Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.5s

Alternatives: 6

Speedup: 2.0×

• 43.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{0.5} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (sqrt 0.5) (cos th)) (+ (* a1 a1) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return (sqrt(0.5) * cos(th)) * ((a1 * a1) + (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (sqrt(0.5d0) * cos(th)) * ((a1 * a1) + (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return (Math.sqrt(0.5) * Math.cos(th)) * ((a1 * a1) + (a2 * a2));
}

Python

def code(a1, a2, th):
	return (math.sqrt(0.5) * math.cos(th)) * ((a1 * a1) + (a2 * a2))

Julia

function code(a1, a2, th)
	return Float64(Float64(sqrt(0.5) * cos(th)) * Float64(Float64(a1 * a1) + Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (sqrt(0.5) * cos(th)) * ((a1 * a1) + (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. associate-/r/ 99.5%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. pow1/2 99.5%

      \[\leadsto \left(\frac{1}{\color{blue}{{2}^{0.5}}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. pow-flip 99.7%

      \[\leadsto \left(\color{blue}{{2}^{\left(-0.5\right)}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   5. metadata-eval 99.7%

      \[\leadsto \left({2}^{\color{blue}{-0.5}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left({2}^{-0.5} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Taylor expanded in th around inf 99.7%

   \[\leadsto \color{blue}{\left(\cos th \cdot \sqrt{0.5}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
8. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

9. Simplified 99.7%

   \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

10. Final simplification 99.7%

    \[\leadsto \left(\sqrt{0.5} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
11. Add Preprocessing

Alternative 2: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2 \cdot \left(\frac{a2}{\sqrt{2}} \cdot \left(-\cos th\right)\right)}{-1} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ (* a2 (* (/ a2 (sqrt 2.0)) (- (cos th)))) -1.0))

C

double code(double a1, double a2, double th) {
	return (a2 * ((a2 / sqrt(2.0)) * -cos(th))) / -1.0;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * ((a2 / sqrt(2.0d0)) * -cos(th))) / (-1.0d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a2 * ((a2 / Math.sqrt(2.0)) * -Math.cos(th))) / -1.0;
}

Python

def code(a1, a2, th):
	return (a2 * ((a2 / math.sqrt(2.0)) * -math.cos(th))) / -1.0

Julia

function code(a1, a2, th)
	return Float64(Float64(a2 * Float64(Float64(a2 / sqrt(2.0)) * Float64(-cos(th)))) / -1.0)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a2 * ((a2 / sqrt(2.0)) * -cos(th))) / -1.0;
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Cos[th], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
4. Add Preprocessing

5. Taylor expanded in a1 around 0 54.9%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. associate-*l/ 54.8%

      \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}} \cdot \cos th} \]

7. Simplified 54.8%

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}} \cdot \cos th} \]
8. Step-by-step derivation

   1. pow2 40.3%

      \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   2. *-un-lft-identity 40.3%

      \[\leadsto \frac{a2 \cdot a2}{\color{blue}{1 \cdot \sqrt{2}}} \]

   3. times-frac 40.3%

      \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]

9. Applied egg-rr 54.9%

   \[\leadsto \color{blue}{\left(\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}\right)} \cdot \cos th \]
10. Step-by-step derivation

    1. /-rgt-identity 54.9%

       \[\leadsto \left(\color{blue}{a2} \cdot \frac{a2}{\sqrt{2}}\right) \cdot \cos th \]

    2. associate-*l* 54.9%

       \[\leadsto \color{blue}{a2 \cdot \left(\frac{a2}{\sqrt{2}} \cdot \cos th\right)} \]

    3. /-rgt-identity 54.9%

       \[\leadsto \color{blue}{\frac{a2}{1}} \cdot \left(\frac{a2}{\sqrt{2}} \cdot \cos th\right) \]

    4. frac-2neg 54.9%

       \[\leadsto \color{blue}{\frac{-a2}{-1}} \cdot \left(\frac{a2}{\sqrt{2}} \cdot \cos th\right) \]

    5. associate-*l/ 54.9%

       \[\leadsto \color{blue}{\frac{\left(-a2\right) \cdot \left(\frac{a2}{\sqrt{2}} \cdot \cos th\right)}{-1}} \]

    6. *-commutative 54.9%

       \[\leadsto \frac{\left(-a2\right) \cdot \color{blue}{\left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)}}{-1} \]

    7. metadata-eval 54.9%

       \[\leadsto \frac{\left(-a2\right) \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)}{\color{blue}{-1}} \]

11. Applied egg-rr 54.9%

    \[\leadsto \color{blue}{\frac{\left(-a2\right) \cdot \left(\cos th \cdot \frac{a2}{\sqrt{2}}\right)}{-1}} \]

12. Final simplification 54.9%

    \[\leadsto \frac{a2 \cdot \left(\frac{a2}{\sqrt{2}} \cdot \left(-\cos th\right)\right)}{-1} \]
13. Add Preprocessing

Alternative 3: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left(a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (cos th) (* a2 (* (sqrt 0.5) a2))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (a2 * (sqrt(0.5) * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * (a2 * (sqrt(0.5d0) * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (a2 * (Math.sqrt(0.5) * a2));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (a2 * (math.sqrt(0.5) * a2))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(a2 * Float64(sqrt(0.5) * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * (a2 * (sqrt(0.5) * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
4. Add Preprocessing

5. Taylor expanded in a1 around 0 54.9%

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. associate-*l/ 54.8%

      \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}} \cdot \cos th} \]

7. Simplified 54.8%

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}} \cdot \cos th} \]
8. Step-by-step derivation

   1. *-un-lft-identity 40.3%

      \[\leadsto \frac{\color{blue}{1 \cdot {a2}^{2}}}{\sqrt{2}} \]

   2. pow2 40.3%

      \[\leadsto \frac{1 \cdot \color{blue}{\left(a2 \cdot a2\right)}}{\sqrt{2}} \]

   3. associate-*l/ 40.3%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   4. associate-*r* 40.3%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot a2\right) \cdot a2} \]

   5. add-sqr-sqrt 40.3%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}\right)} \cdot a2\right) \cdot a2 \]

   6. sqrt-unprod 40.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}} \cdot a2\right) \cdot a2 \]

   7. frac-times 40.3%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}} \cdot a2\right) \cdot a2 \]

   8. metadata-eval 40.3%

      \[\leadsto \left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}} \cdot a2\right) \cdot a2 \]

   9. rem-square-sqrt 40.3%

      \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{2}}} \cdot a2\right) \cdot a2 \]

   10. metadata-eval 40.3%

       \[\leadsto \left(\sqrt{\color{blue}{0.5}} \cdot a2\right) \cdot a2 \]

9. Applied egg-rr 54.9%

   \[\leadsto \color{blue}{\left(\left(\sqrt{0.5} \cdot a2\right) \cdot a2\right)} \cdot \cos th \]

10. Final simplification 54.9%

    \[\leadsto \cos th \cdot \left(a2 \cdot \left(\sqrt{0.5} \cdot a2\right)\right) \]
11. Add Preprocessing

Alternative 4: 66.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return sqrt(0.5) * ((a1 * a1) + (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.sqrt(0.5) * ((a1 * a1) + (a2 * a2));
}

Python

def code(a1, a2, th):
	return math.sqrt(0.5) * ((a1 * a1) + (a2 * a2))

Julia

function code(a1, a2, th)
	return Float64(sqrt(0.5) * Float64(Float64(a1 * a1) + Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. associate-/r/ 99.5%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. pow1/2 99.5%

      \[\leadsto \left(\frac{1}{\color{blue}{{2}^{0.5}}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. pow-flip 99.7%

      \[\leadsto \left(\color{blue}{{2}^{\left(-0.5\right)}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   5. metadata-eval 99.7%

      \[\leadsto \left({2}^{\color{blue}{-0.5}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left({2}^{-0.5} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Taylor expanded in th around 0 69.0%

   \[\leadsto \color{blue}{\sqrt{0.5}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Final simplification 69.0%

   \[\leadsto \sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
9. Add Preprocessing

Alternative 5: 39.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \left(\sqrt{0.5} \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (* (sqrt 0.5) a2)))

C

double code(double a1, double a2, double th) {
	return a2 * (sqrt(0.5) * a2);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (sqrt(0.5d0) * a2)
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (Math.sqrt(0.5) * a2);
}

Python

def code(a1, a2, th):
	return a2 * (math.sqrt(0.5) * a2)

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(sqrt(0.5) * a2))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (sqrt(0.5) * a2);
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
4. Add Preprocessing

5. Taylor expanded in th around 0 69.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Taylor expanded in a1 around 0 40.3%

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 40.3%

      \[\leadsto \frac{\color{blue}{1 \cdot {a2}^{2}}}{\sqrt{2}} \]

   2. pow2 40.3%

      \[\leadsto \frac{1 \cdot \color{blue}{\left(a2 \cdot a2\right)}}{\sqrt{2}} \]

   3. associate-*l/ 40.3%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)} \]

   4. associate-*r* 40.3%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot a2\right) \cdot a2} \]

   5. add-sqr-sqrt 40.3%

      \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}\right)} \cdot a2\right) \cdot a2 \]

   6. sqrt-unprod 40.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}} \cdot a2\right) \cdot a2 \]

   7. frac-times 40.3%

      \[\leadsto \left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}} \cdot a2\right) \cdot a2 \]

   8. metadata-eval 40.3%

      \[\leadsto \left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}} \cdot a2\right) \cdot a2 \]

   9. rem-square-sqrt 40.3%

      \[\leadsto \left(\sqrt{\frac{1}{\color{blue}{2}}} \cdot a2\right) \cdot a2 \]

   10. metadata-eval 40.3%

       \[\leadsto \left(\sqrt{\color{blue}{0.5}} \cdot a2\right) \cdot a2 \]

8. Applied egg-rr 40.3%

   \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot a2\right) \cdot a2} \]

9. Final simplification 40.3%

   \[\leadsto a2 \cdot \left(\sqrt{0.5} \cdot a2\right) \]
10. Add Preprocessing

Alternative 6: 39.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (/ a2 (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return a2 * (a2 / sqrt(2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 / sqrt(2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 / Math.sqrt(2.0));
}

Python

def code(a1, a2, th):
	return a2 * (a2 / math.sqrt(2.0))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 / sqrt(2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 / sqrt(2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
4. Add Preprocessing

5. Taylor expanded in th around 0 69.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Taylor expanded in a1 around 0 40.3%

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
7. Step-by-step derivation

   1. pow2 40.3%

      \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   2. *-un-lft-identity 40.3%

      \[\leadsto \frac{a2 \cdot a2}{\color{blue}{1 \cdot \sqrt{2}}} \]

   3. times-frac 40.3%

      \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]

8. Applied egg-rr 40.3%

   \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]

9. Final simplification 40.3%

   \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))