math.exp on complex, imaginary part

Percentage Accurate: 100.0% → 100.0%

Time: 13.1s

Alternatives: 18

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 86.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+23}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (sin im) (fma re (fma re 0.5 1.0) 1.0))))
   (if (<= t_1 (- INFINITY))
     (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
     (if (<= t_1 -0.05)
       t_2
       (if (<= t_1 2e-71) t_0 (if (<= t_1 1e+23) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double t_2 = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(im, (im * -0.16666666666666666), 1.0) * fma(re, im, im);
	} else if (t_1 <= -0.05) {
		tmp = t_2;
	} else if (t_1 <= 2e-71) {
		tmp = t_0;
	} else if (t_1 <= 1e+23) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, im, im));
	elseif (t_1 <= -0.05)
		tmp = t_2;
	elseif (t_1 <= 2e-71)
		tmp = t_0;
	elseif (t_1 <= 1e+23)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$2, If[LessEqual[t$95$1, 2e-71], t$95$0, If[LessEqual[t$95$1, 1e+23], 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 87.5

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

   5. Simplified 87.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 29.3%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 21.4

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

   14. Simplified 21.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.9999999999999998e-71 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999992e22

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 97.5

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

   5. Simplified 97.5%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999998e-71 or 9.9999999999999992e22 < (*.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

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

   5. Simplified 95.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-71}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{+23}:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ t_1 := e^{re} \cdot \sin im\\ t_2 := \sin im \cdot \left(re + 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+23}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (sin im) (+ re 1.0))))
   (if (<= t_1 (- INFINITY))
     (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
     (if (<= t_1 -0.05)
       t_2
       (if (<= t_1 2e-71) t_0 (if (<= t_1 1e+23) t_2 t_0))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double t_1 = exp(re) * sin(im);
	double t_2 = sin(im) * (re + 1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(im, (im * -0.16666666666666666), 1.0) * fma(re, im, im);
	} else if (t_1 <= -0.05) {
		tmp = t_2;
	} else if (t_1 <= 2e-71) {
		tmp = t_0;
	} else if (t_1 <= 1e+23) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(sin(im) * Float64(re + 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, im, im));
	elseif (t_1 <= -0.05)
		tmp = t_2;
	elseif (t_1 <= 2e-71)
		tmp = t_0;
	elseif (t_1 <= 1e+23)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[im], $MachinePrecision] * N[(re + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], t$95$2, If[LessEqual[t$95$1, 2e-71], t$95$0, If[LessEqual[t$95$1, 1e+23], 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 87.5

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

   5. Simplified 87.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 29.3%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 21.4

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

   14. Simplified 21.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1.9999999999999998e-71 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999992e22

   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. +-commutative N/A

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

      2. +-lowering-+.f64 97.2

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

   5. Simplified 97.2%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999998e-71 or 9.9999999999999992e22 < (*.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

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

   5. Simplified 95.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-71}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{+23}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ t_1 := e^{re} \cdot im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\sin im\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))) (t_1 (* (exp re) im)))
   (if (<= t_0 (- INFINITY))
     (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
     (if (<= t_0 -0.05)
       (sin im)
       (if (<= t_0 5e-19) t_1 (if (<= t_0 1e+23) (sin im) t_1))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double t_1 = exp(re) * im;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, (im * -0.16666666666666666), 1.0) * fma(re, im, im);
	} else if (t_0 <= -0.05) {
		tmp = sin(im);
	} else if (t_0 <= 5e-19) {
		tmp = t_1;
	} else if (t_0 <= 1e+23) {
		tmp = sin(im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	t_1 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, im, im));
	elseif (t_0 <= -0.05)
		tmp = sin(im);
	elseif (t_0 <= 5e-19)
		tmp = t_1;
	elseif (t_0 <= 1e+23)
		tmp = sin(im);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Sin[im], $MachinePrecision], If[LessEqual[t$95$0, 5e-19], t$95$1, If[LessEqual[t$95$0, 1e+23], 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

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 87.5

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

   5. Simplified 87.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 29.3%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 21.4

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

   14. Simplified 21.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 5.0000000000000004e-19 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999992e22

   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. sin-lowering-sin.f64 94.8

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

   5. Simplified 94.8%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 5.0000000000000004e-19 or 9.9999999999999992e22 < (*.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

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

   5. Simplified 95.5%

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

Alternative 5: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ t_1 := e^{re} \cdot \sin im\\ t_2 := e^{re} \cdot im\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (*
          (sin im)
          (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)))
        (t_1 (* (exp re) (sin im)))
        (t_2 (* (exp re) im)))
   (if (<= t_1 -0.05)
     t_0
     (if (<= t_1 1e-100) t_2 (if (<= t_1 1e+23) t_0 t_2)))))

C

double code(double re, double im) {
	double t_0 = sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	double t_1 = exp(re) * sin(im);
	double t_2 = exp(re) * im;
	double tmp;
	if (t_1 <= -0.05) {
		tmp = t_0;
	} else if (t_1 <= 1e-100) {
		tmp = t_2;
	} else if (t_1 <= 1e+23) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sin(im) * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0))
	t_1 = Float64(exp(re) * sin(im))
	t_2 = Float64(exp(re) * im)
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = t_0;
	elseif (t_1 <= 1e-100)
		tmp = t_2;
	elseif (t_1 <= 1e+23)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], t$95$0, If[LessEqual[t$95$1, 1e-100], t$95$2, If[LessEqual[t$95$1, 1e+23], t$95$0, t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003 or 1e-100 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999992e22

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 94.6

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

   5. Simplified 94.6%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (sin.f64 im)) < 1e-100 or 9.9999999999999992e22 < (*.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

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

   5. Simplified 95.4%

      \[\leadsto e^{re} \cdot \color{blue}{im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -0.05:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{-100}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{+23}:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, 0.16666666666666666 \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 (- INFINITY))
     (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
     (if (<= t_0 1e+23)
       (sin im)
       (*
        im
        (*
         (* re re)
         (*
          re
          (fma
           (* im im)
           (*
            0.16666666666666666
            (fma (* im im) 0.008333333333333333 -0.16666666666666666))
           0.16666666666666666))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(im, (im * -0.16666666666666666), 1.0) * fma(re, im, im);
	} else if (t_0 <= 1e+23) {
		tmp = sin(im);
	} else {
		tmp = im * ((re * re) * (re * fma((im * im), (0.16666666666666666 * fma((im * im), 0.008333333333333333, -0.16666666666666666)), 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, im, im));
	elseif (t_0 <= 1e+23)
		tmp = sin(im);
	else
		tmp = Float64(im * Float64(Float64(re * re) * Float64(re * fma(Float64(im * im), Float64(0.16666666666666666 * fma(Float64(im * im), 0.008333333333333333, -0.16666666666666666)), 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+23], N[Sin[im], $MachinePrecision], N[(im * N[(N[(re * re), $MachinePrecision] * N[(re * N[(N[(im * im), $MachinePrecision] * N[(0.16666666666666666 * N[(N[(im * im), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)} \cdot \sin im \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 87.5

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

   5. Simplified 87.5%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 29.3%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 28.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 21.4

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

   14. Simplified 21.4%

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

   if -inf.0 < (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999992e22

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 68.1

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

   5. Simplified 68.1%

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

   if 9.9999999999999992e22 < (*.f64 (exp.f64 re) (sin.f64 im))

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 35.0%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around inf

      \[\leadsto im \cdot \color{blue}{\left({re}^{3} \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(\left(re \cdot re\right) \cdot re\right)} \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto im \cdot \left(\left(\color{blue}{{re}^{2}} \cdot re\right) \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto im \cdot \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \color{blue}{\left({re}^{2} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot \left(re \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \color{blue}{\left(re \cdot \left(\frac{1}{6} + \frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)\right)}\right) \]

       8. +-commutative N/A

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \frac{1}{6}\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(\color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \frac{1}{6}} + \frac{1}{6}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \left(\color{blue}{{im}^{2} \cdot \left(\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot \frac{1}{6}\right)} + \frac{1}{6}\right)\right)\right) \]

       11. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot \frac{1}{6}, \frac{1}{6}\right)}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot \frac{1}{6}, \frac{1}{6}\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot \frac{1}{6}, \frac{1}{6}\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) \cdot \frac{1}{6}}, \frac{1}{6}\right)\right)\right) \]

       15. sub-neg N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)} \cdot \frac{1}{6}, \frac{1}{6}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \left(\color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right) \cdot \frac{1}{6}, \frac{1}{6}\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \left({im}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right) \cdot \frac{1}{6}, \frac{1}{6}\right)\right)\right) \]

       18. accelerator-lowering-fma.f64 N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)} \cdot \frac{1}{6}, \frac{1}{6}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right) \cdot \frac{1}{6}, \frac{1}{6}\right)\right)\right) \]

       20. *-lowering-*.f64 63.2

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

   11. Simplified 63.2%

       \[\leadsto im \cdot \color{blue}{\left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right) \cdot 0.16666666666666666, 0.16666666666666666\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;e^{re} \cdot \sin im \leq 10^{+23}:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\left(re \cdot re\right) \cdot \left(re \cdot \mathsf{fma}\left(im \cdot im, 0.16666666666666666 \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \sin im\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im \cdot im - re \cdot \left(im \cdot \left(re \cdot im\right)\right)}{im - re \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.05)
     (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
     (if (<= t_0 0.0)
       (/ (- (* im im) (* re (* im (* re im)))) (- im (* re im)))
       (*
        (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0)
        (*
         im
         (fma
          im
          (* im (fma (* im im) 0.008333333333333333 -0.16666666666666666))
          1.0)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * sin(im);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = fma(im, (im * -0.16666666666666666), 1.0) * fma(re, im, im);
	} else if (t_0 <= 0.0) {
		tmp = ((im * im) - (re * (im * (re * im)))) / (im - (re * im));
	} else {
		tmp = fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * (im * fma(im, (im * fma((im * im), 0.008333333333333333, -0.16666666666666666)), 1.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * sin(im))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, im, im));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(im * im) - Float64(re * Float64(im * Float64(re * im)))) / Float64(im - Float64(re * im)));
	else
		tmp = Float64(fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0) * Float64(im * fma(im, Float64(im * fma(Float64(im * im), 0.008333333333333333, -0.16666666666666666)), 1.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(im * im), $MachinePrecision] - N[(re * N[(im * N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(im - N[(re * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * N[(im * N[(N[(im * im), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 92.3

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

   5. Simplified 92.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 16.7%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 16.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 12.4

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

   14. Simplified 12.4%

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

   if -0.050000000000000003 < (*.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

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

   5. Simplified 98.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 38.8

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

   8. Simplified 38.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 38.5

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

   11. Simplified 38.5%

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

       1. +-commutative N/A

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

       2. flip-+ N/A

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

       3. /-lowering-/.f64 N/A

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

       4. --lowering--.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. --lowering--.f64 N/A

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

       13. *-lowering-*.f64 19.0

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

   13. Applied egg-rr 19.0%

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

   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 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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right) + 1\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \left(\color{blue}{im \cdot \left(im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} + 1\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right), 1\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, \color{blue}{im \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)}, 1\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\left(\frac{1}{120} \cdot {im}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, 1\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \left(\color{blue}{{im}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), 1\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right), 1\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right) \]

      12. *-lowering-*.f64 53.4

          \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right) \]

   8. Simplified 53.4%

      \[\leadsto \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\left(im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 32.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (sin im))))
   (if (<= t_0 -0.05)
     (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
     (if (<= t_0 0.0)
       (/ (- (* im im) (* re (* im (* re im)))) (- im (* re im)))
       (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < -0.050000000000000003

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 92.3

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

   5. Simplified 92.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 16.7%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 16.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 12.4

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

   14. Simplified 12.4%

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

   if -0.050000000000000003 < (*.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

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

   5. Simplified 98.9%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 38.8

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

   8. Simplified 38.8%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 38.5

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

   11. Simplified 38.5%

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

       1. +-commutative N/A

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

       2. flip-+ N/A

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

       3. /-lowering-/.f64 N/A

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

       4. --lowering--.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 N/A

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

       11. *-lowering-*.f64 N/A

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

       12. --lowering--.f64 N/A

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

       13. *-lowering-*.f64 19.0

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

   13. Applied egg-rr 19.0%

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

   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 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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 53.7%

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

4. Final simplification 31.4%

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

Alternative 9: 35.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 0.0)
   (*
    im
    (fma
     (* im im)
     (fma
      (* im im)
      (fma im (* im -0.0001984126984126984) 0.008333333333333333)
      -0.16666666666666666)
     1.0))
   (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 0.0) {
		tmp = im * fma((im * im), fma((im * im), fma(im, (im * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0);
	} else {
		tmp = im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 0.0)
		tmp = Float64(im * fma(Float64(im * im), fma(Float64(im * im), fma(im, Float64(im * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), 1.0));
	else
		tmp = Float64(im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 0.0], N[(im * N[(N[(im * im), $MachinePrecision] * N[(N[(im * im), $MachinePrecision] * N[(im * N[(im * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $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

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

      1. sin-lowering-sin.f64 41.6

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

   5. Simplified 41.6%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto im \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}\right) + 1\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \color{blue}{\mathsf{fma}\left({im}^{2}, {im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, 1\right)} \]

      4. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, 1\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(\color{blue}{im \cdot im}, {im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) - \frac{1}{6}, 1\right) \]

      6. sub-neg N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, {im}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left({im}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}, \frac{-1}{6}\right)}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}, \frac{-1}{6}\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(\color{blue}{im \cdot im}, \frac{1}{120} + \frac{-1}{5040} \cdot {im}^{2}, \frac{-1}{6}\right), 1\right) \]

      11. +-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\frac{-1}{5040} \cdot {im}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{{im}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      13. unpow2 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\left(im \cdot im\right)} \cdot \frac{-1}{5040} + \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      14. associate-*l* N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{im \cdot \left(im \cdot \frac{-1}{5040}\right)} + \frac{1}{120}, \frac{-1}{6}\right), 1\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \color{blue}{\mathsf{fma}\left(im, im \cdot \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right) \]

      16. *-lowering-*.f64 27.8

          \[\leadsto im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, \color{blue}{im \cdot -0.0001984126984126984}, 0.008333333333333333\right), -0.16666666666666666\right), 1\right) \]

   8. Simplified 27.8%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im \cdot im, \mathsf{fma}\left(im, im \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \]

   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 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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 53.7%

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

4. Final simplification 38.2%

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

Alternative 10: 35.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 5e-305)
   (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
   (* im (fma re (fma re (fma re 0.16666666666666666 0.5) 1.0) 1.0))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 5e-305) {
		tmp = fma(im, (im * -0.16666666666666666), 1.0) * fma(re, im, im);
	} else {
		tmp = im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 5e-305)
		tmp = Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, im, im));
	else
		tmp = Float64(im * fma(re, fma(re, fma(re, 0.16666666666666666, 0.5), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 5e-305], N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], N[(im * N[(re * N[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 4.99999999999999985e-305

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 30.8%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 30.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 29.2

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

   14. Simplified 29.2%

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

   if 4.99999999999999985e-305 < (*.f64 (exp.f64 re) (sin.f64 im))

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 90.5

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

   5. Simplified 90.5%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 52.8%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 5 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 2e-7)
   (* (fma im (* im -0.16666666666666666) 1.0) (fma re im im))
   (* im (* (fma re 0.16666666666666666 0.5) (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 2e-7) {
		tmp = fma(im, (im * -0.16666666666666666), 1.0) * fma(re, im, im);
	} else {
		tmp = im * (fma(re, 0.16666666666666666, 0.5) * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 2e-7)
		tmp = Float64(fma(im, Float64(im * -0.16666666666666666), 1.0) * fma(re, im, im));
	else
		tmp = Float64(im * Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * im + im), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-7

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 67.1

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

   5. Simplified 67.1%

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

   6. Taylor expanded in im around 0

      \[\leadsto \color{blue}{im \cdot \left(1 + \left(re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right) + {im}^{2} \cdot \left(\frac{-1}{6} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right) + \frac{1}{120} \cdot \left({im}^{2} \cdot \left(1 + re \cdot \left(1 + re \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot re\right)\right)\right)\right)\right)\right)\right)} \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(\mathsf{fma}\left(re, \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), 1\right), 1\right), im \cdot \left(im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right)\right), 1\right)\right)} \]

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + im \cdot \left(re \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto im \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) + \color{blue}{\left(im \cdot re\right) \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right)} \]

       3. distribute-rgt-out N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{120} \cdot {im}^{2} - \frac{1}{6}\right)\right) \cdot \left(im + im \cdot re\right)} \]

   11. Simplified 42.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im \cdot im, 0.008333333333333333, -0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(re, im, im\right)} \]

   12. Taylor expanded in im around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 41.2

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

   14. Simplified 41.2%

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

   if 1.9999999999999999e-7 < (*.f64 (exp.f64 re) (sin.f64 im))

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 86.1

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

   5. Simplified 86.1%

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

   6. Taylor expanded in re around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 36.4

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

   8. Simplified 36.4%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 31.4%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right) \cdot \mathsf{fma}\left(re, im, im\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 2e-7)
   (* im (fma im (* im -0.16666666666666666) 1.0))
   (* im (* (fma re 0.16666666666666666 0.5) (* re re)))))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 2e-7) {
		tmp = im * fma(im, (im * -0.16666666666666666), 1.0);
	} else {
		tmp = im * (fma(re, 0.16666666666666666, 0.5) * (re * re));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 2e-7)
		tmp = Float64(im * fma(im, Float64(im * -0.16666666666666666), 1.0));
	else
		tmp = Float64(im * Float64(fma(re, 0.16666666666666666, 0.5) * Float64(re * re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 2e-7], N[(im * N[(im * N[(im * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(im * N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 51.9

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

   5. Simplified 51.9%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 40.1

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

   8. Simplified 40.1%

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

   if 1.9999999999999999e-7 < (*.f64 (exp.f64 re) (sin.f64 im))

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 86.1

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

   5. Simplified 86.1%

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

   6. Taylor expanded in re around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 36.4

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

   8. Simplified 36.4%

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

   9. Taylor expanded in im around 0

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

   11. Simplified 31.4%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 2 \cdot 10^{-7}:\\ \;\;\;\;im \cdot \mathsf{fma}\left(im, im \cdot -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right) \cdot \left(re \cdot re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. sin-lowering-sin.f64 41.6

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

   5. Simplified 41.6%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 27.1

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

   8. Simplified 27.1%

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

   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 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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 84.2

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

   5. Simplified 84.2%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 51.8%

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

4. Final simplification 37.1%

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

Alternative 14: 29.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. sin-lowering-sin.f64 41.6

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

   5. Simplified 41.6%

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

   6. Taylor expanded in im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 27.1

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

   8. Simplified 27.1%

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

   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

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

   5. Simplified 59.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 42.3

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

   8. Simplified 42.3%

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

Alternative 15: 27.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 10^{+23}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (exp re) (sin im)) 1e+23) im (* re im)))

C

double code(double re, double im) {
	double tmp;
	if ((exp(re) * sin(im)) <= 1e+23) {
		tmp = im;
	} else {
		tmp = 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 ((exp(re) * sin(im)) <= 1d+23) then
        tmp = im
    else
        tmp = re * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((Math.exp(re) * Math.sin(im)) <= 1e+23) {
		tmp = im;
	} else {
		tmp = re * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (math.exp(re) * math.sin(im)) <= 1e+23:
		tmp = im
	else:
		tmp = re * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(exp(re) * sin(im)) <= 1e+23)
		tmp = im;
	else
		tmp = Float64(re * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((exp(re) * sin(im)) <= 1e+23)
		tmp = im;
	else
		tmp = re * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[Exp[re], $MachinePrecision] * N[Sin[im], $MachinePrecision]), $MachinePrecision], 1e+23], im, N[(re * im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 re) (sin.f64 im)) < 9.9999999999999992e22

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. sin-lowering-sin.f64 59.1

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

   5. Simplified 59.1%

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

   6. Taylor expanded in im around 0

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

   8. Simplified 32.2%

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

   if 9.9999999999999992e22 < (*.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

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

   5. Simplified 81.3%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. distribute-lft-out N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 54.3

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

   8. Simplified 54.3%

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

   9. Taylor expanded in re around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 27.2

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

   11. Simplified 27.2%

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

   12. Taylor expanded in re around inf

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

       1. *-lowering-*.f64 27.2

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

   14. Simplified 27.2%

       \[\leadsto \color{blue}{im \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \sin im \leq 10^{+23}:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot im\\ \mathbf{if}\;re \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;re \leq 3300000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) im)))
   (if (<= re -9.5e-6)
     t_0
     (if (<= re 3300000.0)
       (* (sin im) (fma re (fma re 0.5 1.0) 1.0))
       (if (<= re 5.6e+102)
         t_0
         (* (sin im) (* 0.16666666666666666 (* re (* re re)))))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * im;
	double tmp;
	if (re <= -9.5e-6) {
		tmp = t_0;
	} else if (re <= 3300000.0) {
		tmp = sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	} else if (re <= 5.6e+102) {
		tmp = t_0;
	} else {
		tmp = sin(im) * (0.16666666666666666 * (re * (re * re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * im)
	tmp = 0.0
	if (re <= -9.5e-6)
		tmp = t_0;
	elseif (re <= 3300000.0)
		tmp = Float64(sin(im) * fma(re, fma(re, 0.5, 1.0), 1.0));
	elseif (re <= 5.6e+102)
		tmp = t_0;
	else
		tmp = Float64(sin(im) * Float64(0.16666666666666666 * Float64(re * Float64(re * re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * im), $MachinePrecision]}, If[LessEqual[re, -9.5e-6], t$95$0, If[LessEqual[re, 3300000.0], N[(N[Sin[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.6e+102], t$95$0, N[(N[Sin[im], $MachinePrecision] * N[(0.16666666666666666 * N[(re * N[(re * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if re < -9.5000000000000005e-6 or 3.3e6 < re < 5.60000000000000037e102

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

   5. Simplified 97.3%

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

   if -9.5000000000000005e-6 < re < 3.3e6

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 98.0

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

   5. Simplified 98.0%

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

   if 5.60000000000000037e102 < 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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {re}^{3}\right)} \cdot \sin im \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 100.0

         \[\leadsto \left(0.16666666666666666 \cdot \left(re \cdot \color{blue}{\left(re \cdot re\right)}\right)\right) \cdot \sin im \]

   8. Simplified 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{elif}\;re \leq 3300000:\\ \;\;\;\;\sin im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;re \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(0.16666666666666666 \cdot \left(re \cdot \left(re \cdot re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 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

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

5. Simplified 70.0%

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

6. Taylor expanded in re around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 33.4

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

8. Simplified 33.4%

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

Alternative 18: 26.3% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (re im) :precision binary64 im)

C

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

Fortran

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

Java

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

Python

def code(re, im):
	return im

Julia

function code(re, im)
	return im
end

MATLAB

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

Wolfram

code[re_, im_] := im

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in re around 0

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

   1. sin-lowering-sin.f64 52.0

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

5. Simplified 52.0%

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

6. Taylor expanded in im around 0

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

8. Simplified 28.5%

   \[\leadsto \color{blue}{im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))