math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 24

Speedup: 1.0×

• 25.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. remove-double-neg N/A

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

   4. rec-exp N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. lift-cos.f64 N/A

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

   8. sin-PI/2 N/A

      \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

   10. lift-cos.f64 N/A

       \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

   11. sin-PI/2 N/A

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

   12. *-commutative N/A

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

   13. *-lft-identity N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999918194367\right)\right):\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 0.0) (not (<= t_0 0.999999918194367)))))
       (/
        (cos im)
        (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367))) {
		tmp = cos(im) / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367)))
		tmp = Float64(cos(im) / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999918194367]], $MachinePrecision]]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999918194366955

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.1

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

   7. Applied rewrites 98.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999918194366955 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999918194367\right)\right):\\ \;\;\;\;\frac{\cos im}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999918194367\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 0.0) (not (<= t_0 0.999999918194367)))))
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (cos im))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367))) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367)))
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999918194367]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999918194366955

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999918194366955 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999918194367\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999918194367\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 0.0) (not (<= t_0 0.999999918194367)))))
       (* (fma (fma 0.5 re 1.0) re 1.0) (cos im))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367))) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367)))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999918194367]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999918194366955

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999918194366955 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999918194367\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999918194367\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (exp re) (* (* im im) -0.5))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 0.0) (not (<= t_0 0.999999918194367)))))
       (/ (cos im) (- 1.0 re))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367))) {
		tmp = cos(im) / (1.0 - re);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(re) * Math.cos(im);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.exp(re) * ((im * im) * -0.5);
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367))) {
		tmp = Math.cos(im) / (1.0 - re);
	} else {
		tmp = Math.exp(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(re) * math.cos(im)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = math.exp(re) * ((im * im) * -0.5)
	elif (t_0 <= -0.02) or not ((t_0 <= 0.0) or not (t_0 <= 0.999999918194367)):
		tmp = math.cos(im) / (1.0 - re)
	else:
		tmp = math.exp(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(exp(re) * Float64(Float64(im * im) * -0.5));
	elseif ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367)))
		tmp = Float64(cos(im) / Float64(1.0 - re));
	else
		tmp = exp(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(re) * cos(im);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = exp(re) * ((im * im) * -0.5);
	elseif ((t_0 <= -0.02) || ~(((t_0 <= 0.0) || ~((t_0 <= 0.999999918194367)))))
		tmp = cos(im) / (1.0 - re);
	else
		tmp = exp(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Exp[re], $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999918194367]], $MachinePrecision]]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999918194366955

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999918194366955 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;e^{re} \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999918194367\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999918194367\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (fma
        (* (fma re im im) -0.001388888888888889)
        im
        (fma 0.041666666666666664 re 0.041666666666666664))
       (* im im)
       (- (* -0.5 re) 0.5))
      (* im im)
      (+ 1.0 re))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 0.0) (not (<= t_0 0.999999918194367)))))
       (/ (cos im) (- 1.0 re))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), (im * im), ((-0.5 * re) - 0.5)), (im * im), (1.0 + re));
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367))) {
		tmp = cos(im) / (1.0 - re);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(Float64(fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), Float64(im * im), Float64(Float64(-0.5 * re) - 0.5)), Float64(im * im), Float64(1.0 + re));
	elseif ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367)))
		tmp = Float64(cos(im) / Float64(1.0 - re));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * im + im), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * im + N[(0.041666666666666664 * re + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(-0.5 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999918194367]], $MachinePrecision]]], $MachinePrecision]], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-0.001388888888888889 \cdot re - 0.001388888888888889\right) \cdot im, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999918194366955

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 95.9

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

   7. Applied rewrites 95.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999918194366955 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999918194367\right)\right):\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999918194367\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (fma
        (* (fma re im im) -0.001388888888888889)
        im
        (fma 0.041666666666666664 re 0.041666666666666664))
       (* im im)
       (- (* -0.5 re) 0.5))
      (* im im)
      (+ 1.0 re))
     (if (or (<= t_0 -0.02)
             (not (or (<= t_0 0.0) (not (<= t_0 0.999999918194367)))))
       (* (+ 1.0 re) (cos im))
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), (im * im), ((-0.5 * re) - 0.5)), (im * im), (1.0 + re));
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367))) {
		tmp = (1.0 + re) * cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(Float64(fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), Float64(im * im), Float64(Float64(-0.5 * re) - 0.5)), Float64(im * im), Float64(1.0 + re));
	elseif ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.999999918194367)))
		tmp = Float64(Float64(1.0 + re) * cos(im));
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * im + im), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * im + N[(0.041666666666666664 * re + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(-0.5 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999918194367]], $MachinePrecision]]], $MachinePrecision]], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-0.001388888888888889 \cdot re - 0.001388888888888889\right) \cdot im, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999918194366955

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999918194366955 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999918194367\right)\right):\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;t\_0 \leq -0.02 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.9999\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (fma
        (* (fma re im im) -0.001388888888888889)
        im
        (fma 0.041666666666666664 re 0.041666666666666664))
       (* im im)
       (- (* -0.5 re) 0.5))
      (* im im)
      (+ 1.0 re))
     (if (or (<= t_0 -0.02) (not (or (<= t_0 0.0) (not (<= t_0 0.9999)))))
       (cos im)
       (exp re)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), (im * im), ((-0.5 * re) - 0.5)), (im * im), (1.0 + re));
	} else if ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.9999))) {
		tmp = cos(im);
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(Float64(fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), Float64(im * im), Float64(Float64(-0.5 * re) - 0.5)), Float64(im * im), Float64(1.0 + re));
	elseif ((t_0 <= -0.02) || !((t_0 <= 0.0) || !(t_0 <= 0.9999)))
		tmp = cos(im);
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * im + im), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * im + N[(0.041666666666666664 * re + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(-0.5 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.02], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.9999]], $MachinePrecision]]], $MachinePrecision]], N[Cos[im], $MachinePrecision], N[Exp[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-0.001388888888888889 \cdot re - 0.001388888888888889\right) \cdot im, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.0200000000000000004 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99990000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -0.0200000000000000004 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99990000000000001 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.02 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.9999\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (fma
        (* (fma re im im) -0.001388888888888889)
        im
        (fma 0.041666666666666664 re 0.041666666666666664))
       (* im im)
       (- (* -0.5 re) 0.5))
      (* im im)
      (+ 1.0 re))
     (if (<= t_0 0.0)
       (/
        (fma (* im im) -0.5 1.0)
        (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
       (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), (im * im), ((-0.5 * re) - 0.5)), (im * im), (1.0 + re));
	} else if (t_0 <= 0.0) {
		tmp = fma((im * im), -0.5, 1.0) / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(Float64(fma(re, im, im) * -0.001388888888888889), im, fma(0.041666666666666664, re, 0.041666666666666664)), Float64(im * im), Float64(Float64(-0.5 * re) - 0.5)), Float64(im * im), Float64(1.0 + re));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(im * im), -0.5, 1.0) / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(re * im + im), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * im + N[(0.041666666666666664 * re + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(-0.5 * re), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(1.0 + re), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-0.001388888888888889 \cdot re - 0.001388888888888889\right) \cdot im, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 78.9

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.9

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

   10. Applied rewrites 36.9%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, im, im\right) \cdot -0.001388888888888889, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), im \cdot im, 1 + re\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.001388888888888889 \cdot im\right) \cdot im - 0.041666666666666664, im \cdot im, 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (*
      (-
       (*
        (fma
         (- (* (* 0.001388888888888889 im) im) 0.041666666666666664)
         (* im im)
         0.5)
        (* im im))
       1.0)
      (- re))
     (if (<= t_0 0.0)
       (/
        (fma (* im im) -0.5 1.0)
        (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
       (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma((((0.001388888888888889 * im) * im) - 0.041666666666666664), (im * im), 0.5) * (im * im)) - 1.0) * -re;
	} else if (t_0 <= 0.0) {
		tmp = fma((im * im), -0.5, 1.0) / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(0.001388888888888889 * im) * im) - 0.041666666666666664), Float64(im * im), 0.5) * Float64(im * im)) - 1.0) * Float64(-re));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(im * im), -0.5, 1.0) / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	else
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(0.001388888888888889 * im), $MachinePrecision] * im), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * (-re)), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 5.3

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\left(re + {im}^{2} \cdot \left(\left(\frac{-1}{2} \cdot re + {im}^{2} \cdot \left(\frac{1}{24} + \left(\frac{1}{24} \cdot re + {im}^{2} \cdot \left(\frac{-1}{720} \cdot re - \frac{1}{720}\right)\right)\right)\right) - \frac{1}{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 90.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-0.001388888888888889 \cdot re - 0.001388888888888889\right) \cdot im, im, \mathsf{fma}\left(0.041666666666666664, re, 0.041666666666666664\right)\right), im \cdot im, -0.5 \cdot re - 0.5\right), \color{blue}{im \cdot im}, 1 + re\right) \]

   9. Taylor expanded in re around -inf

      \[\leadsto -1 \cdot \left(re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{720} \cdot {im}^{2} - \frac{1}{24}\right)\right) - 1\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(0.001388888888888889 \cdot im\right) \cdot im - 0.041666666666666664, im \cdot im, 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right) \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 78.9

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 36.9

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

   10. Applied rewrites 36.9%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(0.001388888888888889 \cdot im\right) \cdot im - 0.041666666666666664, im \cdot im, 0.5\right) \cdot \left(im \cdot im\right) - 1\right) \cdot \left(-re\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_0 \leq -0.95:\\ \;\;\;\;t\_1 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_0 -0.95)
     (* t_1 (* (* im im) -0.5))
     (if (<= t_0 0.0)
       (/
        (fma (* im im) -0.5 1.0)
        (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
       t_1))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = t_1 * ((im * im) * -0.5);
	} else if (t_0 <= 0.0) {
		tmp = fma((im * im), -0.5, 1.0) / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(t_1 * Float64(Float64(im * im) * -0.5));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(im * im), -0.5, 1.0) / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(t$95$1 * N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 68.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 61.6

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

   11. Applied rewrites 61.6%

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

   if -0.94999999999999996 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 77.1

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

   7. Applied rewrites 77.1%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 40.5

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

   10. Applied rewrites 40.5%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.95:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(im \cdot im\right) \cdot -0.5\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* im im) -0.5))
        (t_1 (* (exp re) (cos im)))
        (t_2 (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)))
   (if (<= t_1 (- INFINITY)) (* t_2 t_0) (if (<= t_1 0.0) t_0 t_2))))

C

double code(double re, double im) {
	double t_0 = (im * im) * -0.5;
	double t_1 = exp(re) * cos(im);
	double t_2 = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2 * t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, t$95$2]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 90.5

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

   11. Applied rewrites 90.5%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 34.5

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 3.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 17.6%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 51.6%

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

Alternative 13: 53.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 90.4

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

   8. Applied rewrites 90.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 90.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 34.5

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 3.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 17.6%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 51.6%

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

Alternative 14: 53.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 90.4

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

   8. Applied rewrites 90.4%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 90.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 34.5

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 3.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 17.6%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.2%

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

4. Final simplification 51.6%

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

Alternative 15: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;e^{re} \leq 0.99997683441022:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \leq 1.5:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)))
   (if (<= (exp re) 0.99997683441022)
     (/ t_0 (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
     (if (<= (exp re) 1.5)
       (cos im)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) t_0)))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double tmp;
	if (exp(re) <= 0.99997683441022) {
		tmp = t_0 / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else if (exp(re) <= 1.5) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	tmp = 0.0
	if (exp(re) <= 0.99997683441022)
		tmp = Float64(t_0 / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	elseif (exp(re) <= 1.5)
		tmp = cos(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[re], $MachinePrecision], 0.99997683441022], N[(t$95$0 / N[(N[(N[(N[(-0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[re], $MachinePrecision], 1.5], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.99997683441022001

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 68.7

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 54.8

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

   10. Applied rewrites 54.8%

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

   if 0.99997683441022001 < (exp.f64 re) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 1.5 < (exp.f64 re)

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 59.2%

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

Alternative 16: 51.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in im around inf

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in re around 0

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

       1. lower-+.f64 25.6

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

   11. Applied rewrites 25.6%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-cos.f64 N/A

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

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

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

      12. *-commutative N/A

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

      13. *-lft-identity N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in im around 0

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

      1. exp-neg N/A

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

      2. remove-double-div N/A

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

      3. lower-exp.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in re around 0

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

   10. Applied rewrites 71.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

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

Alternative 17: 50.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 9.6%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 21.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. remove-double-neg N/A

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

      4. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \cdot \cos im \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \cos im}{e^{\mathsf{neg}\left(re\right)}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos im \cdot 1}}{e^{\mathsf{neg}\left(re\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos im} \cdot 1}{e^{\mathsf{neg}\left(re\right)}} \]

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

          \[\leadsto \frac{\cos im \cdot \color{blue}{1}}{e^{\mathsf{neg}\left(re\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{\cos im}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \]

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 84.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (* im im) -0.5)
   (fma (fma (* 0.16666666666666666 re) re 1.0) re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma((0.16666666666666666 * re), re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = fma(fma(Float64(0.16666666666666666 * re), re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(0.16666666666666666 * re), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 29.3

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 29.3%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot \cos im} \]

      2. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{re}} \cdot \cos im \]

      3. remove-double-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)}} \cdot \cos im \]

      4. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \cdot \cos im \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \cos im}{e^{\mathsf{neg}\left(re\right)}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos im \cdot 1}}{e^{\mathsf{neg}\left(re\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos im} \cdot 1}{e^{\mathsf{neg}\left(re\right)}} \]

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

          \[\leadsto \frac{\cos im \cdot \color{blue}{1}}{e^{\mathsf{neg}\left(re\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{\cos im}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \]

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 84.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot re, re, 1\right), re, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 70.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot re, re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 50.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (* im im) -0.5)
   (fma (* (* re re) 0.16666666666666666) re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 29.3

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 29.3%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot \cos im} \]

      2. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{re}} \cdot \cos im \]

      3. remove-double-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)}} \cdot \cos im \]

      4. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \cdot \cos im \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \cos im}{e^{\mathsf{neg}\left(re\right)}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos im \cdot 1}}{e^{\mathsf{neg}\left(re\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos im} \cdot 1}{e^{\mathsf{neg}\left(re\right)}} \]

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

          \[\leadsto \frac{\cos im \cdot \color{blue}{1}}{e^{\mathsf{neg}\left(re\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{\cos im}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \]

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 84.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), \color{blue}{re}, 1\right) \]

   11. Taylor expanded in re around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot {re}^{2}, re, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 70.6%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 47.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (* im im) -0.5)
   (fma (fma 0.5 re 1.0) re 1.0)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = fma(fma(0.5, re, 1.0), re, 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 29.3

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 29.3%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot \cos im} \]

      2. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{re}} \cdot \cos im \]

      3. remove-double-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)}} \cdot \cos im \]

      4. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \cdot \cos im \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \cos im}{e^{\mathsf{neg}\left(re\right)}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos im \cdot 1}}{e^{\mathsf{neg}\left(re\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos im} \cdot 1}{e^{\mathsf{neg}\left(re\right)}} \]

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

          \[\leadsto \frac{\cos im \cdot \color{blue}{1}}{e^{\mathsf{neg}\left(re\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{\cos im}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \]

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 84.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{re \cdot \left(1 + \frac{1}{2} \cdot re\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), \color{blue}{re}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 38.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0) (* (* im im) -0.5) (+ 1.0 re)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = 1.0 + re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((exp(re) * cos(im)) <= 0.0d0) then
        tmp = (im * im) * (-0.5d0)
    else
        tmp = 1.0d0 + re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = 1.0 + re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.cos(im)) <= 0.0:
		tmp = (im * im) * -0.5
	else:
		tmp = 1.0 + re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * cos(im)) <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	else
		tmp = Float64(1.0 + re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * cos(im)) <= 0.0)
		tmp = (im * im) * -0.5;
	else
		tmp = 1.0 + re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], N[(1.0 + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\cos im} \]
   4. Step-by-step derivation

      1. lower-cos.f64 29.3

         \[\leadsto \color{blue}{\cos im} \]

   5. Applied rewrites 29.3%

      \[\leadsto \color{blue}{\cos im} \]

   6. Taylor expanded in im around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {im}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.6%

      \[\leadsto \mathsf{fma}\left(im \cdot im, \color{blue}{-0.5}, 1\right) \]

   9. Taylor expanded in im around inf

      \[\leadsto \frac{-1}{2} \cdot {im}^{\color{blue}{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.8%

       \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{e^{re} \cdot \cos im} \]

      2. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{re}} \cdot \cos im \]

      3. remove-double-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)}} \cdot \cos im \]

      4. rec-exp N/A

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \cdot \cos im \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \cos im}{e^{\mathsf{neg}\left(re\right)}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos im \cdot 1}}{e^{\mathsf{neg}\left(re\right)}} \]

      7. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos im} \cdot 1}{e^{\mathsf{neg}\left(re\right)}} \]

      8. sin-PI/2 N/A

         \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

      10. lift-cos.f64 N/A

          \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

      11. sin-PI/2 N/A

          \[\leadsto \frac{\cos im \cdot \color{blue}{1}}{e^{\mathsf{neg}\left(re\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\color{blue}{1 \cdot \cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{\cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{\cos im}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \]

      15. lower-neg.f64 100.0

          \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
   6. Step-by-step derivation

      1. exp-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

      2. remove-double-div N/A

         \[\leadsto \color{blue}{e^{re}} \]

      3. lower-exp.f64 84.1

         \[\leadsto \color{blue}{e^{re}} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{e^{re}} \]

   8. Taylor expanded in re around 0

      \[\leadsto 1 + \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.2%

       \[\leadsto 1 + \color{blue}{re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + re\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{re} \cdot \cos im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (exp re) (cos im)))

C

double code(double re, double im) {
	return exp(re) * cos(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = exp(re) * cos(im)
end function

Java

public static double code(double re, double im) {
	return Math.exp(re) * Math.cos(im);
}

Python

def code(re, im):
	return math.exp(re) * math.cos(im)

Julia

function code(re, im)
	return Float64(exp(re) * cos(im))
end

MATLAB

function tmp = code(re, im)
	tmp = exp(re) * cos(im);
end

Wolfram

code[re_, im_] := N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 23: 28.6% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ 1 + re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (+ 1.0 re))

C

double code(double re, double im) {
	return 1.0 + re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0 + re
end function

Java

public static double code(double re, double im) {
	return 1.0 + re;
}

Python

def code(re, im):
	return 1.0 + re

Julia

function code(re, im)
	return Float64(1.0 + re)
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0 + re;
end

Wolfram

code[re_, im_] := N[(1.0 + re), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{e^{re} \cdot \cos im} \]

   2. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{re}} \cdot \cos im \]

   3. remove-double-neg N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)}} \cdot \cos im \]

   4. rec-exp N/A

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \cdot \cos im \]

   5. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \cos im}{e^{\mathsf{neg}\left(re\right)}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\cos im \cdot 1}}{e^{\mathsf{neg}\left(re\right)}} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos im} \cdot 1}{e^{\mathsf{neg}\left(re\right)}} \]

   8. sin-PI/2 N/A

      \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

   10. lift-cos.f64 N/A

       \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

   11. sin-PI/2 N/A

       \[\leadsto \frac{\cos im \cdot \color{blue}{1}}{e^{\mathsf{neg}\left(re\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\color{blue}{1 \cdot \cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

   13. *-lft-identity N/A

       \[\leadsto \frac{\color{blue}{\cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{\cos im}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \]

   15. lower-neg.f64 100.0

       \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
6. Step-by-step derivation

   1. exp-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

   2. remove-double-div N/A

      \[\leadsto \color{blue}{e^{re}} \]

   3. lower-exp.f64 70.3

      \[\leadsto \color{blue}{e^{re}} \]

7. Applied rewrites 70.3%

   \[\leadsto \color{blue}{e^{re}} \]

8. Taylor expanded in re around 0

   \[\leadsto 1 + \color{blue}{re} \]
9. Step-by-step derivation

10. Applied rewrites 27.8%

    \[\leadsto 1 + \color{blue}{re} \]

11. Final simplification 27.8%

    \[\leadsto 1 + re \]
12. Add Preprocessing

Alternative 24: 28.1% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 1.0)

C

double code(double re, double im) {
	return 1.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 1.0d0
end function

Java

public static double code(double re, double im) {
	return 1.0;
}

Python

def code(re, im):
	return 1.0

Julia

function code(re, im)
	return 1.0
end

MATLAB

function tmp = code(re, im)
	tmp = 1.0;
end

Wolfram

code[re_, im_] := 1.0

Derivation

1. Initial program 100.0%

   \[e^{re} \cdot \cos im \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{e^{re} \cdot \cos im} \]

   2. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{re}} \cdot \cos im \]

   3. remove-double-neg N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(re\right)\right)\right)}} \cdot \cos im \]

   4. rec-exp N/A

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \cdot \cos im \]

   5. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \cos im}{e^{\mathsf{neg}\left(re\right)}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\cos im \cdot 1}}{e^{\mathsf{neg}\left(re\right)}} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos im} \cdot 1}{e^{\mathsf{neg}\left(re\right)}} \]

   8. sin-PI/2 N/A

      \[\leadsto \frac{\cos im \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}{e^{\mathsf{neg}\left(re\right)}} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos im \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}}} \]

   10. lift-cos.f64 N/A

       \[\leadsto \frac{\color{blue}{\cos im} \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)}{e^{\mathsf{neg}\left(re\right)}} \]

   11. sin-PI/2 N/A

       \[\leadsto \frac{\cos im \cdot \color{blue}{1}}{e^{\mathsf{neg}\left(re\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\color{blue}{1 \cdot \cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

   13. *-lft-identity N/A

       \[\leadsto \frac{\color{blue}{\cos im}}{e^{\mathsf{neg}\left(re\right)}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{\cos im}{\color{blue}{e^{\mathsf{neg}\left(re\right)}}} \]

   15. lower-neg.f64 100.0

       \[\leadsto \frac{\cos im}{e^{\color{blue}{-re}}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\cos im}{e^{-re}}} \]

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{neg}\left(re\right)}}} \]
6. Step-by-step derivation

   1. exp-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{re}}}} \]

   2. remove-double-div N/A

      \[\leadsto \color{blue}{e^{re}} \]

   3. lower-exp.f64 70.3

      \[\leadsto \color{blue}{e^{re}} \]

7. Applied rewrites 70.3%

   \[\leadsto \color{blue}{e^{re}} \]

8. Taylor expanded in re around 0

   \[\leadsto 1 \]
9. Step-by-step derivation

10. Applied rewrites 27.5%

    \[\leadsto 1 \]

11. Final simplification 27.5%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))