math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 16.7s

Alternatives: 24

Speedup: 1.0×

• 24.6% 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: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ t_2 := \left(1 + re\right) \cdot \sin im\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_1 \leq -0.002:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re)))
        (t_1 (* (sin im) (exp re)))
        (t_2 (* (+ 1.0 re) (sin im))))
   (if (<= t_1 (- INFINITY))
     (fma
      (fma
       (fma
        (fma
         (* (fma -0.0001984126984126984 re -0.0001984126984126984) im)
         im
         (fma 0.008333333333333333 re 0.008333333333333333))
        (* im im)
        (fma -0.16666666666666666 re -0.16666666666666666))
       (* im im)
       re)
      im
      im)
     (if (<= t_1 -0.002)
       t_2
       (if (<= t_1 2e-121) t_0 (if (<= t_1 1.0) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double t_2 = (1.0 + re) * sin(im);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(fma((fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), (im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), (im * im), re), im, im);
	} else if (t_1 <= -0.002) {
		tmp = t_2;
	} else if (t_1 <= 2e-121) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	t_2 = Float64(Float64(1.0 + re) * sin(im))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(fma(fma(fma(Float64(fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), Float64(im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), Float64(im * im), re), im, im);
	elseif (t_1 <= -0.002)
		tmp = t_2;
	elseif (t_1 <= 2e-121)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * re + -0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision] * im + N[(0.008333333333333333 * re + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$1, -0.002], t$95$2, If[LessEqual[t$95$1, 2e-121], t$95$0, If[LessEqual[t$95$1, 1.0], t$95$2, t$95$0]]]]]]]

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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 17.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -2e-3 or 2e-121 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-121 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%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-121}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.002:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (fma
        (fma
         (* (fma -0.0001984126984126984 re -0.0001984126984126984) im)
         im
         (fma 0.008333333333333333 re 0.008333333333333333))
        (* im im)
        (fma -0.16666666666666666 re -0.16666666666666666))
       (* im im)
       re)
      im
      im)
     (if (<= t_0 -0.002)
       (sin im)
       (if (<= t_0 2e-121) t_1 (if (<= t_0 1.0) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(fma(fma(fma((fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), (im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), (im * im), re), im, im);
	} else if (t_0 <= -0.002) {
		tmp = sin(im);
	} else if (t_0 <= 2e-121) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(fma(fma(fma(Float64(fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), Float64(im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), Float64(im * im), re), im, im);
	elseif (t_0 <= -0.002)
		tmp = sin(im);
	elseif (t_0 <= 2e-121)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = t_1;
	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[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * re + -0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision] * im + N[(0.008333333333333333 * re + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, -0.002], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 2e-121], t$95$1, If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 17.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -2e-3 or 2e-121 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   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 96.9

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

   5. Applied rewrites 96.9%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-121 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%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-121}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.002:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 10^{-244}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im} \cdot re, -0.25, \frac{0.5}{im}\right), re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 (- INFINITY))
     (fma
      (fma
       (fma
        (fma
         (* (fma -0.0001984126984126984 re -0.0001984126984126984) im)
         im
         (fma 0.008333333333333333 re 0.008333333333333333))
        (* im im)
        (fma -0.16666666666666666 re -0.16666666666666666))
       (* im im)
       re)
      im
      im)
     (if (<= t_0 -0.002)
       (sin im)
       (if (<= t_0 1e-244)
         (/
          1.0
          (fma
           (fma (fma (* (/ re im) re) -0.25 (/ 0.5 im)) re (/ -1.0 im))
           re
           (/ 1.0 im)))
         (if (<= t_0 1.0)
           (sin im)
           (* (* (* (* 0.16666666666666666 re) re) re) 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(fma((fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), (im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), (im * im), re), im, im);
	} else if (t_0 <= -0.002) {
		tmp = sin(im);
	} else if (t_0 <= 1e-244) {
		tmp = 1.0 / fma(fma(fma(((re / im) * re), -0.25, (0.5 / im)), re, (-1.0 / im)), re, (1.0 / im));
	} else if (t_0 <= 1.0) {
		tmp = sin(im);
	} else {
		tmp = (((0.16666666666666666 * re) * re) * re) * 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(fma(Float64(fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), Float64(im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), Float64(im * im), re), im, im);
	elseif (t_0 <= -0.002)
		tmp = sin(im);
	elseif (t_0 <= 1e-244)
		tmp = Float64(1.0 / fma(fma(fma(Float64(Float64(re / im) * re), -0.25, Float64(0.5 / im)), re, Float64(-1.0 / im)), re, Float64(1.0 / im)));
	elseif (t_0 <= 1.0)
		tmp = sin(im);
	else
		tmp = Float64(Float64(Float64(Float64(0.16666666666666666 * re) * 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * re + -0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision] * im + N[(0.008333333333333333 * re + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, -0.002], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 1e-244], N[(1.0 / N[(N[(N[(N[(N[(re / im), $MachinePrecision] * re), $MachinePrecision] * -0.25 + N[(0.5 / im), $MachinePrecision]), $MachinePrecision] * re + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[im], $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * re), $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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 17.9%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -2e-3 or 9.9999999999999993e-245 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1

   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 96.4

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

   5. Applied rewrites 96.4%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999993e-245

   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 50.5%

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

   10. Applied rewrites 50.3%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 94.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot \frac{re}{im}, -0.25, \frac{0.5}{im}\right), re, \frac{-1}{im}\right), re, \frac{1}{im}\right)} \]

   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 76.3

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

   5. Applied rewrites 76.3%

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

   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 63.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 63.6%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 63.6%

       \[\leadsto \left(\left(\left(0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\right) \cdot im \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 10^{-244}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im} \cdot re, -0.25, \frac{0.5}{im}\right), re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ t_1 := im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))) (t_1 (* im (exp re))))
   (if (<= t_0 -0.002)
     (* (fma (fma (fma 0.16666666666666666 re 0.5) re 1.0) re 1.0) (sin im))
     (if (<= t_0 2e-121) t_1 (if (<= t_0 1.0) (* (+ 1.0 re) (sin im)) t_1)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * exp(re);
	double t_1 = im * exp(re);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im);
	} else if (t_0 <= 2e-121) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * exp(re))
	t_1 = Float64(im * exp(re))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = Float64(fma(fma(fma(0.16666666666666666, re, 0.5), re, 1.0), re, 1.0) * sin(im));
	elseif (t_0 <= 2e-121)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(Float64(1.0 + re) * sin(im));
	else
		tmp = t_1;
	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[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-121], t$95$1, If[LessEqual[t$95$0, 1.0], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-121 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 2e-121 < (*.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 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-121}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ t_1 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_1 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re))) (t_1 (* (sin im) (exp re))))
   (if (<= t_1 -0.002)
     (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
     (if (<= t_1 2e-121) t_0 (if (<= t_1 1.0) (* (+ 1.0 re) (sin im)) t_0)))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double t_1 = sin(im) * exp(re);
	double tmp;
	if (t_1 <= -0.002) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (t_1 <= 2e-121) {
		tmp = t_0;
	} else if (t_1 <= 1.0) {
		tmp = (1.0 + re) * sin(im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	t_1 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_1 <= -0.002)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (t_1 <= 2e-121)
		tmp = t_0;
	elseif (t_1 <= 1.0)
		tmp = Float64(Float64(1.0 + re) * sin(im));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.002], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-121], t$95$0, If[LessEqual[t$95$1, 1.0], N[(N[(1.0 + re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 2e-121 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 2e-121 < (*.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 3 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-121}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 1:\\ \;\;\;\;\left(1 + re\right) \cdot \sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot e^{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im} \cdot re, -0.25, \frac{0.5}{im}\right), re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.002)
     (fma
      (fma
       (fma
        (fma
         (* (fma -0.0001984126984126984 re -0.0001984126984126984) im)
         im
         (fma 0.008333333333333333 re 0.008333333333333333))
        (* im im)
        (fma -0.16666666666666666 re -0.16666666666666666))
       (* im im)
       re)
      im
      im)
     (if (<= t_0 0.0)
       (/
        1.0
        (fma
         (fma (fma (* (/ re im) re) -0.25 (/ 0.5 im)) re (/ -1.0 im))
         re
         (/ 1.0 im)))
       (* (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 <= -0.002) {
		tmp = fma(fma(fma(fma((fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), (im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), (im * im), re), im, im);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / fma(fma(fma(((re / im) * re), -0.25, (0.5 / im)), re, (-1.0 / im)), re, (1.0 / 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)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = fma(fma(fma(fma(Float64(fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), Float64(im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), Float64(im * im), re), im, im);
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / fma(fma(fma(Float64(Float64(re / im) * re), -0.25, Float64(0.5 / im)), re, Float64(-1.0 / im)), re, Float64(1.0 / 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_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * re + -0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision] * im + N[(0.008333333333333333 * re + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(N[(N[(N[(N[(re / im), $MachinePrecision] * re), $MachinePrecision] * -0.25 + N[(0.5 / im), $MachinePrecision]), $MachinePrecision] * re + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[(1.0 / im), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 10.5%

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

   if -2e-3 < (*.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 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 46.7%

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

   10. Applied rewrites 46.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 93.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(re \cdot \frac{re}{im}, -0.25, \frac{0.5}{im}\right), re, \frac{-1}{im}\right), re, \frac{1}{im}\right)} \]

   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 59.1

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

   5. Applied rewrites 59.1%

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

   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 53.6%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{re}{im} \cdot re, -0.25, \frac{0.5}{im}\right), re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0.44:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{im}, re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.002)
     (fma
      (fma
       (fma
        (fma
         (* (fma -0.0001984126984126984 re -0.0001984126984126984) im)
         im
         (fma 0.008333333333333333 re 0.008333333333333333))
        (* im im)
        (fma -0.16666666666666666 re -0.16666666666666666))
       (* im im)
       re)
      im
      im)
     (if (<= t_0 0.44)
       (/ 1.0 (fma (fma (/ 0.5 im) re (/ -1.0 im)) re (/ 1.0 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 tmp;
	if (t_0 <= -0.002) {
		tmp = fma(fma(fma(fma((fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), (im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), (im * im), re), im, im);
	} else if (t_0 <= 0.44) {
		tmp = 1.0 / fma(fma((0.5 / im), re, (-1.0 / im)), re, (1.0 / 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))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = fma(fma(fma(fma(Float64(fma(-0.0001984126984126984, re, -0.0001984126984126984) * im), im, fma(0.008333333333333333, re, 0.008333333333333333)), Float64(im * im), fma(-0.16666666666666666, re, -0.16666666666666666)), Float64(im * im), re), im, im);
	elseif (t_0 <= 0.44)
		tmp = Float64(1.0 / fma(fma(Float64(0.5 / im), re, Float64(-1.0 / im)), re, Float64(1.0 / 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]}, If[LessEqual[t$95$0, -0.002], N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * re + -0.0001984126984126984), $MachinePrecision] * im), $MachinePrecision] * im + N[(0.008333333333333333 * re + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, 0.44], N[(1.0 / N[(N[(N[(0.5 / im), $MachinePrecision] * re + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 10.5%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.440000000000000002

   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 92.1

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

   5. Applied rewrites 92.1%

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

   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 56.5%

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

   10. Applied rewrites 56.4%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 77.2%

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

   if 0.440000000000000002 < (*.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 47.0

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

   5. Applied rewrites 47.0%

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

   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 39.4%

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

   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 39.7%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, re, -0.0001984126984126984\right) \cdot im, im, \mathsf{fma}\left(0.008333333333333333, re, 0.008333333333333333\right)\right), im \cdot im, \mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right)\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{im}, re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0.44:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{im}, re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.002)
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 0.44)
       (/ 1.0 (fma (fma (/ 0.5 im) re (/ -1.0 im)) re (/ 1.0 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 tmp;
	if (t_0 <= -0.002) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if (t_0 <= 0.44) {
		tmp = 1.0 / fma(fma((0.5 / im), re, (-1.0 / im)), re, (1.0 / 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))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif (t_0 <= 0.44)
		tmp = Float64(1.0 / fma(fma(Float64(0.5 / im), re, Float64(-1.0 / im)), re, Float64(1.0 / 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]}, If[LessEqual[t$95$0, -0.002], N[(N[(N[(-0.16666666666666666 * re + -0.16666666666666666), $MachinePrecision] * N[(im * im), $MachinePrecision] + re), $MachinePrecision] * im + im), $MachinePrecision], If[LessEqual[t$95$0, 0.44], N[(1.0 / N[(N[(N[(0.5 / im), $MachinePrecision] * re + N[(-1.0 / im), $MachinePrecision]), $MachinePrecision] * re + N[(1.0 / im), $MachinePrecision]), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 48.2

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

   5. Applied rewrites 48.2%

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

   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 9.3%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.440000000000000002

   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 92.1

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

   5. Applied rewrites 92.1%

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

   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 56.5%

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

   10. Applied rewrites 56.4%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 77.2%

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

   if 0.440000000000000002 < (*.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 47.0

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

   5. Applied rewrites 47.0%

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

   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 39.4%

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

   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 39.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{im}, re, \frac{-1}{im}\right), re, \frac{1}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{im} - \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.002)
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 0.0)
       (/ 1.0 (- (/ 1.0 im) (/ re im)))
       (* (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 <= -0.002) {
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), (im * im), re), im, im);
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / ((1.0 / im) - (re / 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)
	t_0 = Float64(sin(im) * exp(re))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = fma(fma(fma(-0.16666666666666666, re, -0.16666666666666666), Float64(im * im), re), im, im);
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / im) - Float64(re / 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_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], 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[(1.0 / N[(N[(1.0 / im), $MachinePrecision] - N[(re / im), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 48.2

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

   5. Applied rewrites 48.2%

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

   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 9.3%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 0.0

   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 46.7%

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

   10. Applied rewrites 46.6%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 63.4%

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

   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 59.1

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

   5. Applied rewrites 59.1%

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

   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 53.6%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\frac{1}{\frac{1}{im} - \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot e^{re}\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im}{1 - re} \cdot \left(1 - re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (sin im) (exp re))))
   (if (<= t_0 -0.002)
     (fma
      (fma (fma -0.16666666666666666 re -0.16666666666666666) (* im im) re)
      im
      im)
     (if (<= t_0 0.0)
       (* (/ im (- 1.0 re)) (- 1.0 (* re re)))
       (* (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 <= -0.002) {
		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)) * (1.0 - (re * re));
	} 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 <= -0.002)
		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)) * Float64(1.0 - Float64(re * re)));
	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, -0.002], 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[(1.0 - N[(re * re), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 48.2

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

   5. Applied rewrites 48.2%

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

   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 9.3%

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

   if -2e-3 < (*.f64 (exp.f64 re) (sin.f64 im)) < 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 25.1%

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

   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 52.4%

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

   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 59.1

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

   5. Applied rewrites 59.1%

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

   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 53.6%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{elif}\;\sin im \cdot e^{re} \leq 0:\\ \;\;\;\;\frac{im}{1 - re} \cdot \left(1 - re \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right), re, 1\right), re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.0)
   (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)) <= 0.0) {
		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)) <= 0.0)
		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], 0.0], 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)) < 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 47.4

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

   5. Applied rewrites 47.4%

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

   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 28.9%

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

   if 0.0 < (*.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 59.1

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

   5. Applied rewrites 59.1%

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

   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 53.6%

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

4. Final simplification 39.0%

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

Alternative 13: 35.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 2e-121)
   (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)) <= 2e-121) {
		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)) <= 2e-121)
		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], 2e-121], 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-121

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 52.7

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

   5. Applied rewrites 52.7%

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

   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 36.2%

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

   if 2e-121 < (*.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 50.0

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

   5. Applied rewrites 50.0%

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

   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 44.4%

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

   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 44.4%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, re, -0.16666666666666666\right), im \cdot im, re\right), im, im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(re \cdot re\right) \cdot 0.16666666666666666, re, 1\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.44)
   (fma (* (fma 0.5 re 1.0) 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)) <= 0.44) {
		tmp = fma((fma(0.5, re, 1.0) * 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)) <= 0.44)
		tmp = fma(Float64(fma(0.5, re, 1.0) * 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], 0.44], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * 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)) < 0.440000000000000002

   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 74.8

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

   5. Applied rewrites 74.8%

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

   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 46.1%

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

   if 0.440000000000000002 < (*.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 47.0

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

   5. Applied rewrites 47.0%

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

   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 39.4%

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

   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 39.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.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 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 66.5

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

   5. Applied rewrites 66.5%

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

   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 41.1%

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

   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 76.3

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

   5. Applied rewrites 76.3%

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

   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 63.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 63.6%

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

   12. Taylor expanded in re around inf

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

   14. Applied rewrites 63.6%

       \[\leadsto \left(\left(\left(0.16666666666666666 \cdot re\right) \cdot re\right) \cdot re\right) \cdot im \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

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

Alternative 16: 35.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.44)
   (fma (* (fma 0.5 re 1.0) im) re im)
   (* (* (* 0.5 re) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.44) {
		tmp = fma((fma(0.5, re, 1.0) * im), re, im);
	} else {
		tmp = ((0.5 * re) * re) * im;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.44)
		tmp = fma(Float64(fma(0.5, re, 1.0) * im), re, im);
	else
		tmp = Float64(Float64(Float64(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], 0.44], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * im), $MachinePrecision] * re + im), $MachinePrecision], N[(N[(N[(0.5 * re), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.440000000000000002

   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 74.8

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

   5. Applied rewrites 74.8%

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

   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 46.1%

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

   if 0.440000000000000002 < (*.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 47.0

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

   5. Applied rewrites 47.0%

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

   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 39.4%

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

   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 39.7%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 29.1%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right) \cdot im, re, im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 31.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.44) (* 1.0 im) (* (* (* 0.5 re) re) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.44) {
		tmp = 1.0 * im;
	} else {
		tmp = ((0.5 * re) * re) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((sin(im) * exp(re)) <= 0.44d0) then
        tmp = 1.0d0 * im
    else
        tmp = ((0.5d0 * re) * re) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sin(im) * Math.exp(re)) <= 0.44) {
		tmp = 1.0 * im;
	} else {
		tmp = ((0.5 * re) * re) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sin(im) * math.exp(re)) <= 0.44:
		tmp = 1.0 * im
	else:
		tmp = ((0.5 * re) * re) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.44)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(Float64(Float64(0.5 * re) * re) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sin(im) * exp(re)) <= 0.44)
		tmp = 1.0 * im;
	else
		tmp = ((0.5 * re) * re) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.44], N[(1.0 * im), $MachinePrecision], N[(N[(N[(0.5 * re), $MachinePrecision] * re), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.440000000000000002

   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 74.8

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

   5. Applied rewrites 74.8%

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

   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 0.440000000000000002 < (*.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 47.0

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

   5. Applied rewrites 47.0%

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

   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 39.4%

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

   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 39.7%

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

   12. Taylor expanded in re around 0

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

   14. Applied rewrites 29.1%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.5 \cdot re\right) \cdot re\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (sin im) (exp re)) 0.44) (* 1.0 im) (* (* (* re re) 0.5) im)))

C

double code(double re, double im) {
	double tmp;
	if ((sin(im) * exp(re)) <= 0.44) {
		tmp = 1.0 * im;
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((sin(im) * exp(re)) <= 0.44d0) then
        tmp = 1.0d0 * im
    else
        tmp = ((re * re) * 0.5d0) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.sin(im) * Math.exp(re)) <= 0.44) {
		tmp = 1.0 * im;
	} else {
		tmp = ((re * re) * 0.5) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.sin(im) * math.exp(re)) <= 0.44:
		tmp = 1.0 * im
	else:
		tmp = ((re * re) * 0.5) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(sin(im) * exp(re)) <= 0.44)
		tmp = Float64(1.0 * im);
	else
		tmp = Float64(Float64(Float64(re * re) * 0.5) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((sin(im) * exp(re)) <= 0.44)
		tmp = 1.0 * im;
	else
		tmp = ((re * re) * 0.5) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Sin[im], $MachinePrecision] * N[Exp[re], $MachinePrecision]), $MachinePrecision], 0.44], N[(1.0 * im), $MachinePrecision], N[(N[(N[(re * re), $MachinePrecision] * 0.5), $MachinePrecision] * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 0.440000000000000002

   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 74.8

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

   5. Applied rewrites 74.8%

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

   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 0.440000000000000002 < (*.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 47.0

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

   5. Applied rewrites 47.0%

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

   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 24.3%

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

   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 29.1%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin im \cdot e^{re} \leq 0.44:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;\left(\left(re \cdot re\right) \cdot 0.5\right) \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := im \cdot e^{re}\\ \mathbf{if}\;re \leq -0.0055:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 60000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* im (exp re))))
   (if (<= re -0.0055)
     t_0
     (if (<= re 60000.0)
       (* (fma (fma 0.5 re 1.0) re 1.0) (sin im))
       (if (<= re 1.05e+103)
         t_0
         (* (* (* (fma 0.16666666666666666 re 0.5) re) re) (sin im)))))))

C

double code(double re, double im) {
	double t_0 = im * exp(re);
	double tmp;
	if (re <= -0.0055) {
		tmp = t_0;
	} else if (re <= 60000.0) {
		tmp = fma(fma(0.5, re, 1.0), re, 1.0) * sin(im);
	} else if (re <= 1.05e+103) {
		tmp = t_0;
	} else {
		tmp = ((fma(0.16666666666666666, re, 0.5) * re) * re) * sin(im);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(im * exp(re))
	tmp = 0.0
	if (re <= -0.0055)
		tmp = t_0;
	elseif (re <= 60000.0)
		tmp = Float64(fma(fma(0.5, re, 1.0), re, 1.0) * sin(im));
	elseif (re <= 1.05e+103)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, re, 0.5) * re) * re) * sin(im));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(im * N[Exp[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -0.0055], t$95$0, If[LessEqual[re, 60000.0], N[(N[(N[(0.5 * re + 1.0), $MachinePrecision] * re + 1.0), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.05e+103], t$95$0, N[(N[(N[(N[(0.16666666666666666 * re + 0.5), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0054999999999999997 or 6e4 < re < 1.0500000000000001e103

   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 95.3

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

   5. Applied rewrites 95.3%

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

   if -0.0054999999999999997 < re < 6e4

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1.0500000000000001e103 < re

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0055:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{elif}\;re \leq 60000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, re, 1\right), re, 1\right) \cdot \sin im\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;im \cdot e^{re}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, re, 0.5\right) \cdot re\right) \cdot re\right) \cdot \sin im\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 39.0% 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 68.0

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

5. Applied rewrites 68.0%

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

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 46.5%

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

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 46.2%

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

Alternative 21: 36.9% 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 68.0

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

5. Applied rewrites 68.0%

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

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 42.5%

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

Alternative 22: 28.0% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 490000:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 490000.0) (* 1.0 im) (* im re)))

C

double code(double re, double im) {
	double tmp;
	if (im <= 490000.0) {
		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 <= 490000.0d0) 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 <= 490000.0) {
		tmp = 1.0 * im;
	} else {
		tmp = im * re;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 490000.0:
		tmp = 1.0 * im
	else:
		tmp = im * re
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 490000.0)
		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 <= 490000.0)
		tmp = 1.0 * im;
	else
		tmp = im * re;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 490000.0], N[(1.0 * im), $MachinePrecision], N[(im * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 4.9e5

   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 77.6

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in re around 0

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

   8. Applied rewrites 35.9%

      \[\leadsto 1 \cdot im \]

   if 4.9e5 < im

   1. Initial program 99.9%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in im around 0

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

   8. Applied rewrites 13.6%

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

   9. Taylor expanded in re around inf

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

   11. Applied rewrites 14.6%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 490000:\\ \;\;\;\;1 \cdot im\\ \mathbf{else}:\\ \;\;\;\;im \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 29.4% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[re_, im_] := N[(im * re + 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 68.0

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

5. Applied rewrites 68.0%

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

6. Taylor expanded in re around 0

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

8. Applied rewrites 32.9%

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

Alternative 24: 6.6% 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-sin.f64 54.5

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in im around 0

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

8. Applied rewrites 32.9%

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

9. Taylor expanded in re around inf

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

11. Applied rewrites 8.7%

    \[\leadsto re \cdot im \]

12. Final simplification 8.7%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))