math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

Alternatives: 19

Speedup: 1.0×

• 24.9% 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

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

FPCore

(FPCore (re im) :precision binary64 (/ (sin im) (exp (- re))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

      \[\leadsto \color{blue}{\frac{\cosh re \cdot \cosh re - \sinh re \cdot \sinh re}{\cosh re - \sinh re}} \cdot \sin im \]

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

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

   6. associate-*l/ N/A

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

   7. sinh-cosh N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\sin im}{e^{-re}}} \]
5. Add Preprocessing

Alternative 2: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)) (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* im im) -0.16666666666666666 1.0) t_0)
     (if (<= t_1 -0.1)
       (sin im)
       (if (<= t_1 2e-15)
         (/ im (exp (- re)))
         (if (<= t_1 1.0)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (sin im))
           t_0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * t_0;
	} else if (t_1 <= -0.1) {
		tmp = sin(im);
	} else if (t_1 <= 2e-15) {
		tmp = im / exp(-re);
	} else if (t_1 <= 1.0) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * t_0);
	elseif (t_1 <= -0.1)
		tmp = sin(im);
	elseif (t_1 <= 2e-15)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_1 <= 1.0)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$1, 2e-15], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.5%

      \[\leadsto \frac{im \cdot 1}{\color{blue}{e^{-re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.5%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 73.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

      \[\leadsto \frac{im \cdot 1}{\color{blue}{e^{-re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

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

   if 2.0000000000000002e-15 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. remove-double-neg N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, -0.16666666666666666, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)) (t_1 (* (exp re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (* im im) -0.16666666666666666 1.0) t_0)
     (if (<= t_1 -0.1)
       (sin im)
       (if (<= t_1 2e-15)
         (/ im (exp (- re)))
         (if (<= t_1 1.0) (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)) t_0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((im * im), -0.16666666666666666, 1.0) * t_0;
	} else if (t_1 <= -0.1) {
		tmp = sin(im);
	} else if (t_1 <= 2e-15) {
		tmp = im / exp(-re);
	} else if (t_1 <= 1.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(im * im), -0.16666666666666666, 1.0) * t_0);
	elseif (t_1 <= -0.1)
		tmp = sin(im);
	elseif (t_1 <= 2e-15)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_1 <= 1.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(im * im), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$1, 2e-15], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 76.5

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.5%

      \[\leadsto \frac{im \cdot 1}{\color{blue}{e^{-re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.5%

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

   10. Taylor expanded in im around 0

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

   12. Applied rewrites 73.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

      \[\leadsto \frac{im \cdot 1}{\color{blue}{e^{-re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

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

   if 2.0000000000000002e-15 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* (* im im) im) -0.16666666666666666 im))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 2e-15)
         (/ im (exp (- re)))
         (if (<= t_0 1.0)
           (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
           (* (exp re) im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma(((im * im) * im), -0.16666666666666666, im);
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 2e-15) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(im * im) * im), -0.16666666666666666, im));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 2e-15)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-15], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 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

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.9

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 19.9%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

      \[\leadsto \frac{im \cdot 1}{\color{blue}{e^{-re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

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

   if 2.0000000000000002e-15 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* (* im im) im) -0.16666666666666666 im))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 2e-15)
         (/ im (exp (- re)))
         (if (<= t_0 1.0) (/ (sin im) (- 1.0 re)) (* (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 = (1.0 + re) * fma(((im * im) * im), -0.16666666666666666, im);
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 2e-15) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = sin(im) / (1.0 - re);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(im * im) * im), -0.16666666666666666, im));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 2e-15)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im) / Float64(1.0 - re));
	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[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-15], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 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

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.9

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 19.9%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

      \[\leadsto \frac{im \cdot 1}{\color{blue}{e^{-re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

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

   if 2.0000000000000002e-15 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

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

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

      4. flip-+ N/A

         \[\leadsto \color{blue}{\frac{\cosh re \cdot \cosh re - \sinh re \cdot \sinh re}{\cosh re - \sinh re}} \cdot \sin im \]

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

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

      6. associate-*l/ N/A

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

      7. sinh-cosh N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

      \[\leadsto \frac{\sin im}{\color{blue}{1 + -1 \cdot re}} \]
   6. Step-by-step derivation

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

         \[\leadsto \frac{\sin im}{1 - \color{blue}{re}} \]

      4. lower--.f64 100.0

         \[\leadsto \frac{\sin im}{\color{blue}{1 - re}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin im}{\color{blue}{1 - re}} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-15} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* (* im im) im) -0.16666666666666666 im))
     (if (<= t_0 -0.1)
       (sin im)
       (if (or (<= t_0 2e-15) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (+ 1.0 re) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma(((im * im) * im), -0.16666666666666666, im);
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if ((t_0 <= 2e-15) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = (1.0 + re) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(im * im) * im), -0.16666666666666666, im));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif ((t_0 <= 2e-15) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(Float64(1.0 + re) * 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[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 2e-15], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(1.0 + re), $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

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.9

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 19.9%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-15 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if 2.0000000000000002e-15 < (*.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. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-15} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{im}{e^{-re}}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* (* im im) im) -0.16666666666666666 im))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 2e-15)
         (/ im (exp (- re)))
         (if (<= t_0 1.0) (* (+ 1.0 re) (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 = (1.0 + re) * fma(((im * im) * im), -0.16666666666666666, im);
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 2e-15) {
		tmp = im / exp(-re);
	} else if (t_0 <= 1.0) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(im * im) * im), -0.16666666666666666, im));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 2e-15)
		tmp = Float64(im / exp(Float64(-re)));
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(1.0 + re) * 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[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-15], N[(im / N[Exp[(-re)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 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

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.9

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 19.9%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

      \[\leadsto \frac{im \cdot 1}{\color{blue}{e^{-re}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

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

   if 2.0000000000000002e-15 < (*.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. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.8

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 85.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-15} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* (* im im) im) -0.16666666666666666 im))
     (if (or (<= t_0 -0.1) (not (or (<= t_0 2e-15) (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 = (1.0 + re) * fma(((im * im) * im), -0.16666666666666666, im);
	} else if ((t_0 <= -0.1) || !((t_0 <= 2e-15) || !(t_0 <= 1.0))) {
		tmp = sin(im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(im * im) * im), -0.16666666666666666, im));
	elseif ((t_0 <= -0.1) || !((t_0 <= 2e-15) || !(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[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.1], N[Not[Or[LessEqual[t$95$0, 2e-15], 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

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.9

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 19.9%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 2.0000000000000002e-15 < (*.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. lower-sin.f64 97.0

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

   5. Applied rewrites 97.0%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2.0000000000000002e-15 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 94.2

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

   5. Applied rewrites 94.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.1 \lor \neg \left(e^{re} \cdot \sin im \leq 2 \cdot 10^{-15} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (* (* im im) im) -0.16666666666666666 im))
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 0.0)
         0.0
         (if (<= t_0 1.0)
           (sin im)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma(((im * im) * im), -0.16666666666666666, im);
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 0.0) {
		tmp = 0.0;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(im * im) * im), -0.16666666666666666, im));
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 0.0)
		tmp = 0.0;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], 0.0, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]]]

Derivation

1. Split input into 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

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

      1. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 19.9

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 19.9%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 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. lower-sin.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 38.9

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

   5. Applied rewrites 38.9%

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

   7. Applied rewrites 5.3%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.2%

       \[\leadsto 0 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 55.3%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.4)
     (* (+ 1.0 re) (fma (* (* im im) im) -0.16666666666666666 im))
     (if (<= t_0 0.0)
       0.0
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = (1.0 + re) * fma(((im * im) * im), -0.16666666666666666, im);
	} else if (t_0 <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.4)
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(im * im) * im), -0.16666666666666666, im));
	elseif (t_0 <= 0.0)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.4], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], 0.0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.40000000000000002

   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. lower-+.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 13.0

          \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left({im}^{\color{blue}{3}}, -0.16666666666666666, im\right) \]

   8. Applied rewrites 13.0%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.0%

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

   if -0.40000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 8.3%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 60.4%

       \[\leadsto 0 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.1%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.4:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(\left(im \cdot im\right) \cdot im, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.4)
     (fma (* im im) (* im -0.16666666666666666) im)
     (if (<= t_0 0.0)
       0.0
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = fma((im * im), (im * -0.16666666666666666), im);
	} else if (t_0 <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.4)
		tmp = fma(Float64(im * im), Float64(im * -0.16666666666666666), im);
	elseif (t_0 <= 0.0)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.4], N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], 0.0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.40000000000000002

   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. lower-sin.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 10.9%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 10.9%

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

   if -0.40000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 8.3%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 60.4%

       \[\leadsto 0 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 56.1%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.4)
     (fma (* im im) (* im -0.16666666666666666) im)
     (if (<= t_0 0.0) 0.0 (* (fma (fma 0.5 re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = fma((im * im), (im * -0.16666666666666666), im);
	} else if (t_0 <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.4)
		tmp = fma(Float64(im * im), Float64(im * -0.16666666666666666), im);
	elseif (t_0 <= 0.0)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.4], N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], 0.0, N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.40000000000000002

   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. lower-sin.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 10.9%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 10.9%

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

   if -0.40000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 8.3%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 60.4%

       \[\leadsto 0 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot im, 0.5, im\right), re, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.4)
     (fma (* im im) (* im -0.16666666666666666) im)
     (if (<= t_0 0.0) 0.0 (fma (fma (* re im) 0.5 im) re im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = fma((im * im), (im * -0.16666666666666666), im);
	} else if (t_0 <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = fma(fma((re * im), 0.5, im), re, im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.4)
		tmp = fma(Float64(im * im), Float64(im * -0.16666666666666666), im);
	elseif (t_0 <= 0.0)
		tmp = 0.0;
	else
		tmp = fma(fma(Float64(re * im), 0.5, im), 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, -0.4], N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], 0.0, N[(N[(N[(re * im), $MachinePrecision] * 0.5 + im), $MachinePrecision] * re + im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.40000000000000002

   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. lower-sin.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 10.9%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 10.9%

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

   if -0.40000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 8.3%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 60.4%

       \[\leadsto 0 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 50.6%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(re \cdot im, 0.5, im\right), re, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.4)
     (fma (* im im) (* im -0.16666666666666666) im)
     (if (<= t_0 0.0) 0.0 (fma re im im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.4) {
		tmp = fma((im * im), (im * -0.16666666666666666), im);
	} else if (t_0 <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = fma(re, im, im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.4)
		tmp = fma(Float64(im * im), Float64(im * -0.16666666666666666), im);
	elseif (t_0 <= 0.0)
		tmp = 0.0;
	else
		tmp = fma(re, im, 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, -0.4], N[(N[(im * im), $MachinePrecision] * N[(im * -0.16666666666666666), $MachinePrecision] + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], 0.0, N[(re * im + im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.40000000000000002

   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. lower-sin.f64 38.2

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 10.9%

      \[\leadsto \mathsf{fma}\left({im}^{3}, \color{blue}{-0.16666666666666666}, im\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 10.9%

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

   if -0.40000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 8.3%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 60.4%

       \[\leadsto 0 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.4:\\ \;\;\;\;\mathsf{fma}\left(im \cdot im, im \cdot -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq 0.98:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 0.0) 0.0 (if (<= t_0 0.98) im (* re im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.0;
	} else if (t_0 <= 0.98) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(re) * sin(im)
    if (t_0 <= 0.0d0) then
        tmp = 0.0d0
    else if (t_0 <= 0.98d0) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.sin(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.0;
	} else if (t_0 <= 0.98) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.sin(im)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 0.0
	elif t_0 <= 0.98:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = 0.0;
	elseif (t_0 <= 0.98)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * sin(im);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 0.0;
	elseif (t_0 <= 0.98)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = 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, 0.0], 0.0, If[LessEqual[t$95$0, 0.98], im, N[(re * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 39.8

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

   5. Applied rewrites 39.8%

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

   7. Applied rewrites 18.7%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 38.9%

       \[\leadsto 0 \]

   if -0.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.97999999999999998

   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. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 42.4%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{im \cdot \cos \left(2 \cdot \mathsf{PI}\left(\right)\right)} \]

   10. Applied rewrites 59.7%

       \[\leadsto im \]

   if 0.97999999999999998 < (*.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 10.8%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 11.0%

       \[\leadsto re \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0.98:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 39.8

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

   5. Applied rewrites 39.8%

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

   7. Applied rewrites 18.7%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 38.9%

       \[\leadsto 0 \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 42.1%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= (* (exp re) (sin im)) 0.0) 0.0 im))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = 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)) <= 0.0d0) then
        tmp = 0.0d0
    else
        tmp = 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)) <= 0.0) {
		tmp = 0.0;
	} else {
		tmp = im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 0.0:
		tmp = 0.0
	else:
		tmp = im
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 39.8

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

   5. Applied rewrites 39.8%

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

   7. Applied rewrites 18.7%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 38.9%

       \[\leadsto 0 \]

   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

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

      1. lower-sin.f64 68.3

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 31.8%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

   8. Taylor expanded in im around 0

      \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{im \cdot \cos \left(2 \cdot \mathsf{PI}\left(\right)\right)} \]

   10. Applied rewrites 38.8%

       \[\leadsto im \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 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 19: 26.6% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. lower-sin.f64 51.4

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

5. Applied rewrites 51.4%

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

7. Applied rewrites 24.0%

   \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) + im\right) + \mathsf{PI}\left(\right)\right) \]

8. Taylor expanded in im around 0

   \[\leadsto \sin \left(2 \cdot \mathsf{PI}\left(\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 24.5%

    \[\leadsto 0 \]

11. Final simplification 24.5%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024364 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))