math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 18

Speedup: 1.0×

• 24.7% 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: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 10^{-141} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0)
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 1e-141) (not (<= t_0 1.0)))))
       (* (- re -1.0) (sin im))
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0) * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if ((t_0 <= -0.02) || !((t_0 <= 1e-141) || !(t_0 <= 1.0))) {
		tmp = (re - -1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 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 <= -0.02) || !((t_0 <= 1e-141) || !(t_0 <= 1.0)))
		tmp = Float64(Float64(re - -1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $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, -0.02], N[Not[Or[LessEqual[t$95$0, 1e-141], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. 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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in re around inf

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

      1. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 67.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 1e-141 < (*.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 98.7

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

   5. Applied rewrites 98.7%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-141 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 10^{-141} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-141} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* (* im im) -0.16666666666666666) im))
     (if (<= t_0 -0.02)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
       (if (or (<= t_0 1e-141) (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 = exp(re) * (((im * im) * -0.16666666666666666) * im);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else if ((t_0 <= 1e-141) || !(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(exp(re) * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	elseif (t_0 <= -0.02)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	elseif ((t_0 <= 1e-141) || !(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[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-141], 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

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]
   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 85.2

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

   5. Applied rewrites 85.2%

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

   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 37.0

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

   8. Applied rewrites 37.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-141 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.3%

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

   if 1e-141 < (*.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 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-141} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-141} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* (* im im) -0.16666666666666666) im))
     (if (<= t_0 -0.02)
       (* (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0) (sin im))
       (if (or (<= t_0 1e-141) (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 = exp(re) * (((im * im) * -0.16666666666666666) * im);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0) * sin(im);
	} else if ((t_0 <= 1e-141) || !(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(exp(re) * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	elseif (t_0 <= -0.02)
		tmp = Float64(fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 1.0) * sin(im));
	elseif ((t_0 <= 1e-141) || !(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[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-141], 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

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]
   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 85.2

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

   5. Applied rewrites 85.2%

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

   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 37.0

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

   8. Applied rewrites 37.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in re around inf

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

      1. lower-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-141 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.3%

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

   if 1e-141 < (*.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 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-141} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-141} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* (* im im) -0.16666666666666666) im))
     (if (<= t_0 -0.02)
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (if (or (<= t_0 1e-141) (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 = exp(re) * (((im * im) * -0.16666666666666666) * im);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if ((t_0 <= 1e-141) || !(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(exp(re) * Float64(Float64(Float64(im * im) * -0.16666666666666666) * im));
	elseif (t_0 <= -0.02)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif ((t_0 <= 1e-141) || !(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[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-141], 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

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + \frac{-1}{6} \cdot {im}^{2}\right)\right)} \]
   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 85.2

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

   5. Applied rewrites 85.2%

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

   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 37.0

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

   8. Applied rewrites 37.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-141 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.3%

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

   if 1e-141 < (*.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 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot -0.16666666666666666\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-141} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-141} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0)
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (<= t_0 -0.02)
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (if (or (<= t_0 1e-141) (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 = fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0) * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if (t_0 <= -0.02) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if ((t_0 <= 1e-141) || !(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(fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 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 <= -0.02)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif ((t_0 <= 1e-141) || !(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[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $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, -0.02], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e-141], 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 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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in re around inf

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

      1. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 67.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-141 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.3%

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

   if 1e-141 < (*.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 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-141} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 10^{-141} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0)
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 1e-141) (not (<= t_0 1.0)))))
       (sin im)
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0) * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if ((t_0 <= -0.02) || !((t_0 <= 1e-141) || !(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(fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 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 <= -0.02) || !((t_0 <= 1e-141) || !(t_0 <= 1.0)))
		tmp = sin(im);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $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, -0.02], N[Not[Or[LessEqual[t$95$0, 1e-141], 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 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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in re around inf

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

      1. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 67.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 1e-141 < (*.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 98.2

         \[\leadsto \sin im \]

   5. Applied rewrites 98.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-141 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Applied rewrites 94.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.02 \lor \neg \left(e^{re} \cdot \sin im \leq 10^{-141} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, im \cdot im, 0.008333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(im \cdot im\right) \cdot \left(\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right) \cdot -0.16666666666666666 - 0.16666666666666666\right)\right) \cdot im\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(im \cdot im\right) - 0.16666666666666666, im \cdot im, 1\right) \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0)
      (*
       (fma
        (-
         (*
          (fma -0.0001984126984126984 (* im im) 0.008333333333333333)
          (* im im))
         0.16666666666666666)
        (* im im)
        1.0)
       im))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 0.0)
         (*
          (*
           (* im im)
           (-
            (* (* (fma 0.5 re 1.0) re) -0.16666666666666666)
            0.16666666666666666))
          im)
         (if (<= t_0 1.0)
           (sin im)
           (*
            (* (* re re) 0.5)
            (*
             (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 tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0) * (fma(((fma(-0.0001984126984126984, (im * im), 0.008333333333333333) * (im * im)) - 0.16666666666666666), (im * im), 1.0) * im);
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) * (((fma(0.5, re, 1.0) * re) * -0.16666666666666666) - 0.16666666666666666)) * im;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = ((re * re) * 0.5) * (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))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 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 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(Float64(Float64(fma(0.5, re, 1.0) * re) * -0.16666666666666666) - 0.16666666666666666)) * im);
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $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, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $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 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 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 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in re around inf

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

      1. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 67.3%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 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 98.5

         \[\leadsto \sin im \]

   5. Applied rewrites 98.5%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 22.1

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

   5. Applied rewrites 22.1%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 20.4%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. +-commutative N/A

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

       8. lower--.f64 N/A

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

   11. Applied rewrites 12.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \sin im \]
   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 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 43.4

         \[\leadsto \left(\left(re \cdot re\right) \cdot 0.5\right) \cdot \sin im \]

   8. Applied rewrites 43.4%

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

   9. Taylor expanded in im around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 50.9

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

   11. Applied rewrites 50.9%

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

Alternative 9: 25.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma 0.5 re 1.0) re)))
   (if (<= (* (exp re) (sin im)) 0.0)
     (* (* (* im im) (- (* t_0 -0.16666666666666666) 0.16666666666666666)) im)
     (fma im 1.0 (* im t_0)))))

C

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

Julia

function code(re, im)
	t_0 = Float64(fma(0.5, re, 1.0) * re)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(Float64(Float64(im * im) * Float64(Float64(t_0 * -0.16666666666666666) - 0.16666666666666666)) * im);
	else
		tmp = fma(im, 1.0, Float64(im * t_0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision]}, If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(im * im), $MachinePrecision] * N[(N[(t$95$0 * -0.16666666666666666), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(im * 1.0 + N[(im * t$95$0), $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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 15.7%

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

   9. Taylor expanded in im around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. +-commutative N/A

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

       8. lower--.f64 N/A

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

   11. Applied rewrites 13.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 43.0%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 54.6

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

   13. Applied rewrites 54.6%

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

Alternative 10: 25.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot \left(\left(im \cdot im\right) \cdot -0.08333333333333333\right)\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(im, 1, im \cdot \left(\mathsf{fma}\left(0.5, re, 1\right) \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (* (* (* re re) (* (* im im) -0.08333333333333333)) im)
   (fma im 1.0 (* im (* (fma 0.5 re 1.0) re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = ((re * re) * ((im * im) * -0.08333333333333333)) * im;
	} else {
		tmp = fma(im, 1.0, (im * (fma(0.5, re, 1.0) * re)));
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(re * re), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(im * 1.0 + N[(im * N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision]), $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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 15.7%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 13.1

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

   11. Applied rewrites 13.1%

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

   12. Taylor expanded in im around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 13.9

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

   14. Applied rewrites 13.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 43.0%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 54.6

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

   13. Applied rewrites 54.6%

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

Alternative 11: 34.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision], N[(im * 1.0 + N[(im * N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision]), $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 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 20.1

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

   8. Applied rewrites 20.1%

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

   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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 43.0%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 54.6

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

   13. Applied rewrites 54.6%

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

Alternative 12: 33.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(im * 1.0 + N[(im * N[(N[(0.5 * re + 1.0), $MachinePrecision] * re), $MachinePrecision]), $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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 15.7%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lift-fma.f64 18.7

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

   11. Applied rewrites 18.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 43.0%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-*.f64 54.6

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

   13. Applied rewrites 54.6%

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

Alternative 13: 33.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 15.7%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lift-fma.f64 18.7

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

   11. Applied rewrites 18.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 54.6

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

   8. Applied rewrites 54.6%

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

Alternative 14: 33.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-9)
   (* (fma (* im im) -0.16666666666666666 1.0) im)
   (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 5e-9) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * im;
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 5e-9)
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * im);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000001e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 29.5%

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

   9. Taylor expanded in re around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lift-fma.f64 32.0

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

   11. Applied rewrites 32.0%

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

   if 5.0000000000000001e-9 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 8.8%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 23.2

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

   11. Applied rewrites 23.2%

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

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 27.6%

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

Alternative 15: 32.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 1.0) {
		tmp = 1.0 * im;
	} else {
		tmp = ((re * re) * 0.5) * 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 * re) * 0.5d0) * 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 * re) * 0.5) * 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 * re) * 0.5) * 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(Float64(Float64(re * re) * 0.5) * 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 * re) * 0.5) * 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[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * 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 71.4%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 26.6%

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

   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 re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 37.7

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 13.5%

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

   9. Taylor expanded in re around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 38.5

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

   11. Applied rewrites 38.5%

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

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 44.9%

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

Alternative 16: 28.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 5.4 \cdot 10^{+40}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 5.4e+40) (* 1.0 im) (* re im)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 5.4e+40) {
		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 (im <= 5.4d+40) 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 (im <= 5.4e+40) {
		tmp = 1.0 * im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 5.4e+40:
		tmp = 1.0 * im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 5.4e+40)
		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 (im <= 5.4e+40)
		tmp = 1.0 * im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 5.4e+40], N[(1.0 * im), $MachinePrecision], N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 5.40000000000000019e40

   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 80.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 29.8%

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

   if 5.40000000000000019e40 < 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 42.4%

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

   6. Taylor expanded in re around 0

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

      1. lower-+.f64 9.1

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

   8. Applied rewrites 9.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 10.3%

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

Alternative 17: 29.8% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + re\right) \cdot im \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (1.0d0 + re) * im
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(N[(1.0 + re), $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 71.9%

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

6. Taylor expanded in re around 0

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

   1. lower-+.f64 25.7

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

8. Applied rewrites 25.7%

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

Alternative 18: 26.8% 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 71.9%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 23.6%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))