Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 9.5s

Alternatives: 8

Speedup: 2.0×

• 42.7% 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: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return (cos(th) / sqrt(2.0)) * ((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 = (cos(th) / sqrt(2.0d0)) * ((a1 * a1) + (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. pow1/2 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. sqrt-pow1 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
7. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. *-lft-identity 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\cos th}{{2}^{0.25}}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
9. Step-by-step derivation

   1. associate-/l/ 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{{2}^{0.25} \cdot {2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. pow-prod-up 99.7%

      \[\leadsto \frac{\cos th}{\color{blue}{{2}^{\left(0.25 + 0.25\right)}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. metadata-eval 99.7%

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

   4. pow1/2 99.7%

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

   5. clear-num 99.7%

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

   6. associate-/r/ 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. sqrt-unprod 99.6%

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

   9. frac-times 99.6%

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

   10. metadata-eval 99.6%

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

   11. rem-square-sqrt 99.6%

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

   12. metadata-eval 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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) * (a2 * sqrt(0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a1 around 0 51.0%

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

   1. associate-/l* 51.0%

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

7. Simplified 51.0%

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

   1. pow2 51.0%

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

   2. *-un-lft-identity 51.0%

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

   3. times-frac 51.0%

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

9. Applied egg-rr 51.0%

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

    1. associate-*l/ 51.0%

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

    2. associate-/l* 51.0%

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

    3. clear-num 51.0%

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

    4. associate-/l/ 51.0%

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

11. Applied egg-rr 51.0%

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

    1. associate-/r/ 51.0%

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

    2. div-inv 51.0%

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

    3. pow1/2 51.0%

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

    4. pow-flip 51.0%

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

    5. metadata-eval 51.0%

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

    6. add-sqr-sqrt 50.8%

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

    7. sqrt-unprod 51.0%

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

    8. pow-prod-up 51.0%

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

    9. metadata-eval 51.0%

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

    10. metadata-eval 51.0%

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

13. Applied egg-rr 51.0%

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

14. Final simplification 51.0%

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

Alternative 4: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 * ((cos(th) / sqrt(2.0)) * 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 * ((cos(th) / sqrt(2.0d0)) * a2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a1 around 0 51.0%

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

   1. associate-/l* 51.0%

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

7. Simplified 51.0%

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

   1. pow2 51.0%

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

   2. *-un-lft-identity 51.0%

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

   3. times-frac 51.0%

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

9. Applied egg-rr 51.0%

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

    1. associate-*l/ 51.0%

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

    2. associate-/l* 51.0%

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

    3. clear-num 51.0%

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

    4. associate-/l/ 51.0%

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

11. Applied egg-rr 51.0%

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

    1. clear-num 50.7%

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

    2. associate-/r/ 51.0%

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

    3. clear-num 51.0%

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

    4. *-un-lft-identity 51.0%

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

    5. times-frac 51.0%

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

    6. /-rgt-identity 51.0%

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

13. Applied egg-rr 51.0%

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

14. Final simplification 51.0%

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

Alternative 5: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 / (sqrt(2.0) / (cos(th) * 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(2.0d0) / (cos(th) * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a1 around 0 51.0%

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

   1. associate-/l* 51.0%

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

7. Simplified 51.0%

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

   1. pow2 51.0%

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

   2. *-un-lft-identity 51.0%

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

   3. times-frac 51.0%

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

9. Applied egg-rr 51.0%

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

    1. associate-*l/ 51.0%

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

    2. associate-/l* 51.0%

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

    3. clear-num 51.0%

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

    4. associate-/l/ 51.0%

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

11. Applied egg-rr 51.0%

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

12. Final simplification 51.0%

    \[\leadsto \frac{a2}{\frac{\sqrt{2}}{\cos th \cdot a2}} \]
13. Add Preprocessing

Alternative 6: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in a1 around 0 51.0%

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

   1. associate-/l* 51.0%

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

7. Simplified 51.0%

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

   1. pow2 51.0%

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

   2. *-un-lft-identity 51.0%

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

   3. times-frac 51.0%

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

9. Applied egg-rr 51.0%

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

10. Final simplification 51.0%

    \[\leadsto a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}} \]
11. Add Preprocessing

Alternative 7: 39.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in th around 0 68.2%

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

6. Taylor expanded in a1 around inf 42.0%

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} \]
7. Step-by-step derivation

   1. div-inv 42.0%

      \[\leadsto \color{blue}{{a1}^{2} \cdot \frac{1}{\sqrt{2}}} \]

   2. pow1/2 42.0%

      \[\leadsto {a1}^{2} \cdot \frac{1}{\color{blue}{{2}^{0.5}}} \]

   3. pow-flip 42.0%

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

   4. metadata-eval 42.0%

      \[\leadsto {a1}^{2} \cdot {2}^{\color{blue}{-0.5}} \]

   5. unpow2 42.0%

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

   6. associate-*l* 42.0%

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

   7. add-sqr-sqrt 41.9%

      \[\leadsto a1 \cdot \left(a1 \cdot \color{blue}{\left(\sqrt{{2}^{-0.5}} \cdot \sqrt{{2}^{-0.5}}\right)}\right) \]

   8. sqrt-unprod 42.0%

      \[\leadsto a1 \cdot \left(a1 \cdot \color{blue}{\sqrt{{2}^{-0.5} \cdot {2}^{-0.5}}}\right) \]

   9. pow-prod-up 42.0%

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

   10. metadata-eval 42.0%

       \[\leadsto a1 \cdot \left(a1 \cdot \sqrt{{2}^{\color{blue}{-1}}}\right) \]

   11. metadata-eval 42.0%

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

8. Applied egg-rr 42.0%

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

9. Final simplification 42.0%

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

Alternative 8: 40.1% 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 99.7%

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

   1. distribute-lft-out 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in th around 0 68.2%

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

6. Taylor expanded in a1 around 0 37.7%

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

   1. pow2 37.7%

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

   2. *-un-lft-identity 37.7%

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

   3. times-frac 37.7%

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

   4. /-rgt-identity 37.7%

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

8. Applied egg-rr 37.7%

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

9. Final simplification 37.7%

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

Reproduce

herbie shell --seed 2024021 
(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))))