math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 22

Speedup: 1.0×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := \cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot 0.5, re\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999999:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (* (cos im) (fma re (fma re 0.5 1.0) 1.0))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (* re 0.5) re)
      (fma
       im
       (*
        im
        (fma
         im
         (* im (fma im (* im -0.001388888888888889) 0.041666666666666664))
         -0.5))
       1.0))
     (if (<= t_0 -0.05)
       t_1
       (if (<= t_0 0.0)
         (exp re)
         (if (<= t_0 0.9999999999999999) t_1 (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (re * 0.5), re) * fma(im, (im * fma(im, (im * fma(im, (im * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0);
	} else if (t_0 <= -0.05) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999999999999) {
		tmp = t_1;
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(cos(im) * fma(re, fma(re, 0.5, 1.0), 1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(re * 0.5), re) * fma(im, Float64(im * fma(im, Float64(im * fma(im, Float64(im * -0.001388888888888889), 0.041666666666666664)), -0.5)), 1.0));
	elseif (t_0 <= -0.05)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999999999999)
		tmp = t_1;
	else
		tmp = exp(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[re], $MachinePrecision] * N[Cos[im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[im], $MachinePrecision] * N[(re * N[(re * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(re * 0.5), $MachinePrecision] + re), $MachinePrecision] * N[(im * N[(im * N[(im * N[(im * N[(im * N[(im * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], t$95$1, If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999999], t$95$1, N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in re around inf

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(re, re \cdot \frac{1}{2}, re\right) \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \frac{1}{2}, re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1\right)} \]

       2. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \frac{1}{2}, re\right) \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right) + 1\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \frac{1}{2}, re\right) \cdot \left(\color{blue}{im \cdot \left(im \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right)\right)} + 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(re, re \cdot \frac{1}{2}, re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \left({im}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {im}^{2}\right) - \frac{1}{2}\right), 1\right)} \]

   11. Applied rewrites 96.1%

       \[\leadsto \mathsf{fma}\left(re, re \cdot 0.5, re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   if -inf.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < -0.050000000000000003 or 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.999999999999999889

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.999999999999999889 < (*.f64 (exp.f64 re) (cos.f64 im))

   1. Initial program 100.0%

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

   3. Taylor expanded in im around 0

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

      1. lower-exp.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, re \cdot 0.5, re\right) \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot \mathsf{fma}\left(im, im \cdot -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999999999999:\\ \;\;\;\;\cos im \cdot \mathsf{fma}\left(re, \mathsf{fma}\left(re, 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))