math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 17.6s

Alternatives: 16

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right)\\ t_1 := e^{re} \cdot \sin im\\ t_2 := \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right)\\ t_3 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;t\_1 \leq -0.005:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, t\_2, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-128}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin im \cdot \frac{\mathsf{fma}\left(t\_0, t\_0, -1\right)}{\mathsf{fma}\left(re, t\_2, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (fma (fma re 0.16666666666666666 0.5) (* re re) re))
        (t_1 (* (exp re) (sin im)))
        (t_2 (fma re (fma re 0.16666666666666666 0.5) 1.0))
        (t_3 (* (exp re) im)))
   (if (<= t_1 -1e+185)
     (* (* im (* re re)) (fma (* im im) -0.08333333333333333 0.5))
     (if (<= t_1 -0.005)
       (* (sin im) (fma re t_2 1.0))
       (if (<= t_1 1e-128)
         t_3
         (if (<= t_1 1.0)
           (* (sin im) (/ (fma t_0 t_0 -1.0) (fma re t_2 -1.0)))
           t_3))))))

C

double code(double re, double im) {
	double t_0 = fma(fma(re, 0.16666666666666666, 0.5), (re * re), re);
	double t_1 = exp(re) * sin(im);
	double t_2 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0);
	double t_3 = exp(re) * im;
	double tmp;
	if (t_1 <= -1e+185) {
		tmp = (im * (re * re)) * fma((im * im), -0.08333333333333333, 0.5);
	} else if (t_1 <= -0.005) {
		tmp = sin(im) * fma(re, t_2, 1.0);
	} else if (t_1 <= 1e-128) {
		tmp = t_3;
	} else if (t_1 <= 1.0) {
		tmp = sin(im) * (fma(t_0, t_0, -1.0) / fma(re, t_2, -1.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), re)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = fma(re, fma(re, 0.16666666666666666, 0.5), 1.0)
	t_3 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= -1e+185)
		tmp = Float64(Float64(im * Float64(re * re)) * fma(Float64(im * im), -0.08333333333333333, 0.5));
	elseif (t_1 <= -0.005)
		tmp = Float64(sin(im) * fma(re, t_2, 1.0));
	elseif (t_1 <= 1e-128)
		tmp = t_3;
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(im) * Float64(fma(t_0, t_0, -1.0) / fma(re, t_2, -1.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+185], N[(N[(im * N[(re * re), $MachinePrecision]), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.005], N[(N[Sin[im], $MachinePrecision] * N[(re * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-128], t$95$3, If[LessEqual[t$95$1, 1.0], N[(N[Sin[im], $MachinePrecision] * N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] / N[(re * t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -9.9999999999999998e184

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 53.1

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

   5. Simplified 53.1%

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

   6. Taylor expanded in re around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 53.3

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

   8. Simplified 53.3%

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

   9. Taylor expanded in im around 0

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

       1. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{12} \cdot \left({im}^{2} \cdot {re}^{2}\right)\right) \cdot im + \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot {re}^{2}\right)} \cdot im + \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot \left({re}^{2} \cdot im\right)} + \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im \]

       4. *-commutative N/A

          \[\leadsto \left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot \color{blue}{\left(im \cdot {re}^{2}\right)} + \left(\frac{1}{2} \cdot {re}^{2}\right) \cdot im \]

       5. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{2}\right) + \color{blue}{\frac{1}{2} \cdot \left({re}^{2} \cdot im\right)} \]

       6. *-commutative N/A

          \[\leadsto \left(\frac{-1}{12} \cdot {im}^{2}\right) \cdot \left(im \cdot {re}^{2}\right) + \frac{1}{2} \cdot \color{blue}{\left(im \cdot {re}^{2}\right)} \]

       7. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(im \cdot {re}^{2}\right) \cdot \left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(im \cdot {re}^{2}\right) \cdot \left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(im \cdot {re}^{2}\right)} \cdot \left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right) \]

       10. unpow2 N/A

           \[\leadsto \left(im \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(im \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{-1}{12} \cdot {im}^{2} + \frac{1}{2}\right) \]

       12. *-commutative N/A

           \[\leadsto \left(im \cdot \left(re \cdot re\right)\right) \cdot \left(\color{blue}{{im}^{2} \cdot \frac{-1}{12}} + \frac{1}{2}\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto \left(im \cdot \left(re \cdot re\right)\right) \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{-1}{12}, \frac{1}{2}\right)} \]

       14. unpow2 N/A

           \[\leadsto \left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{-1}{12}, \frac{1}{2}\right) \]

       15. lower-*.f64 49.1

           \[\leadsto \left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, -0.08333333333333333, 0.5\right) \]

   11. Simplified 49.1%

       \[\leadsto \color{blue}{\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)} \]

   if -9.9999999999999998e184 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.2

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

   5. Simplified 99.2%

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

   if -0.0050000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.00000000000000005e-128 or 1 < (*.f64 (exp.f64 re) (sin.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 94.2

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

   5. Simplified 94.2%

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

   if 1.00000000000000005e-128 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.1

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

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \sin im \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   7. Applied egg-rr 99.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\left(im \cdot \left(re \cdot re\right)\right) \cdot \mathsf{fma}\left(im \cdot im, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.005:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-128}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 1:\\ \;\;\;\;\sin im \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, re\right), -1\right)}{\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))