Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 10.4s

Alternatives: 6

Speedup: 2.0×

• 45.2% 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: 99.6% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (cos th) (pow (hypot a1 a2) 2.0)) (sqrt 0.5)))

C

double code(double a1, double a2, double th) {
	return (cos(th) * pow(hypot(a1, a2), 2.0)) * sqrt(0.5);
}

Java

public static double code(double a1, double a2, double th) {
	return (Math.cos(th) * Math.pow(Math.hypot(a1, a2), 2.0)) * Math.sqrt(0.5);
}

Python

def code(a1, a2, th):
	return (math.cos(th) * math.pow(math.hypot(a1, a2), 2.0)) * math.sqrt(0.5)

Julia

function code(a1, a2, th)
	return Float64(Float64(cos(th) * (hypot(a1, a2) ^ 2.0)) * sqrt(0.5))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (cos(th) * (hypot(a1, a2) ^ 2.0)) * sqrt(0.5);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 99.6%

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

   2. clear-num 99.4%

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

   3. add-sqr-sqrt 99.4%

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

   4. pow2 99.4%

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

   5. hypot-def 99.4%

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

6. Applied egg-rr 99.4%

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

   1. clear-num 99.6%

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

   2. div-inv 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. sqrt-unprod 99.6%

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

   5. frac-times 99.6%

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

   6. metadata-eval 99.6%

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

   7. rem-square-sqrt 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto \left(\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}\right) \cdot \sqrt{\color{blue}{0.5}} \]

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}\right) \cdot \sqrt{0.5}} \]

9. Final simplification 99.6%

   \[\leadsto \left(\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}\right) \cdot \sqrt{0.5} \]
10. 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 99.6% 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 56.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 99.6%

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

   2. clear-num 99.4%

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

   3. add-sqr-sqrt 99.4%

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

   4. pow2 99.4%

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

   5. hypot-def 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in a1 around 0 59.3%

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

   1. pow2 59.3%

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

   2. clear-num 59.3%

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

   3. *-commutative 59.3%

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

   4. associate-*r/ 59.3%

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

   5. associate-*l/ 59.3%

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

   6. associate-*r* 59.3%

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

9. Applied egg-rr 59.3%

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

10. Final simplification 59.3%

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

Alternative 5: 56.6% accurate, 2.0× speedup

Math

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

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(Float64(a2 / 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[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 99.6%

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

   2. clear-num 99.4%

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

   3. add-sqr-sqrt 99.4%

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

   4. pow2 99.4%

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

   5. hypot-def 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in a1 around 0 59.3%

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

   1. pow2 59.3%

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

   2. clear-num 59.3%

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

   3. associate-*l/ 59.3%

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

   4. associate-*r/ 59.3%

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

   5. *-commutative 59.3%

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

   6. associate-*l* 59.3%

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

9. Applied egg-rr 59.3%

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

10. Final simplification 59.3%

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

Alternative 6: 39.2% 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. associate-*l/ 99.6%

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

   2. clear-num 99.4%

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

   3. add-sqr-sqrt 99.4%

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

   4. pow2 99.4%

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

   5. hypot-def 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in a1 around 0 59.3%

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

   1. pow2 59.3%

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

   2. clear-num 59.3%

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

   3. associate-*l/ 59.3%

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

   4. associate-*r/ 59.3%

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

   5. *-commutative 59.3%

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

   6. associate-*l* 59.3%

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

9. Applied egg-rr 59.3%

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

10. Taylor expanded in th around 0 42.7%

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

11. Final simplification 42.7%

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

Reproduce

herbie shell --seed 2024041 
(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))))