Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.3%

Time: 10.2s

Alternatives: 15

Speedup: 2.0×

• 44.1% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. associate-/r/ 99.7%

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

   2. add-sqr-sqrt 99.7%

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

   3. pow2 99.7%

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

   4. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. associate-/r/ 99.7%

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

   2. add-sqr-sqrt 99.7%

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

   3. pow2 99.7%

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

   4. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around inf 99.6%

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

   1. associate-/l* 99.7%

      \[\leadsto \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}}} \]

   2. unpow2 99.7%

      \[\leadsto \frac{\color{blue}{a2 \cdot a2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}} \]

   3. +-commutative 99.7%

      \[\leadsto \frac{\color{blue}{{a1}^{2} + a2 \cdot a2}}{\frac{\sqrt{2}}{\cos th}} \]

   4. unpow2 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}} \]

Alternative 5: 58.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] / 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a2 around inf 52.0%

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

   1. unpow2 52.0%

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

   2. associate-*r/ 52.0%

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

   3. associate-*r* 52.1%

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

6. Simplified 52.1%

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

7. Final simplification 52.1%

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

Alternative 6: 58.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return sqrt(0.5) * (a2 * (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 = sqrt(0.5d0) * (a2 * (cos(th) * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * N[(N[Cos[th], $MachinePrecision] * 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in a2 around inf 52.1%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} \cdot \cos th\right)} \]
7. Step-by-step derivation

   1. unpow2 52.1%

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

   2. associate-*r* 52.1%

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

8. Simplified 52.1%

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

9. Final simplification 52.1%

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

Alternative 7: 58.0% accurate, 2.0× speedup

Math

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

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(cos(th) * Float64(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[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 * N[Sqrt[0.5], $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a2 around inf 52.0%

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

   1. unpow2 52.0%

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

   2. associate-*l/ 52.1%

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

6. Simplified 52.1%

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

   1. *-un-lft-identity 37.7%

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

   2. associate-*l/ 37.6%

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

   3. associate-*r* 37.6%

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

   4. add-sqr-sqrt 37.6%

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

   5. sqrt-unprod 37.6%

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

   6. frac-times 37.6%

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

   7. metadata-eval 37.6%

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

   8. add-sqr-sqrt 37.7%

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

   9. metadata-eval 37.7%

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

8. Applied egg-rr 52.1%

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

9. Final simplification 52.1%

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

Alternative 8: 58.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a2 around inf 52.0%

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

   1. unpow2 52.0%

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

   2. associate-*l/ 52.1%

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

6. Simplified 52.1%

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

   1. associate-/l* 52.1%

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

   2. div-inv 52.1%

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

8. Applied egg-rr 52.1%

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

   1. associate-*r/ 52.1%

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

   2. *-rgt-identity 52.1%

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

10. Simplified 52.1%

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

11. Final simplification 52.1%

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

Alternative 9: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2.1 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(th \cdot \left(th \cdot -0.5\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 5e+154)
   (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2.0))
   (if (<= a2 2.1e+188)
     (* (sqrt 0.5) (* (* a2 a2) (+ (* th (* th -0.5)) 1.0)))
     (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 5e+154) {
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	} else if (a2 <= 2.1e+188) {
		tmp = sqrt(0.5) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0));
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

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) :: tmp
    if (a2 <= 5d+154) then
        tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0d0)
    else if (a2 <= 2.1d+188) then
        tmp = sqrt(0.5d0) * ((a2 * a2) * ((th * (th * (-0.5d0))) + 1.0d0))
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 5e+154) {
		tmp = ((a1 * a1) + (a2 * a2)) / Math.sqrt(2.0);
	} else if (a2 <= 2.1e+188) {
		tmp = Math.sqrt(0.5) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 5e+154:
		tmp = ((a1 * a1) + (a2 * a2)) / math.sqrt(2.0)
	elif a2 <= 2.1e+188:
		tmp = math.sqrt(0.5) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 5e+154)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / sqrt(2.0));
	elseif (a2 <= 2.1e+188)
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) * Float64(Float64(th * Float64(th * -0.5)) + 1.0)));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 5e+154)
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	elseif (a2 <= 2.1e+188)
		tmp = sqrt(0.5) * ((a2 * a2) * ((th * (th * -0.5)) + 1.0));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 5e+154], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 2.1e+188], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[(N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 5.00000000000000004e154

   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. +-commutative 99.6%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. associate-/r/ 99.7%

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

      2. add-sqr-sqrt 99.7%

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

      3. pow2 99.7%

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

      4. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around inf 99.6%

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

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}}} \]

      2. unpow2 99.6%

         \[\leadsto \frac{\color{blue}{a2 \cdot a2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}} \]

      3. +-commutative 99.6%

         \[\leadsto \frac{\color{blue}{{a1}^{2} + a2 \cdot a2}}{\frac{\sqrt{2}}{\cos th}} \]

      4. unpow2 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in th around 0 68.2%

      \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\color{blue}{\sqrt{2}}} \]

   if 5.00000000000000004e154 < a2 < 2.09999999999999986e188

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. pow1/2 100.0%

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

      4. pow-flip 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in a2 around inf 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} \cdot \cos th\right)} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in th around 0 0.0%

      \[\leadsto \sqrt{0.5} \cdot \color{blue}{\left({a2}^{2} + -0.5 \cdot \left({th}^{2} \cdot {a2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

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

       2. unpow2 0.0%

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

       3. associate-*r* 0.0%

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

       4. distribute-rgt1-in 100.0%

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

       5. unpow2 100.0%

          \[\leadsto \sqrt{0.5} \cdot \left(\left(-0.5 \cdot \color{blue}{\left(th \cdot th\right)} + 1\right) \cdot \left(a2 \cdot a2\right)\right) \]

       6. associate-*r* 100.0%

          \[\leadsto \sqrt{0.5} \cdot \left(\left(\color{blue}{\left(-0.5 \cdot th\right) \cdot th} + 1\right) \cdot \left(a2 \cdot a2\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \sqrt{0.5} \cdot \color{blue}{\left(\left(\left(-0.5 \cdot th\right) \cdot th + 1\right) \cdot \left(a2 \cdot a2\right)\right)} \]

   if 2.09999999999999986e188 < a2

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in th around 0 81.8%

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

   5. Taylor expanded in a2 around inf 81.8%

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

      1. unpow2 81.8%

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

      2. associate-*r/ 81.8%

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

   7. Simplified 81.8%

      \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 5 \cdot 10^{+154}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 2.1 \cdot 10^{+188}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(th \cdot \left(th \cdot -0.5\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 10: 66.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return ((a1 * a1) + (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 = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 68.9%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
7. Step-by-step derivation

   1. *-commutative 68.9%

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

   2. unpow2 68.9%

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

   3. +-commutative 68.9%

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

   4. unpow2 68.9%

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

8. Simplified 68.9%

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

9. Final simplification 68.9%

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

Alternative 11: 66.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. associate-/r/ 99.7%

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

   2. add-sqr-sqrt 99.7%

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

   3. pow2 99.7%

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

   4. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around inf 99.6%

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

   1. associate-/l* 99.7%

      \[\leadsto \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}}} \]

   2. unpow2 99.7%

      \[\leadsto \frac{\color{blue}{a2 \cdot a2} + {a1}^{2}}{\frac{\sqrt{2}}{\cos th}} \]

   3. +-commutative 99.7%

      \[\leadsto \frac{\color{blue}{{a1}^{2} + a2 \cdot a2}}{\frac{\sqrt{2}}{\cos th}} \]

   4. unpow2 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in th around 0 68.9%

   \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\color{blue}{\sqrt{2}}} \]

10. Final simplification 68.9%

    \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}} \]

Alternative 12: 40.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}} \end{array} \]

FPCore

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

C

double code(double a1, double a2, double th) {
	return 1.0 / (sqrt(2.0) / (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 = 1.0d0 / (sqrt(2.0d0) / (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(1.0 / N[(N[Sqrt[2.0], $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 68.9%

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

5. Taylor expanded in a2 around inf 37.7%

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

   1. unpow2 37.7%

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

   2. associate-*r/ 37.6%

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

7. Simplified 37.6%

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

   1. associate-*r/ 37.7%

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

   2. clear-num 37.7%

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

9. Applied egg-rr 37.7%

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

10. Final simplification 37.7%

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

Alternative 13: 40.5% 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 68.9%

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

5. Taylor expanded in a2 around inf 37.7%

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

   1. unpow2 37.7%

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

   2. associate-*r/ 37.6%

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

7. Simplified 37.6%

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

8. Final simplification 37.6%

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

Alternative 14: 40.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 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 = a2 * (a2 * sqrt(0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * 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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 68.9%

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

5. Taylor expanded in a2 around inf 37.7%

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

   1. unpow2 37.7%

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

7. Simplified 37.7%

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

   1. *-un-lft-identity 37.7%

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

   2. associate-*l/ 37.6%

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

   3. associate-*r* 37.6%

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

   4. add-sqr-sqrt 37.6%

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

   5. sqrt-unprod 37.6%

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

   6. frac-times 37.6%

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

   7. metadata-eval 37.6%

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

   8. add-sqr-sqrt 37.7%

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

   9. metadata-eval 37.7%

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

9. Applied egg-rr 37.7%

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

10. Final simplification 37.7%

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

Alternative 15: 40.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 / N[(N[Sqrt[2.0], $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 68.9%

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

5. Taylor expanded in a2 around inf 37.7%

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

   1. unpow2 37.7%

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

   2. associate-*r/ 37.6%

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

7. Simplified 37.6%

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

   1. clear-num 37.7%

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

   2. un-div-inv 37.7%

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

9. Applied egg-rr 37.7%

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

10. Final simplification 37.7%

    \[\leadsto \frac{a2}{\frac{\sqrt{2}}{a2}} \]

Reproduce

herbie shell --seed 2023273 
(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))))