math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 17

Speedup: 1.0×

• 24.3% 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 99.6%

   \[e^{re} \cdot \sin im \]
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 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (pow im 3.0) -0.16666666666666666 im))
     (if (<= t_0 -0.05)
       (/ (sin im) (- 1.0 re))
       (if (or (<= t_0 5e-305) (not (<= t_0 1.0)))
         (* (exp re) im)
         (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 + re) * fma(pow(im, 3.0), -0.16666666666666666, im);
	} else if (t_0 <= -0.05) {
		tmp = sin(im) / (1.0 - re);
	} else if ((t_0 <= 5e-305) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + re) * fma((im ^ 3.0), -0.16666666666666666, im));
	elseif (t_0 <= -0.05)
		tmp = Float64(sin(im) / Float64(1.0 - re));
	elseif ((t_0 <= 5e-305) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 + re), $MachinePrecision] * N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-305], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 4.5

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

   5. Applied rewrites 4.5%

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

   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 32.6

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

   8. Applied rewrites 32.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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 99.9

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

   4. Applied rewrites 99.9%

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

   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 99.9

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

   7. Applied rewrites 99.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999985e-305 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.3%

      \[e^{re} \cdot \sin im \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if 4.99999999999999985e-305 < (*.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%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-305} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (fma (* -0.16666666666666666 (* im im)) im im)
     (if (or (<= t_0 -0.05) (not (or (<= t_0 5e-305) (not (<= 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 = fma((-0.16666666666666666 * (im * im)), im, im);
	} else if ((t_0 <= -0.05) || !((t_0 <= 5e-305) || !(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 = fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im);
	elseif ((t_0 <= -0.05) || !((t_0 <= 5e-305) || !(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[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.05], N[Not[Or[LessEqual[t$95$0, 5e-305], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 2.8

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

   5. Applied rewrites 2.8%

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

   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 28.8%

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

   10. Applied rewrites 28.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 4.99999999999999985e-305 < (*.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 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999985e-305 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.3%

      \[e^{re} \cdot \sin im \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-305} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (+ 1.0 re) (fma (pow im 3.0) -0.16666666666666666 im))
     (if (<= t_0 -0.05)
       (/ (sin im) (- 1.0 re))
       (if (or (<= t_0 5e-305) (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(pow(im, 3.0), -0.16666666666666666, im);
	} else if (t_0 <= -0.05) {
		tmp = sin(im) / (1.0 - re);
	} else if ((t_0 <= 5e-305) || !(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((im ^ 3.0), -0.16666666666666666, im));
	elseif (t_0 <= -0.05)
		tmp = Float64(sin(im) / Float64(1.0 - re));
	elseif ((t_0 <= 5e-305) || !(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[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666 + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-305], 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 4.5

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

   5. Applied rewrites 4.5%

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

   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 32.6

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

   8. Applied rewrites 32.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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 99.9

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

   4. Applied rewrites 99.9%

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

   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 99.9

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

   7. Applied rewrites 99.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999985e-305 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.3%

      \[e^{re} \cdot \sin im \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if 4.99999999999999985e-305 < (*.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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left({im}^{3}, -0.16666666666666666, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-305} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (fma (* -0.16666666666666666 (* im im)) im im)
     (if (<= t_0 -0.05)
       (/ (sin im) (- 1.0 re))
       (if (or (<= t_0 5e-305) (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 = fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= -0.05) {
		tmp = sin(im) / (1.0 - re);
	} else if ((t_0 <= 5e-305) || !(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 = fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im);
	elseif (t_0 <= -0.05)
		tmp = Float64(sin(im) / Float64(1.0 - re));
	elseif ((t_0 <= 5e-305) || !(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[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-305], 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

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

      1. lower-sin.f64 2.8

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

   5. Applied rewrites 2.8%

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

   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 28.8%

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

   10. Applied rewrites 28.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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 99.9

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

   4. Applied rewrites 99.9%

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

   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 99.9

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

   7. Applied rewrites 99.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999985e-305 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.3%

      \[e^{re} \cdot \sin im \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if 4.99999999999999985e-305 < (*.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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-305} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (fma (* -0.16666666666666666 (* im im)) im im)
     (if (or (<= t_0 -0.05) (not (or (<= t_0 5e-305) (not (<= t_0 1.0)))))
       (sin im)
       (* (exp re) im)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im);
	} else if ((t_0 <= -0.05) || !((t_0 <= 5e-305) || !(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 = fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im);
	elseif ((t_0 <= -0.05) || !((t_0 <= 5e-305) || !(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[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.05], N[Not[Or[LessEqual[t$95$0, 5e-305], 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

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

      1. lower-sin.f64 2.8

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

   5. Applied rewrites 2.8%

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

   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 28.8%

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

   10. Applied rewrites 28.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 4.99999999999999985e-305 < (*.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 98.6

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

   5. Applied rewrites 98.6%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999985e-305 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.3%

      \[e^{re} \cdot \sin im \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05 \lor \neg \left(e^{re} \cdot \sin im \leq 5 \cdot 10^{-305} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im \cdot im - t\_1 \cdot t\_1}{im - t\_1}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (* im im) (* -0.16666666666666666 im))))
   (if (<= t_0 (- INFINITY))
     (fma (* -0.16666666666666666 (* im im)) im im)
     (if (<= t_0 -0.05)
       (sin im)
       (if (<= t_0 0.0)
         (/ (- (* im im) (* t_1 t_1)) (- im t_1))
         (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 t_1 = (im * im) * (-0.16666666666666666 * im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= -0.05) {
		tmp = sin(im);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) - (t_1 * t_1)) / (im - t_1);
	} 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))
	t_1 = Float64(Float64(im * im) * Float64(-0.16666666666666666 * im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im);
	elseif (t_0 <= -0.05)
		tmp = sin(im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) - Float64(t_1 * t_1)) / Float64(im - t_1));
	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]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(im - t$95$1), $MachinePrecision]), $MachinePrecision], 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

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

      1. lower-sin.f64 2.8

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

   5. Applied rewrites 2.8%

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

   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 28.8%

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

   10. Applied rewrites 28.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 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 98.6

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

   5. Applied rewrites 98.6%

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

   if -0.050000000000000003 < (*.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 38.7

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

   5. Applied rewrites 38.7%

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

   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 37.3%

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

   10. Applied rewrites 37.3%

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

   12. Applied rewrites 30.6%

       \[\leadsto \frac{im \cdot im - \left(\left(-im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right)}{im + \color{blue}{\left(-im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)}} \]

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

   1. Initial program 97.7%

      \[e^{re} \cdot \sin im \]
   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 82.3

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

   5. Applied rewrites 82.3%

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

   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 45.3%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\frac{im \cdot im - \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right)}{im - \left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)}\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-305} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.05)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
     (if (or (<= t_0 5e-305) (not (<= t_0 1.0)))
       (* (exp re) im)
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else if ((t_0 <= 5e-305) || !(t_0 <= 1.0)) {
		tmp = exp(re) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	elseif ((t_0 <= 5e-305) || !(t_0 <= 1.0))
		tmp = Float64(exp(re) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-305], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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 82.9

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

   5. Applied rewrites 82.9%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999985e-305 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 99.3%

      \[e^{re} \cdot \sin im \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if 4.99999999999999985e-305 < (*.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%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-305} \lor \neg \left(e^{re} \cdot \sin im \leq 1\right):\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := \left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im \cdot im - t\_1 \cdot t\_1}{im - t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im)))
        (t_1 (* (* im im) (* -0.16666666666666666 im))))
   (if (<= t_0 -0.05)
     (fma (* -0.16666666666666666 (* im im)) im im)
     (if (<= t_0 0.0)
       (/ (- (* im im) (* t_1 t_1)) (- im t_1))
       (* (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 t_1 = (im * im) * (-0.16666666666666666 * im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) - (t_1 * t_1)) / (im - t_1);
	} 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))
	t_1 = Float64(Float64(im * im) * Float64(-0.16666666666666666 * im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) - Float64(t_1 * t_1)) / Float64(im - t_1));
	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]}, Block[{t$95$1 = N[(N[(im * im), $MachinePrecision] * N[(-0.16666666666666666 * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(im - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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 49.5

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

   5. Applied rewrites 49.5%

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

   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 16.0%

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

   10. Applied rewrites 16.0%

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

   if -0.050000000000000003 < (*.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 38.7

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

   5. Applied rewrites 38.7%

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

   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 37.3%

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

   10. Applied rewrites 37.3%

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

   12. Applied rewrites 30.6%

       \[\leadsto \frac{im \cdot im - \left(\left(-im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right) \cdot \left(\left(-im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right)}{im + \color{blue}{\left(-im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)}} \]

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

   1. Initial program 99.1%

      \[e^{re} \cdot \sin im \]
   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 53.4

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

   5. Applied rewrites 53.4%

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

   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 39.6%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 0:\\ \;\;\;\;\frac{im \cdot im - \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right) \cdot \left(\left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)\right)}{im - \left(im \cdot im\right) \cdot \left(-0.16666666666666666 \cdot im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + 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 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 42.7

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

   5. Applied rewrites 42.7%

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

   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 29.4%

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

   10. Applied rewrites 29.4%

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

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

   1. Initial program 99.1%

      \[e^{re} \cdot \sin im \]
   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 53.4

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

   5. Applied rewrites 53.4%

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

   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 39.6%

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

Alternative 11: 35.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.05)
   (fma (* -0.16666666666666666 (* im im)) im im)
   (* (* (* (fma 0.16666666666666666 re 0.5) re) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.05) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.05)
		tmp = fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.050000000000000003

   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 50.9

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

   5. Applied rewrites 50.9%

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

   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 38.0%

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

   10. Applied rewrites 38.0%

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

   if 0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 98.8%

      \[e^{re} \cdot \sin im \]
   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 42.0

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

   5. Applied rewrites 42.0%

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

   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 23.9%

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

   9. Taylor expanded in re around inf

      \[\leadsto \left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{re}\right)\right) \cdot im \]
   10. Step-by-step derivation

   11. Applied rewrites 24.5%

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

Alternative 12: 34.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 42.7

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

   5. Applied rewrites 42.7%

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

   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 29.4%

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

   10. Applied rewrites 29.4%

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

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

   1. Initial program 99.1%

      \[e^{re} \cdot \sin im \]
   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 53.4

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

   5. Applied rewrites 53.4%

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

   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 37.7%

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

Alternative 13: 34.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(im \cdot im\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.05)
   (fma (* -0.16666666666666666 (* im im)) im im)
   (* (* (* re re) im) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.05) {
		tmp = fma((-0.16666666666666666 * (im * im)), im, im);
	} else {
		tmp = ((re * re) * im) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.05)
		tmp = fma(Float64(-0.16666666666666666 * Float64(im * im)), im, im);
	else
		tmp = Float64(Float64(Float64(re * re) * im) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(-0.16666666666666666 * N[(im * im), $MachinePrecision]), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.050000000000000003

   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 50.9

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

   5. Applied rewrites 50.9%

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

   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 38.0%

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

   10. Applied rewrites 38.0%

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

   if 0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 98.8%

      \[e^{re} \cdot \sin im \]
   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 42.0

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

   5. Applied rewrites 42.0%

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

   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 15.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 21.9%

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

Alternative 14: 33.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.05) (fma re im im) (* (* (* re re) im) 0.5)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.05) {
		tmp = fma(re, im, im);
	} else {
		tmp = ((re * re) * im) * 0.5;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.05)
		tmp = fma(re, im, im);
	else
		tmp = Float64(Float64(Float64(re * re) * im) * 0.5);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.05], N[(re * im + im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * im), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.050000000000000003

   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 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 36.7%

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

   if 0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 98.8%

      \[e^{re} \cdot \sin im \]
   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 42.0

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

   5. Applied rewrites 42.0%

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

   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 15.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 21.9%

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

Alternative 15: 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 99.6%

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

Alternative 16: 29.8% accurate, 29.4× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 (fma re im im))

C

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

Julia

function code(re, im)
	return fma(re, im, im)
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[e^{re} \cdot \sin im \]
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 66.8

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

5. Applied rewrites 66.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 27.4%

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

Alternative 17: 7.2% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[e^{re} \cdot \sin im \]
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 66.8

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

5. Applied rewrites 66.8%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 27.4%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 6.7%

    \[\leadsto re \cdot im \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))