MachinePrecision], 0.06], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 im) < 0.059999999999999998

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 70.1

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 70.1%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   11. Taylor expanded in im around 0

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(1 + \frac{-1}{6} \cdot {im}^{2}\right) \cdot \color{blue}{im}\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left(\frac{-1}{6} \cdot {im}^{2} + 1\right) \cdot im\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\left({im}^{2} \cdot \frac{-1}{6} + 1\right) \cdot im\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left({im}^{2}, \frac{-1}{6}, 1\right) \cdot im\right) \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(im \cdot im, \frac{-1}{6}, 1\right) \cdot im\right) \]

       7. lift-*.f64 48.5

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right) \]

   13. Applied rewrites 48.5%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)} \]

   if 0.059999999999999998 < (sin.f64 im)

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 13.7

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

   8. Applied rewrites 13.7%

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

Alternative 14: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.16:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.125 \lor \neg \left(re \leq 7.2 \cdot 10^{+51}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.16)
   (* (exp re) im)
   (if (or (<= re 0.125) (not (<= re 7.2e+51)))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma
        (fma
         (fma 0.001388888888888889 (* re re) 0.041666666666666664)
         (* re re)
         0.5)
        (* re re)
        1.0))
      (sin im))
     (* (exp re) (* (fma (* im im) -0.16666666666666666 1.0) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.16) {
		tmp = exp(re) * im;
	} else if ((re <= 0.125) || !(re <= 7.2e+51)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, (re * re), 0.041666666666666664), (re * re), 0.5), (re * re), 1.0)) * sin(im);
	} else {
		tmp = exp(re) * (fma((im * im), -0.16666666666666666, 1.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.16)
		tmp = Float64(exp(re) * im);
	elseif ((re <= 0.125) || !(re <= 7.2e+51))
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(fma(0.001388888888888889, Float64(re * re), 0.041666666666666664), Float64(re * re), 0.5), Float64(re * re), 1.0)) * sin(im));
	else
		tmp = Float64(exp(re) * Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.16], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[re, 0.125], N[Not[LessEqual[re, 7.2e+51]], $MachinePrecision]], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(N[(0.001388888888888889 * N[(re * re), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.160000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 98.6%

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

   if -0.160000000000000003 < re < 0.125 or 7.20000000000000022e51 < re

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right)\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + {re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right), \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left({re}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}\right) \cdot {re}^{2} + \frac{1}{2}, {re}^{2}, 1\right)\right) \cdot \sin im \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {re}^{2}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      7. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {re}^{2} + \frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, {re}^{2}, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      9. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), {re}^{2}, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      11. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      13. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, re \cdot re, \frac{1}{24}\right), re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      14. lift-*.f64 99.8

          \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 99.8%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   if 0.125 < re < 7.20000000000000022e51

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.16:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.125 \lor \neg \left(re \leq 7.2 \cdot 10^{+51}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, re \cdot re, 0.041666666666666664\right), re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.16:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.125 \lor \neg \left(re \leq 6.3 \cdot 10^{+71}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.16)
   (* (exp re) im)
   (if (or (<= re 0.125) (not (<= re 6.3e+71)))
     (*
      (+
       (* (fma (* re re) 0.16666666666666666 1.0) re)
       (fma (fma 0.041666666666666664 (* re re) 0.5) (* re re) 1.0))
      (sin im))
     (* (exp re) (* (fma (* im im) -0.16666666666666666 1.0) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.16) {
		tmp = exp(re) * im;
	} else if ((re <= 0.125) || !(re <= 6.3e+71)) {
		tmp = ((fma((re * re), 0.16666666666666666, 1.0) * re) + fma(fma(0.041666666666666664, (re * re), 0.5), (re * re), 1.0)) * sin(im);
	} else {
		tmp = exp(re) * (fma((im * im), -0.16666666666666666, 1.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.16)
		tmp = Float64(exp(re) * im);
	elseif ((re <= 0.125) || !(re <= 6.3e+71))
		tmp = Float64(Float64(Float64(fma(Float64(re * re), 0.16666666666666666, 1.0) * re) + fma(fma(0.041666666666666664, Float64(re * re), 0.5), Float64(re * re), 1.0)) * sin(im));
	else
		tmp = Float64(exp(re) * Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.16], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[re, 0.125], N[Not[LessEqual[re, 6.3e+71]], $MachinePrecision]], N[(N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * re), $MachinePrecision] + N[(N[(0.041666666666666664 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.160000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 98.6%

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

   if -0.160000000000000003 < re < 0.125 or 6.3e71 < re

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sinh.f64 N/A

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

      6. lower-cosh.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in re around 0

      \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \color{blue}{\left(1 + {re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {re}^{2}\right)\right)}\right) \cdot \sin im \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left({re}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {re}^{2}\right) + \color{blue}{1}\right)\right) \cdot \sin im \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {re}^{2}\right) \cdot {re}^{2} + 1\right)\right) \cdot \sin im \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {re}^{2}, \color{blue}{{re}^{2}}, 1\right)\right) \cdot \sin im \]

      4. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\frac{1}{24} \cdot {re}^{2} + \frac{1}{2}, {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {re}^{2}, \frac{1}{2}\right), {\color{blue}{re}}^{2}, 1\right)\right) \cdot \sin im \]

      6. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, re \cdot re, \frac{1}{2}\right), {re}^{2}, 1\right)\right) \cdot \sin im \]

      8. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, \frac{1}{6}, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, re \cdot re, \frac{1}{2}\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

      9. lift-*.f64 99.3

         \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, 0.5\right), re \cdot \color{blue}{re}, 1\right)\right) \cdot \sin im \]

   10. Applied rewrites 99.3%

       \[\leadsto \left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, 0.5\right), re \cdot re, 1\right)}\right) \cdot \sin im \]

   if 0.125 < re < 6.3e71

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.16:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.125 \lor \neg \left(re \leq 6.3 \cdot 10^{+71}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(re \cdot re, 0.16666666666666666, 1\right) \cdot re + \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, re \cdot re, 0.5\right), re \cdot re, 1\right)\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.14:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.125 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.14)
   (* (exp re) im)
   (if (or (<= re 0.125) (not (<= re 1.05e+103)))
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
     (* (exp re) (* (fma (* im im) -0.16666666666666666 1.0) im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.14) {
		tmp = exp(re) * im;
	} else if ((re <= 0.125) || !(re <= 1.05e+103)) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * (fma((im * im), -0.16666666666666666, 1.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.14)
		tmp = Float64(exp(re) * im);
	elseif ((re <= 0.125) || !(re <= 1.05e+103))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.14], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], If[Or[LessEqual[re, 0.125], N[Not[LessEqual[re, 1.05e+103]], $MachinePrecision]], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.14000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 98.6%

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

   if -0.14000000000000001 < re < 0.125 or 1.0500000000000001e103 < re

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.125 < re < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.9

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

   5. Applied rewrites 78.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.14:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 0.125 \lor \neg \left(re \leq 1.05 \cdot 10^{+103}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 27.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 7.5e+23) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+23) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 7.5d+23) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 7.5e+23) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 7.5e+23:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 7.5e+23)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 7.5e+23)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 7.5e+23], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 7.49999999999999987e23

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lift-sin.f64 52.4

         \[\leadsto \sin im \]

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 33.7%

      \[\leadsto im \]

   if 7.49999999999999987e23 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 9.7%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 10.6%

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

Alternative 18: 29.4% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \left(re - -1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (- re -1.0) im))

C

double code(double re, double im) {
	return (re - -1.0) * 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 - (-1.0d0)) * im
end function

Java

public static double code(double re, double im) {
	return (re - -1.0) * im;
}

Python

def code(re, im):
	return (re - -1.0) * im

Julia

function code(re, im)
	return Float64(Float64(re - -1.0) * im)
end

MATLAB

function tmp = code(re, im)
	tmp = (re - -1.0) * im;
end

Wolfram

code[re_, im_] := N[(N[(re - -1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower--.f64 N/A

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

   8. metadata-eval 53.5

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

5. Applied rewrites 53.5%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 29.0%

   \[\leadsto \left(re - -1\right) \cdot \color{blue}{im} \]
9. Add Preprocessing

Alternative 19: 26.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 im)

C

double code(double re, double im) {
	return im;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. lift-sin.f64 52.7

      \[\leadsto \sin im \]

5. Applied rewrites 52.7%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 25.4%

   \[\leadsto im \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025082 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))