math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 18

Speedup: 1.0×

• 24.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

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 2: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999999999999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))) (t_1 (fma (fma 0.5 re 1.0) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (* t_1 (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.05)
       (cos im)
       (if (or (<= t_0 0.0) (not (<= t_0 0.999999999999999)))
         (exp re)
         (* t_1 (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(fma(0.5, re, 1.0), re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.05) {
		tmp = cos(im);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.999999999999999)) {
		tmp = exp(re);
	} else {
		tmp = t_1 * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(fma(0.5, re, 1.0), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.999999999999999))
		tmp = exp(re);
	else
		tmp = Float64(t_1 * cos(im));
	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[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Cos[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999999999999]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(t$95$1 * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999999999999001 < (*.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%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999999999999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999999999999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.05)
       (cos im)
       (if (or (<= t_0 0.0) (not (<= t_0 0.999999999999999)))
         (exp re)
         (/ (cos im) (- 1.0 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(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.05) {
		tmp = cos(im);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.999999999999999)) {
		tmp = exp(re);
	} else {
		tmp = cos(im) / (1.0 - 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(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.999999999999999))
		tmp = exp(re);
	else
		tmp = Float64(cos(im) / Float64(1.0 - 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[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Cos[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999999999999]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[Cos[im], $MachinePrecision] / N[(1.0 - re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999999999999001 < (*.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%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999001

   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

      \[\leadsto \frac{\cos im}{\color{blue}{1 + -1 \cdot re}} \]
   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 97.0

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

   7. Applied rewrites 97.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999999999999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos im}{1 - re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999999999999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))
     (if (<= t_0 -0.05)
       (cos im)
       (if (or (<= t_0 0.0) (not (<= t_0 0.999999999999999)))
         (exp re)
         (* (+ 1.0 re) (cos im)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if (t_0 <= -0.05) {
		tmp = cos(im);
	} else if ((t_0 <= 0.0) || !(t_0 <= 0.999999999999999)) {
		tmp = exp(re);
	} else {
		tmp = (1.0 + re) * cos(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif ((t_0 <= 0.0) || !(t_0 <= 0.999999999999999))
		tmp = exp(re);
	else
		tmp = Float64(Float64(1.0 + re) * cos(im));
	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 + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Cos[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999999999999]], $MachinePrecision]], N[Exp[re], $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

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

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999999999999001 < (*.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%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999001

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999999999999\right):\\ \;\;\;\;e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \cos im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05 \lor \neg \left(t\_0 \leq 0 \lor \neg \left(t\_0 \leq 0.999999999999999\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 (- INFINITY))
     (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))
     (if (or (<= t_0 -0.05)
             (not (or (<= t_0 0.0) (not (<= t_0 0.999999999999999)))))
       (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(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	} else if ((t_0 <= -0.05) || !((t_0 <= 0.0) || !(t_0 <= 0.999999999999999))) {
		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 = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 1.0));
	elseif ((t_0 <= -0.05) || !((t_0 <= 0.0) || !(t_0 <= 0.999999999999999)))
		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[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.05], N[Not[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 0.999999999999999]], $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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999001

   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 97.0

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

   5. Applied rewrites 97.0%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999999999999001 < (*.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 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05 \lor \neg \left(e^{re} \cdot \cos im \leq 0 \lor \neg \left(e^{re} \cdot \cos im \leq 0.999999999999999\right)\right):\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.99999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (fma 0.5 re 1.0) re 1.0) t_0)
     (if (<= t_1 -0.05)
       (cos im)
       (if (<= t_1 0.0)
         (/ t_0 (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
         (if (<= t_1 0.99999)
           (cos im)
           (*
            (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
            (fma
             (- (* 0.041666666666666664 (* im im)) 0.5)
             (* im im)
             1.0))))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * t_0;
	} else if (t_1 <= -0.05) {
		tmp = cos(im);
	} else if (t_1 <= 0.0) {
		tmp = t_0 / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else if (t_1 <= 0.99999) {
		tmp = cos(im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * t_0);
	elseif (t_1 <= -0.05)
		tmp = cos(im);
	elseif (t_1 <= 0.0)
		tmp = Float64(t_0 / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	elseif (t_1 <= 0.99999)
		tmp = cos(im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$1, 0.0], 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[t$95$1, 0.99999], N[Cos[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999990000000000046

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if -0.050000000000000003 < (*.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 69.1

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

   7. Applied rewrites 69.1%

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

   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 50.1

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

   10. Applied rewrites 50.1%

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

   if 0.999990000000000046 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}}, {im}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      9. lower-*.f64 82.3

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot re} + 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), re, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right) \cdot re} + 1, re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot re, re, 1\right)}, re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       8. lower-fma.f64 96.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right) \]

   11. Applied rewrites 96.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.99999:\\ \;\;\;\;\cos im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)) (t_1 (* (exp re) (cos im))))
   (if (<= t_1 (- INFINITY))
     (* (fma (fma 0.5 re 1.0) re 1.0) t_0)
     (if (<= t_1 0.0)
       (/ t_0 (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
       (if (<= t_1 0.9995)
         (* (pow im -1.0) im)
         (*
          (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0)
          (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_0 / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else if (t_1 <= 0.9995) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(t_0 / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	elseif (t_1 <= 0.9995)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], 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[t$95$1, 0.9995], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   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. 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 79.2

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

   7. Applied rewrites 79.2%

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

   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 34.9

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

   10. Applied rewrites 34.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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 95.0

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

   5. Applied rewrites 95.0%

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

   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 0.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.8%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 21.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}}, {im}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      9. lower-*.f64 82.5

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

   8. Applied rewrites 82.5%

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

   9. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + 1\right)} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) \cdot re} + 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right), re, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right) + 1}, re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot re\right) \cdot re} + 1, re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot re, re, 1\right)}, re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot re + \frac{1}{2}}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, im \cdot im, 1\right) \]

       8. lower-fma.f64 96.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, re, 0.5\right)}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right) \]

   11. Applied rewrites 96.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right)} \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0))
        (t_1 (* (exp re) (cos im)))
        (t_2 (fma (fma 0.5 re 1.0) re 1.0)))
   (if (<= t_1 (- INFINITY))
     (* t_2 t_0)
     (if (<= t_1 0.0)
       (/ t_0 (fma (- (* (fma -0.16666666666666666 re 0.5) re) 1.0) re 1.0))
       (if (<= t_1 0.9995)
         (* (pow im -1.0) im)
         (*
          t_2
          (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double t_2 = fma(fma(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 / fma(((fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0);
	} else if (t_1 <= 0.9995) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = t_2 * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	t_2 = fma(fma(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 = Float64(t_0 / fma(Float64(Float64(fma(-0.16666666666666666, re, 0.5) * re) - 1.0), re, 1.0));
	elseif (t_1 <= 0.9995)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(t_2 * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * 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], 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[t$95$1, 0.9995], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(t$95$2 * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   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. 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 79.2

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

   7. Applied rewrites 79.2%

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

   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 34.9

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

   10. Applied rewrites 34.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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 95.0

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

   5. Applied rewrites 95.0%

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

   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 0.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.8%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 21.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}}, {im}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      9. lower-*.f64 93.4

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

   8. Applied rewrites 93.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, 0.5\right) \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ t_1 := e^{re} \cdot \cos im\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.16666666666666666 - 1, re, 1\right)}\\ \mathbf{elif}\;t\_1 \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0))
        (t_1 (* (exp re) (cos im)))
        (t_2 (fma (fma 0.5 re 1.0) re 1.0)))
   (if (<= t_1 (- INFINITY))
     (* t_2 t_0)
     (if (<= t_1 0.0)
       (/ t_0 (fma (- (* (* re re) -0.16666666666666666) 1.0) re 1.0))
       (if (<= t_1 0.9995)
         (* (pow im -1.0) im)
         (*
          t_2
          (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = fma((im * im), -0.5, 1.0);
	double t_1 = exp(re) * cos(im);
	double t_2 = fma(fma(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 / fma((((re * re) * -0.16666666666666666) - 1.0), re, 1.0);
	} else if (t_1 <= 0.9995) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = t_2 * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(Float64(im * im), -0.5, 1.0)
	t_1 = Float64(exp(re) * cos(im))
	t_2 = fma(fma(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 = Float64(t_0 / fma(Float64(Float64(Float64(re * re) * -0.16666666666666666) - 1.0), re, 1.0));
	elseif (t_1 <= 0.9995)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(t_2 * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 * 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], N[(t$95$0 / N[(N[(N[(N[(re * re), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9995], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(t$95$2 * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + \frac{1}{2} \cdot re\right)\right)} \cdot \cos im \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   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. 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 79.2

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

   7. Applied rewrites 79.2%

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

   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 34.9

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

   10. Applied rewrites 34.9%

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

   11. Taylor expanded in re around inf

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

   13. Applied rewrites 34.9%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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 95.0

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

   5. Applied rewrites 95.0%

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

   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 0.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.8%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 21.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}}, {im}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      9. lower-*.f64 93.4

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

   8. Applied rewrites 93.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(\left(re \cdot re\right) \cdot -0.16666666666666666 - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right)\\ t_2 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;t\_0 \leq -0.96:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(0.5 \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (fma (fma 0.5 re 1.0) re 1.0))
        (t_2 (fma (* im im) -0.5 1.0)))
   (if (<= t_0 -0.96)
     (* t_1 t_2)
     (if (<= t_0 0.0)
       (/ t_2 (fma (- (* 0.5 re) 1.0) re 1.0))
       (if (<= t_0 0.9995)
         (* (pow im -1.0) im)
         (*
          t_1
          (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = fma(fma(0.5, re, 1.0), re, 1.0);
	double t_2 = fma((im * im), -0.5, 1.0);
	double tmp;
	if (t_0 <= -0.96) {
		tmp = t_1 * t_2;
	} else if (t_0 <= 0.0) {
		tmp = t_2 / fma(((0.5 * re) - 1.0), re, 1.0);
	} else if (t_0 <= 0.9995) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = t_1 * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = fma(fma(0.5, re, 1.0), re, 1.0)
	t_2 = fma(Float64(im * im), -0.5, 1.0)
	tmp = 0.0
	if (t_0 <= -0.96)
		tmp = Float64(t_1 * t_2);
	elseif (t_0 <= 0.0)
		tmp = Float64(t_2 / fma(Float64(Float64(0.5 * re) - 1.0), re, 1.0));
	elseif (t_0 <= 0.9995)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(t_1 * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 1.0));
	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[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.96], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(t$95$2 / N[(N[(N[(0.5 * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(t$95$1 * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (cos.f64 im)) < -0.95999999999999996

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.7

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

   8. Applied rewrites 69.7%

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

   if -0.95999999999999996 < (*.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 78.0

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

   7. Applied rewrites 78.0%

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

   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 36.7

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

   10. Applied rewrites 36.7%

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

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 24.1

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

   13. Applied rewrites 24.1%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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 95.0

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

   5. Applied rewrites 95.0%

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

   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 0.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.8%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 21.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}}, {im}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      9. lower-*.f64 93.4

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

   8. Applied rewrites 93.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -0.96:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(0.5 \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* (* im im) -0.5)
     (if (<= t_0 0.9995)
       (* (pow im -1.0) im)
       (*
        (+ 1.0 re)
        (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 0.9995) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = (1.0 + re) * fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (t_0 <= 0.9995)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(Float64(1.0 + re) * fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 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, 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] * N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 31.5

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

   5. Applied rewrites 31.5%

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

   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 11.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.4%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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 95.0

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

   5. Applied rewrites 95.0%

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

   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 0.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.8%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 21.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 70.8

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

   5. Applied rewrites 70.8%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(1 + re\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}\right) \cdot {im}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}, {im}^{2}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} - \frac{1}{2}}, {im}^{2}, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2}} - \frac{1}{2}, {im}^{2}, 1\right) \]

      6. unpow2 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(im \cdot im\right)} - \frac{1}{2}, {im}^{2}, 1\right) \]

      8. unpow2 N/A

         \[\leadsto \left(1 + re\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot \left(im \cdot im\right) - \frac{1}{2}, \color{blue}{im \cdot im}, 1\right) \]

      9. lower-*.f64 82.5

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

   8. Applied rewrites 82.5%

      \[\leadsto \left(1 + re\right) \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* (* im im) -0.5)
     (if (<= t_0 0.9995)
       (* (pow im -1.0) im)
       (* (fma (fma 0.5 re 1.0) re 1.0) (fma (* im im) -0.5 1.0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 0.9995) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * fma((im * im), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (t_0 <= 0.9995)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * fma(Float64(im * im), -0.5, 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, 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 31.5

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

   5. Applied rewrites 31.5%

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

   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 11.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.4%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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 95.0

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

   5. Applied rewrites 95.0%

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

   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 0.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.8%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 21.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 80.6

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

   8. Applied rewrites 80.6%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im))))
   (if (<= t_0 0.0)
     (* (* im im) -0.5)
     (if (<= t_0 0.9995)
       (* (pow im -1.0) im)
       (fma (- (* 0.041666666666666664 (* im im)) 0.5) (* im im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * im) * -0.5;
	} else if (t_0 <= 0.9995) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = fma(((0.041666666666666664 * (im * im)) - 0.5), (im * im), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (t_0 <= 0.9995)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = fma(Float64(Float64(0.041666666666666664 * Float64(im * im)) - 0.5), Float64(im * im), 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, 0.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 0.9995], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 31.5

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

   5. Applied rewrites 31.5%

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

   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 11.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.4%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99950000000000006

   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 95.0

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

   5. Applied rewrites 95.0%

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

   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 0.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 0.8%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 21.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 0.99950000000000006 < (*.f64 (exp.f64 re) (cos.f64 im))

   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 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 78.4%

      \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, \color{blue}{im \cdot im}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9995:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.041666666666666664 \cdot \left(im \cdot im\right) - 0.5, im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0) (* (* im im) -0.5) (* (pow im -1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * cos(im)) <= 0.0) {
		tmp = (im * im) * -0.5;
	} else {
		tmp = pow(im, -1.0) * im;
	}
	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 = (im ** (-1.0d0)) * im
    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 = Math.pow(im, -1.0) * im;
	}
	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 = math.pow(im, -1.0) * im
	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((im ^ -1.0) * im);
	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 = (im ^ -1.0) * im;
	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[(N[Power[im, -1.0], $MachinePrecision] * im), $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 31.5

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

   5. Applied rewrites 31.5%

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

   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 11.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 75.2

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

   5. Applied rewrites 75.2%

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

   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 52.0%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 37.3%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 56.2%

       \[\leadsto \frac{1}{im} \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

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

Alternative 15: 91.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 re) < 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 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%

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

   if 0.0 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 93.8

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

   5. Applied rewrites 93.8%

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

Alternative 16: 34.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (cos im)) 0.0)
   (* (* im im) -0.5)
   (fma (* im im) -0.5 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((im * im), -0.5, 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(im * im), -0.5, 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[(im * im), $MachinePrecision] * -0.5 + 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 31.5

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

   5. Applied rewrites 31.5%

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

   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 11.2%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 23.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 75.2

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

   5. Applied rewrites 75.2%

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

   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 52.0%

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

4. Final simplification 40.1%

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

Alternative 17: 50.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \mathbf{if}\;re \leq -1.75 \cdot 10^{+151}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(0.5 \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;re \leq -510:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-22}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (* im im) -0.5 1.0)))
   (if (<= re -1.75e+151)
     (/ t_0 (fma (- (* 0.5 re) 1.0) re 1.0))
     (if (<= re -510.0)
       (* (* im im) -0.5)
       (if (<= re 5.2e-22)
         (* (pow im -1.0) im)
         (* (fma (fma 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 (re <= -1.75e+151) {
		tmp = t_0 / fma(((0.5 * re) - 1.0), re, 1.0);
	} else if (re <= -510.0) {
		tmp = (im * im) * -0.5;
	} else if (re <= 5.2e-22) {
		tmp = pow(im, -1.0) * im;
	} else {
		tmp = fma(fma(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 (re <= -1.75e+151)
		tmp = Float64(t_0 / fma(Float64(Float64(0.5 * re) - 1.0), re, 1.0));
	elseif (re <= -510.0)
		tmp = Float64(Float64(im * im) * -0.5);
	elseif (re <= 5.2e-22)
		tmp = Float64((im ^ -1.0) * im);
	else
		tmp = Float64(fma(fma(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[re, -1.75e+151], N[(t$95$0 / N[(N[(N[(0.5 * re), $MachinePrecision] - 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -510.0], N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[re, 5.2e-22], N[(N[Power[im, -1.0], $MachinePrecision] * im), $MachinePrecision], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.7500000000000001e151

   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 100.0

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

   7. Applied rewrites 100.0%

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

   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 68.8

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

   10. Applied rewrites 68.8%

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

   11. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 60.3

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

   13. Applied rewrites 60.3%

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

   if -1.7500000000000001e151 < re < -510

   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 3.1

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

   5. Applied rewrites 3.1%

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

   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 2.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 32.8%

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

   if -510 < re < 5.2e-22

   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 98.4

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

   5. Applied rewrites 98.4%

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

   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 52.8%

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

   9. Taylor expanded in im around inf

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

   11. Applied rewrites 29.3%

       \[\leadsto \left(\left(\frac{1}{im \cdot im} - 0.5\right) \cdot im\right) \cdot im \]

   12. Taylor expanded in im around 0

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

   14. Applied rewrites 56.6%

       \[\leadsto \frac{1}{im} \cdot im \]

   if 5.2e-22 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \cos im \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in im around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 57.5

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

   8. Applied rewrites 57.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(im \cdot im, -0.5, 1\right)}{\mathsf{fma}\left(0.5 \cdot re - 1, re, 1\right)}\\ \mathbf{elif}\;re \leq -510:\\ \;\;\;\;\left(im \cdot im\right) \cdot -0.5\\ \mathbf{elif}\;re \leq 5.2 \cdot 10^{-22}:\\ \;\;\;\;{im}^{-1} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \mathsf{fma}\left(im \cdot im, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 11.1% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \left(im \cdot im\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* im im) -0.5))

C

double code(double re, double im) {
	return (im * im) * -0.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (im * im) * (-0.5d0)
end function

Java

public static double code(double re, double im) {
	return (im * im) * -0.5;
}

Python

def code(re, im):
	return (im * im) * -0.5

Julia

function code(re, im)
	return Float64(Float64(im * im) * -0.5)
end

MATLAB

function tmp = code(re, im)
	tmp = (im * im) * -0.5;
end

Wolfram

code[re_, im_] := N[(N[(im * im), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

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 57.1

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

5. Applied rewrites 57.1%

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

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 35.1%

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

9. Taylor expanded in im around inf

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

11. Applied rewrites 10.7%

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

12. Final simplification 10.7%

    \[\leadsto \left(im \cdot im\right) \cdot -0.5 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))