math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 16

Speedup: 1.0×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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: 86.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-134} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

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 (<= t_0 -0.01)
       (sin im)
       (if (or (<= t_0 5e-134) (not (<= t_0 1.0)))
         (* (exp re) im)
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (sin 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) {
		tmp = sin(im);
	} else if ((t_0 <= 5e-134) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(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)
		tmp = sin(im);
	elseif ((t_0 <= 5e-134) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(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[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-134], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 55.6

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

   5. Applied rewrites 55.6%

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

   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 19.4

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

   8. Applied rewrites 19.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002

   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 97.7

         \[\leadsto \sin im \]

   5. Applied rewrites 97.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000003e-134 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 96.7%

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

   if 5.0000000000000003e-134 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-134} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-134} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

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 (<= t_0 -0.01)
       (sin im)
       (if (or (<= t_0 5e-134) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (fma (fma 0.5 re 1.0) re 1.0) (sin 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) {
		tmp = sin(im);
	} else if ((t_0 <= 5e-134) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(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)
		tmp = sin(im);
	elseif ((t_0 <= 5e-134) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(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[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-134], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 55.6

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

   5. Applied rewrites 55.6%

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

   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 19.4

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

   8. Applied rewrites 19.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002

   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 97.7

         \[\leadsto \sin im \]

   5. Applied rewrites 97.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000003e-134 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 96.7%

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

   if 5.0000000000000003e-134 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-134} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-134} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+ re (fma (* re re) (fma 0.16666666666666666 re 0.5) 1.0))
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (<= t_0 -0.01)
       (sin im)
       (if (or (<= t_0 5e-134) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re + fma((re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if (t_0 <= -0.01) {
		tmp = sin(im);
	} else if ((t_0 <= 5e-134) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(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(re + fma(Float64(re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333) * Float64(im * im)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif (t_0 <= -0.01)
		tmp = sin(im);
	elseif ((t_0 <= 5e-134) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(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[(re + N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-134], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 45.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002

   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 97.7

         \[\leadsto \sin im \]

   5. Applied rewrites 97.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000003e-134 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 96.7%

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

   if 5.0000000000000003e-134 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-134} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 1.5 \cdot 10^{-22} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+ re (fma (* re re) (fma 0.16666666666666666 re 0.5) 1.0))
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (<= t_0 -0.01)
       (sin im)
       (if (or (<= t_0 1.5e-22) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (- re -1.0) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (re + fma((re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if (t_0 <= -0.01) {
		tmp = sin(im);
	} else if ((t_0 <= 1.5e-22) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = (re - -1.0) * sin(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(re + fma(Float64(re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333) * Float64(im * im)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif (t_0 <= -0.01)
		tmp = sin(im);
	elseif ((t_0 <= 1.5e-22) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(Float64(re - -1.0) * sin(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[(re + N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 1.5e-22], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 45.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002

   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 97.7

         \[\leadsto \sin im \]

   5. Applied rewrites 97.7%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.5e-22 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 96.8%

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

   if 1.5e-22 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1.5 \cdot 10^{-22} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01 \lor \neg \left(t\_0 \leq 1.5 \cdot 10^{-22} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (+ re (fma (* re re) (fma 0.16666666666666666 re 0.5) 1.0))
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (or (<= t_0 -0.01) (not (or (<= t_0 1.5e-22) (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 = (re + fma((re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if ((t_0 <= -0.01) || !((t_0 <= 1.5e-22) || !(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(re + fma(Float64(re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333) * Float64(im * im)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif ((t_0 <= -0.01) || !((t_0 <= 1.5e-22) || !(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[(re + N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $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, 1.5e-22], 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. Taylor expanded in im around 0

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 45.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 1.5e-22 < (*.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 98.8

         \[\leadsto \sin im \]

   5. Applied rewrites 98.8%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.5e-22 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 96.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.01 \lor \neg \left(e^{re} \cdot \sin im \leq 1.5 \cdot 10^{-22} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;1 \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (+ re (fma (* re re) (fma 0.16666666666666666 re 0.5) 1.0))))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (<= t_0 -0.01)
       (sin im)
       (if (<= t_0 0.0)
         (* 1.0 (* (* (* im im) -0.16666666666666666) im))
         (if (<= t_0 1.0)
           (sin im)
           (*
            t_1
            (*
             (fma
              (- (* 0.008333333333333333 (* im im)) 0.16666666666666666)
              (* im im)
              1.0)
             im))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = re + fma((re * re), fma(0.16666666666666666, re, 0.5), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if (t_0 <= -0.01) {
		tmp = sin(im);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 * (((im * im) * -0.16666666666666666) * im);
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1 * (fma(((0.008333333333333333 * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(re + fma(Float64(re * re), fma(0.16666666666666666, re, 0.5), 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(im * im), 0.008333333333333333) * Float64(im * im)) - 0.16666666666666666), Float64(im * im), 1.0) * im));
	elseif (t_0 <= -0.01)
		tmp = sin(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(t_1 * Float64(fma(Float64(Float64(0.008333333333333333 * Float64(im * im)) - 0.16666666666666666), Float64(im * im), 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]}, Block[{t$95$1 = N[(re + N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(im * im), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   13. Applied rewrites 45.1%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 0.0 < (*.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 96.6

         \[\leadsto \sin im \]

   5. Applied rewrites 96.6%

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

   if -0.0100000000000000002 < (*.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 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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.6%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 20.2

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

   11. Applied rewrites 20.2%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 63.8

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

   8. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 63.8%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 66.3

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

   13. Applied rewrites 66.3%

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

Alternative 8: 31.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;1 \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re + \mathsf{fma}\left(re \cdot re, \mathsf{fma}\left(0.16666666666666666, re, 0.5\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* 1.0 (* (* (* im im) -0.16666666666666666) im))
   (*
    (+ re (fma (* re re) (fma 0.16666666666666666 re 0.5) 1.0))
    (*
     (fma
      (- (* 0.008333333333333333 (* im im)) 0.16666666666666666)
      (* im im)
      1.0)
     im))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = 1.0 * (((im * im) * -0.16666666666666666) * im);
	} else {
		tmp = (re + fma((re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * (fma(((0.008333333333333333 * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	else
		tmp = Float64(Float64(re + fma(Float64(re * re), fma(0.16666666666666666, re, 0.5), 1.0)) * Float64(fma(Float64(Float64(0.008333333333333333 * Float64(im * im)) - 0.16666666666666666), Float64(im * im), 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[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(re + N[(N[(re * re), $MachinePrecision] * N[(0.16666666666666666 * re + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.008333333333333333 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 14.8

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

   11. Applied rewrites 14.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.2

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

   8. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot im \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. associate-+l+ N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 56.2%

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

   11. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 57.6

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

   13. Applied rewrites 57.6%

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

Alternative 9: 31.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* 1.0 (* (* (* im im) -0.16666666666666666) im))
   (* (fma (fma (fma 0.16666666666666666 re 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 = 1.0 * (((im * im) * -0.16666666666666666) * im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 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(1.0 * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 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[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * 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 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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 14.8

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

   11. Applied rewrites 14.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.2

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

   8. Applied rewrites 56.2%

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

Alternative 10: 31.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = 1.0 * (((im * im) * -0.16666666666666666) * im);
	} else {
		tmp = fma(fma((0.16666666666666666 * re), 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(1.0 * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	else
		tmp = Float64(fma(fma(Float64(0.16666666666666666 * re), 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[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * 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 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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 14.8

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

   11. Applied rewrites 14.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.2

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 55.7

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

   11. Applied rewrites 55.7%

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

Alternative 11: 31.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(1.0 * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	else
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), 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[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $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 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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 14.8

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

   11. Applied rewrites 14.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.2

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in re around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 54.7

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

   11. Applied rewrites 54.7%

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

Alternative 12: 30.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* 1.0 (* (* (* im im) -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 = 1.0 * (((im * im) * -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(1.0 * Float64(Float64(Float64(im * im) * -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[(1.0 * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $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 (*.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 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 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 14.8

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

   11. Applied rewrites 14.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 56.2

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

   8. Applied rewrites 56.2%

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

   9. Taylor expanded in re around 0

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

   11. Applied rewrites 49.8%

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

Alternative 13: 28.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 1.0) (* 1.0 im) (* re im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 1.0) {
		tmp = 1.0 * 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 ((exp(re) * sin(im)) <= 1.0d0) then
        tmp = 1.0d0 * im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 1.0:
		tmp = 1.0 * im
	else:
		tmp = re * im
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 1.0)
		tmp = 1.0 * im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.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 im around 0

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 31.3%

      \[\leadsto \color{blue}{1} \cdot im \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 63.8

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

   8. Applied rewrites 63.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

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

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. lower--.f64 27.7

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

   11. Applied rewrites 27.7%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 27.7%

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

Alternative 14: 37.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (fma (fma 0.5 re 1.0) re 1.0) im))

C

double code(double re, double im) {
	return fma(fma(0.5, re, 1.0), re, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

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 72.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 45.0

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

8. Applied rewrites 45.0%

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

9. Taylor expanded in re around 0

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

11. Applied rewrites 41.5%

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

Alternative 15: 30.2% 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 im around 0

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

5. Applied rewrites 72.4%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 45.0

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

8. Applied rewrites 45.0%

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

9. Taylor expanded in re around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

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

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. lower--.f64 35.1

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

11. Applied rewrites 35.1%

    \[\leadsto \color{blue}{\left(re - -1\right)} \cdot im \]
12. Add Preprocessing

Alternative 16: 26.9% accurate, 34.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 1.0 im))

C

double code(double re, double im) {
	return 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 = 1.0d0 * im
end function

Java

public static double code(double re, double im) {
	return 1.0 * im;
}

Python

def code(re, im):
	return 1.0 * im

Julia

function code(re, im)
	return Float64(1.0 * im)
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 * im;
end

Wolfram

code[re_, im_] := N[(1.0 * im), $MachinePrecision]

Derivation

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 72.4%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 27.5%

   \[\leadsto \color{blue}{1} \cdot im \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025080 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))