math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 15.1s

Alternatives: 17

Speedup: 1.0×

• 24.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-53} \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 -0.1)
       (sin im)
       (if (or (<= t_0 5e-53) (not (<= t_0 1.0)))
         (* im (exp re))
         (* (+ 1.0 re) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if ((t_0 <= 5e-53) || !(t_0 <= 1.0)) {
		tmp = im * exp(re);
	} else {
		tmp = (1.0 + re) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif ((t_0 <= 5e-53) || !(t_0 <= 1.0))
		tmp = Float64(im * exp(re));
	else
		tmp = Float64(Float64(1.0 + re) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[Or[LessEqual[t$95$0, 5e-53], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-sin.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5e-53 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 5e-53 < (*.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

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

      1. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 5 \cdot 10^{-53} \lor \neg \left(\sin im \cdot e^{re} \leq 1\right):\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1 \lor \neg \left(t\_0 \leq 10^{-20} \lor \neg \left(t\_0 \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (or (<= t_0 -0.1) (not (or (<= t_0 1e-20) (not (<= t_0 1.0)))))
       (sin im)
       (* im (exp re))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if ((t_0 <= -0.1) || !((t_0 <= 1e-20) || !(t_0 <= 1.0))) {
		tmp = sin(im);
	} else {
		tmp = im * exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif ((t_0 <= -0.1) || !((t_0 <= 1e-20) || !(t_0 <= 1.0)))
		tmp = sin(im);
	else
		tmp = Float64(im * exp(re));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[Or[LessEqual[t$95$0, -0.1], N[Not[Or[LessEqual[t$95$0, 1e-20], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]]], $MachinePrecision]], N[Sin[im], $MachinePrecision], N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 9.99999999999999945e-21 < (*.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

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

      1. lower-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999945e-21 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 93.5

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

   5. Applied rewrites 93.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.1 \lor \neg \left(\sin im \cdot e^{re} \leq 10^{-20} \lor \neg \left(\sin im \cdot e^{re} \leq 1\right)\right):\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-20}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{re - 1}{t\_1} \cdot \mathsf{fma}\left(-0.019444444444444445, im \cdot im, -0.16666666666666666\right), im \cdot im, \frac{1 - re}{t\_1}\right)}{im}\right)}^{-1}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 -0.1)
       (sin im)
       (if (<= t_0 1e-20)
         (pow
          (/
           (fma
            (*
             (/ (- re 1.0) t_1)
             (fma -0.019444444444444445 (* im im) -0.16666666666666666))
            (* im im)
            (/ (- 1.0 re) t_1))
           im)
          -1.0)
         (if (<= t_0 1.0)
           (sin im)
           (* (* (* (fma 0.16666666666666666 re 0.5) re) re) im)))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if (t_0 <= -0.1) {
		tmp = sin(im);
	} else if (t_0 <= 1e-20) {
		tmp = pow((fma((((re - 1.0) / t_1) * fma(-0.019444444444444445, (im * im), -0.16666666666666666)), (im * im), ((1.0 - re) / t_1)) / im), -1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif (t_0 <= -0.1)
		tmp = sin(im);
	elseif (t_0 <= 1e-20)
		tmp = Float64(fma(Float64(Float64(Float64(re - 1.0) / t_1) * fma(-0.019444444444444445, Float64(im * im), -0.16666666666666666)), Float64(im * im), Float64(Float64(1.0 - re) / t_1)) / im) ^ -1.0;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, -0.1], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-20], N[Power[N[(N[(N[(N[(N[(re - 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(-0.019444444444444445 * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(1.0 - re), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 38.8%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001 or 9.99999999999999945e-21 < (*.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

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

      1. lower-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.99999999999999945e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 54.2

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

   5. Applied rewrites 54.2%

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in im around 0

      \[\leadsto \frac{1}{\frac{\left({im}^{2} \cdot \left(-1 \cdot \left({im}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{1 - re}{1 - {re}^{2}} + \frac{1}{120} \cdot \frac{1 - re}{1 - {re}^{2}}\right)\right) - \frac{-1}{6} \cdot \frac{1 - re}{1 - {re}^{2}}\right) + \frac{1}{1 - {re}^{2}}\right) - \frac{re}{1 - {re}^{2}}}{\color{blue}{im}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.4%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-\frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)} \cdot \mathsf{fma}\left(-0.019444444444444445, im \cdot im, -0.16666666666666666\right), im \cdot im, \frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)}\right)}{\color{blue}{im}}} \]

   if 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 49.2%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 49.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-20}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{re - 1}{\mathsf{fma}\left(-re, re, 1\right)} \cdot \mathsf{fma}\left(-0.019444444444444445, im \cdot im, -0.16666666666666666\right), im \cdot im, \frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)}\right)}{im}\right)}^{-1}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{if}\;t\_0 \leq -0.32:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{re - 1}{t\_1} \cdot \mathsf{fma}\left(-0.019444444444444445, im \cdot im, -0.16666666666666666\right), im \cdot im, \frac{1 - re}{t\_1}\right)}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (fma (- re) re 1.0)))
   (if (<= t_0 -0.32)
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 0.5)
       (pow
        (/
         (fma
          (*
           (/ (- re 1.0) t_1)
           (fma -0.019444444444444445 (* im im) -0.16666666666666666))
          (* im im)
          (/ (- 1.0 re) t_1))
         im)
        -1.0)
       (* (* (* (fma 0.16666666666666666 re 0.5) re) re) im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = fma(-re, re, 1.0);
	double tmp;
	if (t_0 <= -0.32) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if (t_0 <= 0.5) {
		tmp = pow((fma((((re - 1.0) / t_1) * fma(-0.019444444444444445, (im * im), -0.16666666666666666)), (im * im), ((1.0 - re) / t_1)) / im), -1.0);
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = fma(Float64(-re), re, 1.0)
	tmp = 0.0
	if (t_0 <= -0.32)
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif (t_0 <= 0.5)
		tmp = Float64(fma(Float64(Float64(Float64(re - 1.0) / t_1) * fma(-0.019444444444444445, Float64(im * im), -0.16666666666666666)), Float64(im * im), Float64(Float64(1.0 - re) / t_1)) / im) ^ -1.0;
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-re) * re + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.32], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[Power[N[(N[(N[(N[(N[(re - 1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(-0.019444444444444445 * N[(im * im), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(N[(1.0 - re), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.320000000000000007

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 18.2%

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

   if -0.320000000000000007 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 58.1%

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

   8. Taylor expanded in im around 0

      \[\leadsto \frac{1}{\frac{\left({im}^{2} \cdot \left(-1 \cdot \left({im}^{2} \cdot \left(\frac{-1}{36} \cdot \frac{1 - re}{1 - {re}^{2}} + \frac{1}{120} \cdot \frac{1 - re}{1 - {re}^{2}}\right)\right) - \frac{-1}{6} \cdot \frac{1 - re}{1 - {re}^{2}}\right) + \frac{1}{1 - {re}^{2}}\right) - \frac{re}{1 - {re}^{2}}}{\color{blue}{im}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.5%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-\frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)} \cdot \mathsf{fma}\left(-0.019444444444444445, im \cdot im, -0.16666666666666666\right), im \cdot im, \frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)}\right)}{\color{blue}{im}}} \]

   if 0.5 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.5%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.32:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0.5:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(\frac{re - 1}{\mathsf{fma}\left(-re, re, 1\right)} \cdot \mathsf{fma}\left(-0.019444444444444445, im \cdot im, -0.16666666666666666\right), im \cdot im, \frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)}\right)}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(0.16666666666666666, im \cdot im, 1\right) \cdot \frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)}}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.1)
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 0.0)
       (pow
        (/
         (*
          (fma 0.16666666666666666 (* im im) 1.0)
          (/ (- 1.0 re) (fma (- re) re 1.0)))
         im)
        -1.0)
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if (t_0 <= 0.0) {
		tmp = pow(((fma(0.16666666666666666, (im * im), 1.0) * ((1.0 - re) / fma(-re, re, 1.0))) / im), -1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(0.16666666666666666, Float64(im * im), 1.0) * Float64(Float64(1.0 - re) / fma(Float64(-re), re, 1.0))) / im) ^ -1.0;
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Power[N[(N[(N[(0.16666666666666666 * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(1.0 - re), $MachinePrecision] / N[((-re) * re + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / im), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 16.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 38.7

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

   5. Applied rewrites 38.7%

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 52.5%

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

   if 0.0 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.7%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(0.16666666666666666, im \cdot im, 1\right) \cdot \frac{1 - re}{\mathsf{fma}\left(-re, re, 1\right)}}{im}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im}{1 - re} \cdot \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.1)
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 0.0)
       (* (/ im (- 1.0 re)) (fma (- re) re 1.0))
       (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if (t_0 <= 0.0) {
		tmp = (im / (1.0 - re)) * fma(-re, re, 1.0);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(im / Float64(1.0 - re)) * fma(Float64(-re), re, 1.0));
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(im / N[(1.0 - re), $MachinePrecision]), $MachinePrecision] * N[((-re) * re + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 16.9%

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

   if -0.10000000000000001 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 38.7

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

   5. Applied rewrites 38.7%

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in im around 0

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

   10. Applied rewrites 41.7%

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

   if 0.0 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 46.7%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\frac{im}{1 - re} \cdot \mathsf{fma}\left(-re, re, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.002)
   (fma
    (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
    im
    im)
   (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.002) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.002)
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.0%

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

   if 2e-3 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 31.8

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.6%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.002)
   (fma
    (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
    im
    im)
   (* (fma (* (* re re) 0.16666666666666666) re 1.0) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.002) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else {
		tmp = fma(((re * re) * 0.16666666666666666), re, 1.0) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.002)
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	else
		tmp = Float64(fma(Float64(Float64(re * re) * 0.16666666666666666), re, 1.0) * im);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.002], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], N[(N[(N[(N[(re * re), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-3

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 43.0%

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

   if 2e-3 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 31.8

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 22.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 22.6%

       \[\leadsto \mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.5

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.4%

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

   if 0.5 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 25.1%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 25.5%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

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

Alternative 11: 36.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.5

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 40.4%

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

   if 0.5 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 21.1%

       \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

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

Alternative 12: 34.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.5

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 39.7%

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

   if 0.5 < (*.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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 21.1%

       \[\leadsto \left(\left(re \cdot re\right) \cdot im\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

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

Alternative 13: 39.8% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 65.0

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

5. Applied rewrites 65.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 38.1%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 37.5%

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

Alternative 14: 37.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (fma (fma 0.5 re 1.0) re 1.0) im))

C

double code(double re, double im) {
	return fma(fma(0.5, re, 1.0), re, 1.0) * im;
}

Julia

function code(re, im)
	return Float64(fma(fma(0.5, re, 1.0), re, 1.0) * im)
end

Wolfram

code[re_, im_] := N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * im), $MachinePrecision]

Derivation

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 65.0

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

5. Applied rewrites 65.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 36.1%

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

Alternative 15: 28.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{+29}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 1.15e+29) (* 1.0 im) (* im re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.15e+29) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.15d+29) then
        tmp = 1.0d0 * im
    else
        tmp = im * re
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.15e+29) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.15e+29:
		tmp = 1.0 * im
	else:
		tmp = im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.15e+29)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(im * re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.15e+29)
		tmp = 1.0 * im;
	else
		tmp = im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.15e+29], N[(1.0 * im), $MachinePrecision], N[(im * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.1500000000000001e29

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 73.4

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in re around 0

      \[\leadsto 1 \cdot im \]
   7. Step-by-step derivation

   8. Applied rewrites 36.5%

      \[\leadsto 1 \cdot im \]

   if 1.1500000000000001e29 < im

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 53.4

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 5.6%

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

   9. Taylor expanded in re around inf

      \[\leadsto im \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 6.1%

       \[\leadsto re \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.15 \cdot 10^{+29}:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.4% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(re, im, im\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (fma re im im))

C

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

Julia

function code(re, im)
	return fma(re, im, im)
end

Wolfram

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

Derivation

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-exp.f64 65.0

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

5. Applied rewrites 65.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 31.1%

   \[\leadsto \mathsf{fma}\left(re, \color{blue}{im}, im\right) \]
9. Add Preprocessing

Alternative 17: 7.2% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 57.4

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

5. Applied rewrites 57.4%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 31.1%

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

9. Taylor expanded in re around inf

   \[\leadsto im \cdot re \]
10. Step-by-step derivation

11. Applied rewrites 6.6%

    \[\leadsto re \cdot im \]

12. Final simplification 6.6%

    \[\leadsto im \cdot re \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))