math.cos on complex, imaginary part

Percentage Accurate: 66.0% → 90.4%

Time: 9.1s

Alternatives: 6

Speedup: 1.5×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

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

Initial Program: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
end

Wolfram

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

Alternative 1: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (* 0.5 (sin re))
  (+
   (* -2.0 im)
   (+
    (* -0.3333333333333333 (pow im 3.0))
    (* -0.016666666666666666 (pow im 5.0))))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * ((-2.0 * im) + ((-0.3333333333333333 * pow(im, 3.0)) + (-0.016666666666666666 * pow(im, 5.0))));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (((-2.0d0) * im) + (((-0.3333333333333333d0) * (im ** 3.0d0)) + ((-0.016666666666666666d0) * (im ** 5.0d0))))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * ((-2.0 * im) + ((-0.3333333333333333 * Math.pow(im, 3.0)) + (-0.016666666666666666 * Math.pow(im, 5.0))));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * ((-2.0 * im) + ((-0.3333333333333333 * math.pow(im, 3.0)) + (-0.016666666666666666 * math.pow(im, 5.0))))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(Float64(-2.0 * im) + Float64(Float64(-0.3333333333333333 * (im ^ 3.0)) + Float64(-0.016666666666666666 * (im ^ 5.0)))))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * ((-2.0 * im) + ((-0.3333333333333333 * (im ^ 3.0)) + (-0.016666666666666666 * (im ^ 5.0))));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(-2.0 * im), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 89.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

4. Final simplification 89.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right) \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (sin re) (- (* (pow im 3.0) -0.16666666666666666) im)))

C

double code(double re, double im) {
	return sin(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
}

Python

def code(re, im):
	return math.sin(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)

Julia

function code(re, im)
	return Float64(sin(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 82.5%

   \[\leadsto \color{blue}{-1 \cdot \left(im \cdot \sin re\right) + -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. +-commutative 82.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + -1 \cdot \left(im \cdot \sin re\right)} \]

   2. mul-1-neg 82.5%

      \[\leadsto -0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) + \color{blue}{\left(-im \cdot \sin re\right)} \]

   3. unsub-neg 82.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left({im}^{3} \cdot \sin re\right) - im \cdot \sin re} \]

   4. associate-*r* 82.5%

      \[\leadsto \color{blue}{\left(-0.16666666666666666 \cdot {im}^{3}\right) \cdot \sin re} - im \cdot \sin re \]

   5. distribute-rgt-out-- 82.5%

      \[\leadsto \color{blue}{\sin re \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

   6. *-commutative 82.5%

      \[\leadsto \sin re \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

5. Simplified 82.5%

   \[\leadsto \color{blue}{\sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

6. Final simplification 82.5%

   \[\leadsto \sin re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right) \]
7. Add Preprocessing

Alternative 3: 55.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* -0.008333333333333333 (* (sin re) (pow im 5.0))))

C

double code(double re, double im) {
	return -0.008333333333333333 * (sin(re) * pow(im, 5.0));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (-0.008333333333333333d0) * (sin(re) * (im ** 5.0d0))
end function

Java

public static double code(double re, double im) {
	return -0.008333333333333333 * (Math.sin(re) * Math.pow(im, 5.0));
}

Python

def code(re, im):
	return -0.008333333333333333 * (math.sin(re) * math.pow(im, 5.0))

Julia

function code(re, im)
	return Float64(-0.008333333333333333 * Float64(sin(re) * (im ^ 5.0)))
end

MATLAB

function tmp = code(re, im)
	tmp = -0.008333333333333333 * (sin(re) * (im ^ 5.0));
end

Wolfram

code[re_, im_] := N[(-0.008333333333333333 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 89.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.3333333333333333 \cdot {im}^{3} + -0.016666666666666666 \cdot {im}^{5}\right)\right)} \]

4. Taylor expanded in im around inf 57.7%

   \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left({im}^{5} \cdot \sin re\right)} \]

5. Final simplification 57.7%

   \[\leadsto -0.008333333333333333 \cdot \left(\sin re \cdot {im}^{5}\right) \]
6. Add Preprocessing

Alternative 4: 51.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \left(-im\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 48.3%

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

   1. associate-*r* 48.3%

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

   2. neg-mul-1 48.3%

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

5. Simplified 48.3%

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

6. Final simplification 48.3%

   \[\leadsto \sin re \cdot \left(-im\right) \]
7. Add Preprocessing

Alternative 5: 32.7% accurate, 77.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0 48.3%

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

   1. associate-*r* 48.3%

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

   2. neg-mul-1 48.3%

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

5. Simplified 48.3%

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

6. Taylor expanded in re around 0 33.4%

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

   1. mul-1-neg 33.4%

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

   2. distribute-rgt-neg-in 33.4%

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

8. Simplified 33.4%

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

9. Final simplification 33.4%

   \[\leadsto -re \cdot im \]
10. Add Preprocessing

Alternative 6: 3.2% accurate, 102.7× speedup

Math

\[\begin{array}{l} \\ re \cdot 4 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* re 4.0))

C

double code(double re, double im) {
	return re * 4.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re * 4.0d0
end function

Java

public static double code(double re, double im) {
	return re * 4.0;
}

Python

def code(re, im):
	return re * 4.0

Julia

function code(re, im)
	return Float64(re * 4.0)
end

MATLAB

function tmp = code(re, im)
	tmp = re * 4.0;
end

Wolfram

code[re_, im_] := N[(re * 4.0), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in re around 0 54.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(e^{-im} - e^{im}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* 54.0%

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

   2. *-commutative 54.0%

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

5. Simplified 54.0%

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

7. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{8} \cdot \left(0.5 \cdot re\right) \]

8. Taylor expanded in re around 0 3.5%

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

   1. *-commutative 3.5%

      \[\leadsto \color{blue}{re \cdot 4} \]

10. Simplified 3.5%

    \[\leadsto \color{blue}{re \cdot 4} \]

11. Final simplification 3.5%

    \[\leadsto re \cdot 4 \]
12. Add Preprocessing

Developer target: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (sin re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (sin re)) (- (exp (- im)) (exp im)))))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(sin(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * sin(re)) * (exp(-im) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.sin(re)) * (Math.exp(-im) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.sin(re)) * (math.exp(-im) - math.exp(im))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(sin(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * sin(re)) * (exp(-im) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024033 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))