math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 2.1s

Alternatives: 12

Speedup: 1.0×

• 23.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im \cdot \left(re - -1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\frac{\sin im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* im (fma (* -0.16666666666666666 im) im 1.0)))
     (if (<= t_0 -0.01)
       (* (sin im) (- re -1.0))
       (if (<= t_0 4e-29)
         t_1
         (if (<= t_0 1.0) (/ (sin im) (- 1.0 re)) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * (im * fma((-0.16666666666666666 * im), im, 1.0));
	} else if (t_0 <= -0.01) {
		tmp = sin(im) * (re - -1.0);
	} else if (t_0 <= 4e-29) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im) / (1.0 - re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(im * fma(Float64(-0.16666666666666666 * im), im, 1.0)));
	elseif (t_0 <= -0.01)
		tmp = Float64(sin(im) * Float64(re - -1.0));
	elseif (t_0 <= 4e-29)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(sin(im) / Float64(1.0 - re));
	else
		tmp = t_1;
	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[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(N[Sin[im], $MachinePrecision] * N[(re - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-29], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[Sin[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

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. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 61.0

         \[\leadsto e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 61.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 61.0

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

   6. Applied rewrites 61.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.1

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 52.1

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

   6. Applied rewrites 52.1%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.99999999999999977e-29 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

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

   if 3.99999999999999977e-29 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. add-flip N/A

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

      6. cosh-neg-rev N/A

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

      7. sinh-neg-rev N/A

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

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

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

      9. rec-exp N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 58.0

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

   6. Applied rewrites 58.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-out N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

      13. /-rgt-identity 58.0

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

   8. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{\sin im}{1 - re}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := \sin im \cdot \left(re - -1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right)\right)\\ \mathbf{elif}\;t\_1 \leq -0.01:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (sin im) (- re -1.0))))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* im (fma (* -0.16666666666666666 im) im 1.0)))
     (if (<= t_1 -0.01)
       t_2
       (if (<= t_1 4e-41) t_0 (if (<= t_1 1.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double t_2 = sin(im) * (re - -1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (im * fma((-0.16666666666666666 * im), im, 1.0));
	} else if (t_1 <= -0.01) {
		tmp = t_2;
	} else if (t_1 <= 4e-41) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(sin(im) * Float64(re - -1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(im * fma(Float64(-0.16666666666666666 * im), im, 1.0)));
	elseif (t_1 <= -0.01)
		tmp = t_2;
	elseif (t_1 <= 4e-41)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[im], $MachinePrecision] * N[(re - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.01], t$95$2, If[LessEqual[t$95$1, 4e-41], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]]]]

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. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 61.0

         \[\leadsto e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 61.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 61.0

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

   6. Applied rewrites 61.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 4.00000000000000002e-41 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-+.f64 52.1

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

   4. Applied rewrites 52.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.1

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 52.1

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

   6. Applied rewrites 52.1%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000002e-41 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

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

Alternative 4: 93.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* im (fma (* -0.16666666666666666 im) im 1.0)))
     (if (<= t_0 -0.01)
       (sin im)
       (if (<= t_0 4e-29) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * (im * fma((-0.16666666666666666 * im), im, 1.0));
	} else if (t_0 <= -0.01) {
		tmp = sin(im);
	} else if (t_0 <= 4e-29) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(im * fma(Float64(-0.16666666666666666 * im), im, 1.0)));
	elseif (t_0 <= -0.01)
		tmp = sin(im);
	elseif (t_0 <= 4e-29)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	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[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 4e-29], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

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. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 61.0

         \[\leadsto e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 61.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 61.0

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

   6. Applied rewrites 61.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0100000000000000002 or 3.99999999999999977e-29 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 51.5

         \[\leadsto \sin im \]

   4. Applied rewrites 51.5%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 3.99999999999999977e-29 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

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

Alternative 5: 69.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;e^{re} \cdot \left(im \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.01)
   (* (exp re) (* im (fma (* -0.16666666666666666 im) im 1.0)))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.01) {
		tmp = exp(re) * (im * fma((-0.16666666666666666 * im), im, 1.0));
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.01)
		tmp = Float64(exp(re) * Float64(im * fma(Float64(-0.16666666666666666 * im), im, 1.0)));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.01], N[(N[Exp[re], $MachinePrecision] * N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 61.0

         \[\leadsto e^{re} \cdot \left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 61.0%

      \[\leadsto e^{re} \cdot \color{blue}{\left(im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 61.0

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

   6. Applied rewrites 61.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.01:\\ \;\;\;\;im \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.01)
   (* im (fma (* -0.16666666666666666 im) im 1.0))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.01) {
		tmp = im * fma((-0.16666666666666666 * im), im, 1.0);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.01)
		tmp = Float64(im * fma(Float64(-0.16666666666666666 * im), im, 1.0));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.01], N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 51.5

         \[\leadsto \sin im \]

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.1

         \[\leadsto im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right) \]

   7. Applied rewrites 31.1%

      \[\leadsto im \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 31.1

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

   9. Applied rewrites 31.1%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 37.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (* im (fma (* -0.16666666666666666 im) im 1.0))
     (if (<= t_0 0.0) (/ 1.0 (/ (+ 1.0 (* -1.0 re)) im)) (* im (- re -1.0))))))

C

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

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(im * fma(Float64(-0.16666666666666666 * im), im, 1.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(-1.0 * re)) / im));
	else
		tmp = Float64(im * Float64(re - -1.0));
	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, -0.01], N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(1.0 + N[(-1.0 * re), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision], N[(im * N[(re - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 51.5

         \[\leadsto \sin im \]

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.1

         \[\leadsto im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right) \]

   7. Applied rewrites 31.1%

      \[\leadsto im \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 31.1

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

   9. Applied rewrites 31.1%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. add-flip N/A

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

      6. cosh-neg-rev N/A

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

      7. sinh-neg-rev N/A

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

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

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

      9. rec-exp N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 69.3

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

   6. Applied rewrites 69.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 69.2

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

   8. Applied rewrites 69.2%

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

   9. Taylor expanded in re around 0

      \[\leadsto \frac{1}{\frac{1 + -1 \cdot re}{im}} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \frac{1}{\frac{1 + -1 \cdot re}{im}} \]

       2. lower-*.f64 33.7

          \[\leadsto \frac{1}{\frac{1 + -1 \cdot re}{im}} \]

   11. Applied rewrites 33.7%

       \[\leadsto \frac{1}{\frac{1 + -1 \cdot re}{im}} \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 30.2

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

   7. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 30.2

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. /-rgt-identity N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-out N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. distribute-lft-neg-out N/A

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

      19. *-commutative N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. mult-flip N/A

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

      24. /-rgt-identity 30.2

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

   9. Applied rewrites 30.2%

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

Alternative 8: 37.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;im \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-41}:\\ \;\;\;\;\frac{im}{1 + -1 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.01)
     (* im (fma (* -0.16666666666666666 im) im 1.0))
     (if (<= t_0 4e-41) (/ im (+ 1.0 (* -1.0 re))) (* im (- re -1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = im * fma((-0.16666666666666666 * im), im, 1.0);
	} else if (t_0 <= 4e-41) {
		tmp = im / (1.0 + (-1.0 * re));
	} else {
		tmp = im * (re - -1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = Float64(im * fma(Float64(-0.16666666666666666 * im), im, 1.0));
	elseif (t_0 <= 4e-41)
		tmp = Float64(im / Float64(1.0 + Float64(-1.0 * re)));
	else
		tmp = Float64(im * Float64(re - -1.0));
	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, -0.01], N[(im * N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-41], N[(im / N[(1.0 + N[(-1.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. lower-sin.f64 51.5

         \[\leadsto \sin im \]

   4. Applied rewrites 51.5%

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

   5. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 31.1

         \[\leadsto im \cdot \left(1 + -0.16666666666666666 \cdot {im}^{2}\right) \]

   7. Applied rewrites 31.1%

      \[\leadsto im \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 31.1

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

   9. Applied rewrites 31.1%

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

   if -0.0100000000000000002 < (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000002e-41

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. add-flip N/A

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

      6. cosh-neg-rev N/A

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

      7. sinh-neg-rev N/A

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

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

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

      9. rec-exp N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 69.3

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 33.6

         \[\leadsto \frac{im}{1 + -1 \cdot re} \]

   9. Applied rewrites 33.6%

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

   if 4.00000000000000002e-41 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 30.2

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

   7. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 30.2

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. /-rgt-identity N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-out N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. distribute-lft-neg-out N/A

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

      19. *-commutative N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. mult-flip N/A

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

      24. /-rgt-identity 30.2

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

   9. Applied rewrites 30.2%

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

Alternative 9: 35.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-41}:\\ \;\;\;\;\frac{im}{1 + -1 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-41)
   (/ im (+ 1.0 (* -1.0 re)))
   (* im (- re -1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-41) {
		tmp = im / (1.0 + (-1.0 * re));
	} else {
		tmp = im * (re - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 4d-41) then
        tmp = im / (1.0d0 + ((-1.0d0) * re))
    else
        tmp = im * (re - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 4e-41) {
		tmp = im / (1.0 + (-1.0 * re));
	} else {
		tmp = im * (re - -1.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 4e-41:
		tmp = im / (1.0 + (-1.0 * re))
	else:
		tmp = im * (re - -1.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-41)
		tmp = Float64(im / Float64(1.0 + Float64(-1.0 * re)));
	else
		tmp = Float64(im * Float64(re - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 4e-41)
		tmp = im / (1.0 + (-1.0 * re));
	else
		tmp = im * (re - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-41], N[(im / N[(1.0 + N[(-1.0 * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(im * N[(re - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000002e-41

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. add-flip N/A

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

      6. cosh-neg-rev N/A

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

      7. sinh-neg-rev N/A

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

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

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

      9. rec-exp N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 69.3

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 33.6

         \[\leadsto \frac{im}{1 + -1 \cdot re} \]

   9. Applied rewrites 33.6%

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

   if 4.00000000000000002e-41 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 30.2

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

   7. Applied rewrites 30.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 30.2

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. /-rgt-identity N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-out N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. distribute-lft-neg-out N/A

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

      19. *-commutative N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. mult-flip N/A

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

      24. /-rgt-identity 30.2

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

   9. Applied rewrites 30.2%

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

Alternative 10: 35.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 4 \cdot 10^{-167}:\\ \;\;\;\;\frac{-1}{re - 1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;im + im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 4e-167)
   (* (/ -1.0 (- re 1.0)) im)
   (+ im (* im re))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 4e-167) {
		tmp = (-1.0 / (re - 1.0)) * im;
	} else {
		tmp = im + (im * re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= 4d-167) then
        tmp = ((-1.0d0) / (re - 1.0d0)) * im
    else
        tmp = im + (im * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 4e-167) {
		tmp = (-1.0 / (re - 1.0)) * im;
	} else {
		tmp = im + (im * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 4e-167:
		tmp = (-1.0 / (re - 1.0)) * im
	else:
		tmp = im + (im * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 4e-167)
		tmp = Float64(Float64(-1.0 / Float64(re - 1.0)) * im);
	else
		tmp = Float64(im + Float64(im * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 4e-167)
		tmp = (-1.0 / (re - 1.0)) * im;
	else
		tmp = im + (im * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 4e-167], N[(N[(-1.0 / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision], N[(im + N[(im * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.00000000000000001e-167

   1. Initial program 100.0%

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

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 30.2

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

   7. Applied rewrites 30.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. sqr-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 36.7%

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

   10. Taylor expanded in re around 0

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

   12. Applied rewrites 33.6%

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

   if 4.00000000000000001e-167 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. add-flip N/A

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

      6. cosh-neg-rev N/A

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

      7. sinh-neg-rev N/A

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

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

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

      9. rec-exp N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 69.3

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in re around 0

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

      1. lower-+.f64 N/A

         \[\leadsto im + im \cdot \color{blue}{re} \]

      2. lower-*.f64 30.2

         \[\leadsto im + im \cdot re \]

   9. Applied rewrites 30.2%

      \[\leadsto im + \color{blue}{im \cdot re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 30.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in im around 0

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

4. Applied rewrites 69.3%

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

5. Taylor expanded in re around 0

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

   1. lower-+.f64 30.2

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

7. Applied rewrites 30.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 30.2

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. add-flip N/A

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

   7. /-rgt-identity N/A

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

   8. mult-flip N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-neg-out N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower--.f64 N/A

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

   17. metadata-eval N/A

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

   18. distribute-lft-neg-out N/A

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

   19. *-commutative N/A

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

   20. distribute-rgt-neg-in N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. mult-flip N/A

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

   24. /-rgt-identity 30.2

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

9. Applied rewrites 30.2%

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

Alternative 12: 27.3% accurate, 45.8× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-exp.f64 N/A

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

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

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

   5. add-flip N/A

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

   6. cosh-neg-rev N/A

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

   7. sinh-neg-rev N/A

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

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

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

   9. rec-exp N/A

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

   10. mult-flip-rev N/A

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

   11. lower-/.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{im}{e^{\mathsf{neg}\left(re\right)}}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-neg.f64 69.3

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

6. Applied rewrites 69.3%

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

7. Taylor expanded in re around 0

   \[\leadsto im \]
8. Step-by-step derivation

9. Applied rewrites 27.3%

   \[\leadsto im \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))