math.exp on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 16.1s

Alternatives: 23

Speedup: 1.0×

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

Specification

Math

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

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: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{re} \cdot \cos im\\ t_1 := re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \frac{\mathsf{fma}\left(t\_1, t\_1, -1\right)}{\mathsf{fma}\left(re, 0.5, -1\right)}, 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;t\_0 \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;t\_0 \leq 0.9999999999999996:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (exp re) (cos im)))
        (t_1 (* re (fma re 0.16666666666666666 0.5))))
   (if (<= t_0 (- INFINITY))
     (*
      (fma re (/ (fma t_1 t_1 -1.0) (fma re 0.5 -1.0)) 1.0)
      (fma im (* im -0.5) 1.0))
     (if (<= t_0 -0.05)
       (cos im)
       (if (<= t_0 0.0)
         (exp re)
         (if (<= t_0 0.9999999999999996)
           (*
            (cos im)
            (+ re (fma (fma re 0.16666666666666666 0.5) (* re re) 1.0)))
           (exp re)))))))

C

double code(double re, double im) {
	double t_0 = exp(re) * cos(im);
	double t_1 = re * fma(re, 0.16666666666666666, 0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(re, (fma(t_1, t_1, -1.0) / fma(re, 0.5, -1.0)), 1.0) * fma(im, (im * -0.5), 1.0);
	} else if (t_0 <= -0.05) {
		tmp = cos(im);
	} else if (t_0 <= 0.0) {
		tmp = exp(re);
	} else if (t_0 <= 0.9999999999999996) {
		tmp = cos(im) * (re + fma(fma(re, 0.16666666666666666, 0.5), (re * re), 1.0));
	} else {
		tmp = exp(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(exp(re) * cos(im))
	t_1 = Float64(re * fma(re, 0.16666666666666666, 0.5))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(re, Float64(fma(t_1, t_1, -1.0) / fma(re, 0.5, -1.0)), 1.0) * fma(im, Float64(im * -0.5), 1.0));
	elseif (t_0 <= -0.05)
		tmp = cos(im);
	elseif (t_0 <= 0.0)
		tmp = exp(re);
	elseif (t_0 <= 0.9999999999999996)
		tmp = Float64(cos(im) * Float64(re + fma(fma(re, 0.16666666666666666, 0.5), Float64(re * re), 1.0)));
	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[(re * N[(re * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(re * N[(N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision] / N[(re * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(im * N[(im * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.05], N[Cos[im], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[Exp[re], $MachinePrecision], If[LessEqual[t$95$0, 0.9999999999999996], N[(N[Cos[im], $MachinePrecision] * N[(re + N[(N[(re * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(re * re), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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 im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 91.5

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

   8. Applied rewrites 91.5%

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

   10. Applied rewrites 40.2%

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

   11. Taylor expanded in re around 0

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

   13. Applied rewrites 94.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. lower-cos.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -0.050000000000000003 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.0 or 0.99999999999999956 < (*.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 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.0 < (*.f64 (exp.f64 re) (cos.f64 im)) < 0.99999999999999956

   1. Initial program 100.0%

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

   3. Taylor expanded in re around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 97.7

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \cdot \cos im \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(re, \frac{\mathsf{fma}\left(re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot \mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), -1\right)}{\mathsf{fma}\left(re, 0.5, -1\right)}, 1\right) \cdot \mathsf{fma}\left(im, im \cdot -0.5, 1\right)\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq -0.05:\\ \;\;\;\;\cos im\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0:\\ \;\;\;\;e^{re}\\ \mathbf{elif}\;e^{re} \cdot \cos im \leq 0.9999999999999996:\\ \;\;\;\;\cos im \cdot \left(re + \mathsf{fma}\left(\mathsf{fma}\left(re, 0.16666666666666666, 0.5\right), re \cdot re, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(FPCore (re im)
  :name "math.exp on complex, real part"
  :precision binary64
  (* (exp re) (cos im)))