Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.8s

Alternatives: 7

Speedup: 2.0×

• 45.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \cos th \cdot \frac{\mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (cos th) (/ (fma a2 a2 (* a1_m a1_m)) (sqrt 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return cos(th) * (fma(a2, a2, (a1_m * a1_m)) / sqrt(2.0));
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(cos(th) * Float64(fma(a2, a2, Float64(a1_m * a1_m)) / sqrt(2.0)))
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2 + N[(a1$95$m * a1$95$m), $MachinePrecision]), $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. 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/ 99.7%

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

   4. associate-/l* 99.7%

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

   5. cos-neg 99.7%

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

   6. +-commutative 99.7%

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

   7. fma-define 99.7%

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

3. Simplified 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \left(\cos th \cdot \sqrt{0.5}\right) \cdot \left(a1\_m \cdot a1\_m + a2 \cdot a2\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (* (cos th) (sqrt 0.5)) (+ (* a1_m a1_m) (* a2 a2))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return (cos(th) * sqrt(0.5)) * ((a1_m * a1_m) + (a2 * a2));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (cos(th) * sqrt(0.5d0)) * ((a1_m * a1_m) + (a2 * a2))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return (Math.cos(th) * Math.sqrt(0.5)) * ((a1_m * a1_m) + (a2 * a2));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return (math.cos(th) * math.sqrt(0.5)) * ((a1_m * a1_m) + (a2 * a2))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(Float64(cos(th) * sqrt(0.5)) * Float64(Float64(a1_m * a1_m) + Float64(a2 * a2)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = (cos(th) * sqrt(0.5)) * ((a1_m * a1_m) + (a2 * a2));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(a1$95$m * a1$95$m), $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. Step-by-step derivation

   1. add-cbrt-cube 99.1%

      \[\leadsto \frac{\cos th}{\color{blue}{\sqrt[3]{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. pow1/3 99.6%

      \[\leadsto \frac{\cos th}{\color{blue}{{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. rem-square-sqrt 99.6%

      \[\leadsto \frac{\cos th}{{\left(\color{blue}{2} \cdot \sqrt{2}\right)}^{0.3333333333333333}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{\cos th}{\color{blue}{{\left(2 \cdot \sqrt{2}\right)}^{0.3333333333333333}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
7. Step-by-step derivation

   1. unpow1/3 99.1%

      \[\leadsto \frac{\cos th}{\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. *-commutative 99.1%

      \[\leadsto \frac{\cos th}{\sqrt[3]{\color{blue}{\sqrt{2} \cdot 2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. unpow1/2 99.1%

      \[\leadsto \frac{\cos th}{\sqrt[3]{\color{blue}{{2}^{0.5}} \cdot 2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. pow-plus 99.1%

      \[\leadsto \frac{\cos th}{\sqrt[3]{\color{blue}{{2}^{\left(0.5 + 1\right)}}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   5. metadata-eval 99.1%

      \[\leadsto \frac{\cos th}{\sqrt[3]{{2}^{\color{blue}{1.5}}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Simplified 99.1%

   \[\leadsto \frac{\cos th}{\color{blue}{\sqrt[3]{{2}^{1.5}}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
9. Step-by-step derivation

   1. clear-num 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{{2}^{1.5}}}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. pow1/3 99.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{{\left({2}^{1.5}\right)}^{0.3333333333333333}}}{\cos th}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. pow-pow 99.6%

      \[\leadsto \frac{1}{\frac{\color{blue}{{2}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{\cos th}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. metadata-eval 99.6%

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

   5. pow1/2 99.6%

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

   6. associate-/r/ 99.5%

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

   7. add-sqr-sqrt 99.5%

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

   8. sqrt-unprod 99.5%

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

   9. frac-times 99.5%

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

   10. metadata-eval 99.5%

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

   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%

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

Alternative 3: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \frac{\mathsf{fma}\left(a2, a2, a1\_m \cdot a1\_m\right)}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (/ (fma a2 a2 (* a1_m a1_m)) (sqrt 2.0)))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return fma(a2, a2, (a1_m * a1_m)) / sqrt(2.0);
}

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(fma(a2, a2, Float64(a1_m * a1_m)) / sqrt(2.0))
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[(a2 * a2 + N[(a1$95$m * a1$95$m), $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/ 99.7%

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

   4. associate-/l* 99.7%

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

   5. cos-neg 99.7%

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

   6. +-commutative 99.7%

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

   7. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in th around 0 69.4%

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

6. Final simplification 69.4%

   \[\leadsto \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}} \]
7. Add Preprocessing

Alternative 4: 67.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ \sqrt{0.5} \cdot \left(a1\_m \cdot a1\_m + a2 \cdot a2\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th)
 :precision binary64
 (* (sqrt 0.5) (+ (* a1_m a1_m) (* a2 a2))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return sqrt(0.5) * ((a1_m * a1_m) + (a2 * a2));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = sqrt(0.5d0) * ((a1_m * a1_m) + (a2 * a2))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return Math.sqrt(0.5) * ((a1_m * a1_m) + (a2 * a2));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return math.sqrt(0.5) * ((a1_m * a1_m) + (a2 * a2))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(sqrt(0.5) * Float64(Float64(a1_m * a1_m) + Float64(a2 * a2)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = sqrt(0.5) * ((a1_m * a1_m) + (a2 * a2));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a1$95$m * a1$95$m), $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. 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/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   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 0 69.4%

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

Alternative 5: 53.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ a2 \cdot \left(a2 \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* a2 (* a2 (sqrt 0.5))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return a2 * (a2 * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return a2 * (a2 * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return a2 * (a2 * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a2 * Float64(a2 * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = a2 * (a2 * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, 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. 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. Taylor expanded in th around 0 69.3%

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

6. Taylor expanded in a1 around 0 37.0%

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

   1. pow2 37.0%

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

   2. clear-num 37.0%

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

   3. associate-/r/ 37.0%

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

   4. metadata-eval 37.0%

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

   5. metadata-eval 37.0%

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

   6. rem-square-sqrt 37.0%

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

   7. frac-times 37.0%

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

   8. sqrt-unprod 37.0%

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

   9. add-sqr-sqrt 37.0%

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

   10. remove-double-div 37.0%

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

   11. associate-*r* 37.0%

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

8. Applied egg-rr 37.0%

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

9. Final simplification 37.0%

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

Alternative 6: 53.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* a2 (/ a2 (sqrt 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return a2 * (a2 / sqrt(2.0));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return a2 * (a2 / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return a2 * (a2 / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a2 * Float64(a2 / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = a2 * (a2 / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, 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. Taylor expanded in th around 0 69.3%

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

6. Taylor expanded in a1 around 0 37.0%

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

   1. pow2 37.0%

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

   2. associate-/l* 37.0%

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

8. Applied egg-rr 37.0%

   \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]
9. Add Preprocessing

Alternative 7: 27.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ [a1_m, a2, th] = \mathsf{sort}([a1_m, a2, th])\\ \\ a1\_m \cdot \frac{a1\_m}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2 th) :precision binary64 (* a1_m (/ a1_m (sqrt 2.0))))

C

a1_m = fabs(a1);
assert(a1_m < a2 && a2 < th);
double code(double a1_m, double a2, double th) {
	return a1_m * (a1_m / sqrt(2.0));
}

Fortran

a1_m = abs(a1)
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a1_m * (a1_m / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
assert a1_m < a2 && a2 < th;
public static double code(double a1_m, double a2, double th) {
	return a1_m * (a1_m / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
[a1_m, a2, th] = sort([a1_m, a2, th])
def code(a1_m, a2, th):
	return a1_m * (a1_m / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a1_m, a2, th = sort([a1_m, a2, th])
function code(a1_m, a2, th)
	return Float64(a1_m * Float64(a1_m / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a1_m, a2, th = num2cell(sort([a1_m, a2, th])){:}
function tmp = code(a1_m, a2, th)
	tmp = a1_m * (a1_m / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
NOTE: a1_m, a2, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2_, th_] := N[(a1$95$m * N[(a1$95$m / 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. Taylor expanded in th around 0 69.3%

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

6. Taylor expanded in a1 around inf 46.5%

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

   1. pow2 46.5%

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

   2. associate-/l* 46.5%

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

8. Applied egg-rr 46.5%

   \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))