math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 13

Speedup: 1.0×

• 24.5% 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: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-265}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (fma (fma 0.5 re 1.0) re 1.0) (sin im)))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* (* (* im im) im) -0.16666666666666666))
     (if (<= t_1 -0.02)
       t_0
       (if (<= t_1 5e-265) t_2 (if (<= t_1 2.0) t_0 t_2))))))

C

double code(double re, double im) {
	double t_0 = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 5e-265) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im))
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 5e-265)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$0, If[LessEqual[t$95$1, 5e-265], t$95$2, If[LessEqual[t$95$1, 2.0], t$95$0, t$95$2]]]]]]]

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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 5.0000000000000001e-265 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 62.8

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

   4. Applied rewrites 62.8%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000001e-265 or 2 < (*.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 69.2%

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

Alternative 3: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\frac{-1}{re - 1} \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\left(re - -1\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 im) im) -0.16666666666666666))
     (if (<= t_0 -0.02)
       (* (/ -1.0 (- re 1.0)) (sin im))
       (if (<= t_0 1e-22)
         t_1
         (if (<= t_0 2.0) (* (- re -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 * im) * im) * -0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = (-1.0 / (re - 1.0)) * sin(im);
	} else if (t_0 <= 1e-22) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = (re - -1.0) * sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.sin(im);
	double t_1 = Math.exp(re) * im;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = (-1.0 / (re - 1.0)) * Math.sin(im);
	} else if (t_0 <= 1e-22) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = (re - -1.0) * Math.sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.sin(im)
	t_1 = math.exp(re) * im
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * (((im * im) * im) * -0.16666666666666666)
	elif t_0 <= -0.02:
		tmp = (-1.0 / (re - 1.0)) * math.sin(im)
	elif t_0 <= 1e-22:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = (re - -1.0) * math.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(Float64(Float64(im * im) * im) * -0.16666666666666666));
	elseif (t_0 <= -0.02)
		tmp = Float64(Float64(-1.0 / Float64(re - 1.0)) * sin(im));
	elseif (t_0 <= 1e-22)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(re - -1.0) * sin(im));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * sin(im);
	t_1 = exp(re) * im;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	elseif (t_0 <= -0.02)
		tmp = (-1.0 / (re - 1.0)) * sin(im);
	elseif (t_0 <= 1e-22)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = (re - -1.0) * sin(im);
	else
		tmp = t_1;
	end
	tmp_2 = 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[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(-1.0 / N[(re - 1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-22], t$95$1, If[LessEqual[t$95$0, 2.0], N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im], $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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 51.1

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

   4. Applied rewrites 51.1%

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

      1. lift--.f64 N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-+ N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

          \[\leadsto \frac{{re}^{2} - -1 \cdot -1}{re - 1} \cdot \sin im \]

      11. metadata-eval N/A

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

      12. fp-cancel-sign-sub N/A

          \[\leadsto \frac{{re}^{2} + 1 \cdot -1}{\color{blue}{re} - 1} \cdot \sin im \]

      13. unpow2 N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 62.5

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in re around 0

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

   9. Applied rewrites 57.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-22 or 2 < (*.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 69.2%

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

   if 1e-22 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 51.1

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

   4. Applied rewrites 51.1%

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

Alternative 4: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re - -1\right) \cdot \sin im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (- re -1.0) (sin im)))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 (- INFINITY))
     (* (exp re) (* (* (* im im) im) -0.16666666666666666))
     (if (<= t_1 -0.02)
       t_0
       (if (<= t_1 1e-22) t_2 (if (<= t_1 2.0) t_0 t_2))))))

C

double code(double re, double im) {
	double t_0 = (re - -1.0) * sin(im);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 1e-22) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = (re - -1.0) * Math.sin(im);
	double t_1 = Math.exp(re) * Math.sin(im);
	double t_2 = Math.exp(re) * im;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_1 <= -0.02) {
		tmp = t_0;
	} else if (t_1 <= 1e-22) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re - -1.0) * math.sin(im)
	t_1 = math.exp(re) * math.sin(im)
	t_2 = math.exp(re) * im
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.exp(re) * (((im * im) * im) * -0.16666666666666666)
	elif t_1 <= -0.02:
		tmp = t_0
	elif t_1 <= 1e-22:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re - -1.0) * sin(im))
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 1e-22)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re - -1.0) * sin(im);
	t_1 = exp(re) * sin(im);
	t_2 = exp(re) * im;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	elseif (t_1 <= -0.02)
		tmp = t_0;
	elseif (t_1 <= 1e-22)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re - -1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.02], t$95$0, If[LessEqual[t$95$1, 1e-22], t$95$2, If[LessEqual[t$95$1, 2.0], t$95$0, t$95$2]]]]]]]

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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 1e-22 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval 51.1

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

   4. Applied rewrites 51.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-22 or 2 < (*.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 69.2%

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

Alternative 5: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 im) im) -0.16666666666666666))
     (if (<= t_0 -0.02)
       (sin im)
       (if (<= t_0 1e-22) t_1 (if (<= t_0 2.0) (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 * im) * im) * -0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = sin(im);
	} else if (t_0 <= 1e-22) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.sin(im);
	double t_1 = Math.exp(re) * im;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else if (t_0 <= -0.02) {
		tmp = Math.sin(im);
	} else if (t_0 <= 1e-22) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = Math.sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.sin(im)
	t_1 = math.exp(re) * im
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * (((im * im) * im) * -0.16666666666666666)
	elif t_0 <= -0.02:
		tmp = math.sin(im)
	elif t_0 <= 1e-22:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = math.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(Float64(Float64(im * im) * im) * -0.16666666666666666));
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 1e-22)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * sin(im);
	t_1 = exp(re) * im;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	elseif (t_0 <= -0.02)
		tmp = sin(im);
	elseif (t_0 <= 1e-22)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	tmp_2 = 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[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-22], t$95$1, If[LessEqual[t$95$0, 2.0], 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0200000000000000004 or 1e-22 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2

   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. lift-sin.f64 50.5

         \[\leadsto \sin im \]

   4. Applied rewrites 50.5%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-22 or 2 < (*.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 69.2%

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

Alternative 6: 69.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;e^{re} \cdot \left(\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (exp re) (* (fma (* -0.16666666666666666 im) im 1.0) im))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = exp(re) * (fma((-0.16666666666666666 * im), im, 1.0) * im);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(exp(re) * Float64(fma(Float64(-0.16666666666666666 * im), im, 1.0) * im));
	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], -0.02], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * im), $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)) < -0.0200000000000000004

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 60.4

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

   6. Applied rewrites 60.4%

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

   if -0.0200000000000000004 < (*.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 69.2%

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

Alternative 7: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;e^{re} \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (exp re) (* (* (* im im) im) -0.16666666666666666))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= (-0.02d0)) then
        tmp = exp(re) * (((im * im) * im) * (-0.16666666666666666d0))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= -0.02) {
		tmp = Math.exp(re) * (((im * im) * im) * -0.16666666666666666);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= -0.02:
		tmp = math.exp(re) * (((im * im) * im) * -0.16666666666666666)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(exp(re) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= -0.02)
		tmp = exp(re) * (((im * im) * im) * -0.16666666666666666);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[Exp[re], $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $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)) < -0.0200000000000000004

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   if -0.0200000000000000004 < (*.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 69.2%

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

Alternative 8: 62.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (fma (fma 0.5 re 1.0) re 1.0) (* (* (* im im) im) -0.16666666666666666))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * (((im * im) * im) * -0.16666666666666666);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	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], -0.02], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $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)) < -0.0200000000000000004

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 11.4

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

   10. Applied rewrites 11.4%

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

   if -0.0200000000000000004 < (*.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 69.2%

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

Alternative 9: 62.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;{im}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (pow im 7.0) -0.0001984126984126984)
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = pow(im, 7.0) * -0.0001984126984126984;
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= (-0.02d0)) then
        tmp = (im ** 7.0d0) * (-0.0001984126984126984d0)
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= -0.02) {
		tmp = Math.pow(im, 7.0) * -0.0001984126984126984;
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= -0.02:
		tmp = math.pow(im, 7.0) * -0.0001984126984126984
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64((im ^ 7.0) * -0.0001984126984126984);
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= -0.02)
		tmp = (im ^ 7.0) * -0.0001984126984126984;
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[Power[im, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. lift-sin.f64 50.5

         \[\leadsto \sin im \]

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{5040} \cdot {im}^{\color{blue}{7}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {im}^{7} \cdot \frac{-1}{5040} \]

      2. lower-*.f64 N/A

         \[\leadsto {im}^{7} \cdot \frac{-1}{5040} \]

      3. lower-pow.f64 18.2

         \[\leadsto {im}^{7} \cdot -0.0001984126984126984 \]

   10. Applied rewrites 18.2%

       \[\leadsto {im}^{7} \cdot -0.0001984126984126984 \]

   if -0.0200000000000000004 < (*.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 69.2%

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

Alternative 10: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\left(1 + re\right) \cdot \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (+ 1.0 re) (* (* (* im im) im) -0.16666666666666666))
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = (1.0 + re) * (((im * im) * im) * -0.16666666666666666);
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * sin(im)) <= (-0.02d0)) then
        tmp = (1.0d0 + re) * (((im * im) * im) * (-0.16666666666666666d0))
    else
        tmp = exp(re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= -0.02) {
		tmp = (1.0 + re) * (((im * im) * im) * -0.16666666666666666);
	} else {
		tmp = Math.exp(re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= -0.02:
		tmp = (1.0 + re) * (((im * im) * im) * -0.16666666666666666)
	else:
		tmp = math.exp(re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(Float64(1.0 + re) * Float64(Float64(Float64(im * im) * im) * -0.16666666666666666));
	else
		tmp = Float64(exp(re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= -0.02)
		tmp = (1.0 + re) * (((im * im) * im) * -0.16666666666666666);
	else
		tmp = exp(re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(im * im), $MachinePrecision] * im), $MachinePrecision] * -0.16666666666666666), $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)) < -0.0200000000000000004

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in im around inf

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

      1. *-commutative N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto e^{re} \cdot \left({im}^{3} \cdot \frac{-1}{6}\right) \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in re around 0

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

      1. lower-+.f64 15.6

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

   10. Applied rewrites 15.6%

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

   if -0.0200000000000000004 < (*.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 69.2%

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

Alternative 11: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) -0.02)
   (* (fma (* -0.16666666666666666 im) im 1.0) im)
   (* (exp re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= -0.02) {
		tmp = fma((-0.16666666666666666 * im), im, 1.0) * im;
	} else {
		tmp = exp(re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= -0.02)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im), im, 1.0) * im);
	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], -0.02], N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. lift-sin.f64 50.5

         \[\leadsto \sin im \]

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in im around 0

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

      1. lower-*.f64 29.3

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

   10. Applied rewrites 29.3%

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

   if -0.0200000000000000004 < (*.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 69.2%

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

Alternative 12: 33.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 0.12:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.12)
   (* (fma (* -0.16666666666666666 im) im 1.0) im)
   (* (fma (fma 0.5 re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.12) {
		tmp = fma((-0.16666666666666666 * im), im, 1.0) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.12)
		tmp = Float64(fma(Float64(-0.16666666666666666 * im), im, 1.0) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.12], N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.12

   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. lift-sin.f64 50.5

         \[\leadsto \sin im \]

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

   7. Applied rewrites 30.2%

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

   8. Taylor expanded in im around 0

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

      1. lower-*.f64 29.3

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

   10. Applied rewrites 29.3%

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

   if 0.12 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   2. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 59.8

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 36.5

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

   7. Applied rewrites 36.5%

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

Alternative 13: 29.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot im, im, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (fma (* -0.16666666666666666 im) im 1.0) im))

C

double code(double re, double im) {
	return fma((-0.16666666666666666 * im), im, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(Float64(-0.16666666666666666 * im), im, 1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(-0.16666666666666666 * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

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. lift-sin.f64 50.5

      \[\leadsto \sin im \]

4. Applied rewrites 50.5%

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

5. Taylor expanded in im around 0

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

   1. *-commutative N/A

      \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right) \cdot im \]

7. Applied rewrites 30.2%

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

8. Taylor expanded in im around 0

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

   1. lower-*.f64 29.3

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

10. Applied rewrites 29.3%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))