Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.0%

Time: 7.2s

Alternatives: 5

Speedup: N/A×

• 43.9% 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: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ a2\_m \cdot \left(\cos th \cdot \left(a2\_m \cdot \sqrt{0.5}\right)\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* a2_m (* (cos th) (* a2_m (sqrt 0.5)))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return a2_m * (cos(th) * (a2_m * sqrt(0.5)));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = a2_m * (cos(th) * (a2_m * sqrt(0.5d0)))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return a2_m * (Math.cos(th) * (a2_m * Math.sqrt(0.5)));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return a2_m * (math.cos(th) * (a2_m * math.sqrt(0.5)))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(a2_m * Float64(cos(th) * Float64(a2_m * sqrt(0.5))))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = a2_m * (cos(th) * (a2_m * sqrt(0.5)));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(a2$95$m * N[(N[Cos[th], $MachinePrecision] * N[(a2$95$m * 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. 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. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 57.7%

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

   1. clear-num 57.6%

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

   2. pow2 57.6%

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

   3. associate-/r/ 57.7%

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

   4. *-commutative 57.7%

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

   5. associate-*l* 57.7%

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

   6. pow1/2 57.7%

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

   7. pow-flip 57.7%

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

   8. metadata-eval 57.7%

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

   9. associate-*r* 57.8%

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

7. Applied egg-rr 57.8%

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

8. Taylor expanded in th around inf 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*r* 57.8%

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

   3. *-commutative 57.8%

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

10. Simplified 57.8%

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

11. Final simplification 57.8%

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

Alternative 2: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \cos th \cdot \left(a2\_m \cdot \frac{a2\_m}{\sqrt{2}}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (cos th) (* a2_m (/ a2_m (sqrt 2.0)))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return cos(th) * (a2_m * (a2_m / sqrt(2.0)));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = cos(th) * (a2_m * (a2_m / sqrt(2.0d0)))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return Math.cos(th) * (a2_m * (a2_m / Math.sqrt(2.0)));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return math.cos(th) * (a2_m * (a2_m / math.sqrt(2.0)))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(cos(th) * Float64(a2_m * Float64(a2_m / sqrt(2.0))))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = cos(th) * (a2_m * (a2_m / sqrt(2.0)));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2$95$m * N[(a2$95$m / 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%

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

   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. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 57.7%

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

   1. pow2 41.7%

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

   2. associate-/l* 41.7%

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

7. Applied egg-rr 57.7%

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

8. Final simplification 57.7%

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

Alternative 3: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ \cos th \cdot \left(a2\_m \cdot \left(a2\_m \cdot \sqrt{0.5}\right)\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th)
 :precision binary64
 (* (cos th) (* a2_m (* a2_m (sqrt 0.5)))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return cos(th) * (a2_m * (a2_m * sqrt(0.5)));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = cos(th) * (a2_m * (a2_m * sqrt(0.5d0)))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return Math.cos(th) * (a2_m * (a2_m * Math.sqrt(0.5)));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return math.cos(th) * (a2_m * (a2_m * math.sqrt(0.5)))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(cos(th) * Float64(a2_m * Float64(a2_m * sqrt(0.5))))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = cos(th) * (a2_m * (a2_m * sqrt(0.5)));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2$95$m * N[(a2$95$m * 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. 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. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 57.7%

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

   1. *-un-lft-identity 57.7%

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

   2. pow2 57.7%

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

   3. associate-*l/ 57.7%

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

   4. associate-*r* 57.7%

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

   5. add-sqr-sqrt 57.7%

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

   6. sqrt-unprod 57.7%

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

   7. frac-times 57.7%

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

   8. metadata-eval 57.7%

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

   9. rem-square-sqrt 57.8%

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

   10. metadata-eval 57.8%

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

7. Applied egg-rr 57.8%

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

8. Final simplification 57.8%

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

Alternative 4: 67.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ a2\_m \cdot \frac{a2\_m}{\sqrt{2}} \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th) :precision binary64 (* a2_m (/ a2_m (sqrt 2.0))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m / sqrt(2.0));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = a2_m * (a2_m / sqrt(2.0d0))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m / Math.sqrt(2.0));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return a2_m * (a2_m / math.sqrt(2.0))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(a2_m * Float64(a2_m / sqrt(2.0)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = a2_m * (a2_m / sqrt(2.0));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(a2$95$m * N[(a2$95$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 66.0%

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

6. Taylor expanded in a1 around 0 41.7%

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

   1. pow2 41.7%

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

   2. associate-/l* 41.7%

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

8. Applied egg-rr 41.7%

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

9. Final simplification 41.7%

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

Alternative 5: 67.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} a1_m = \left|a1\right| \\ a2_m = \left|a2\right| \\ [a1_m, a2_m, th] = \mathsf{sort}([a1_m, a2_m, th])\\ \\ a2\_m \cdot \left(a2\_m \cdot \sqrt{0.5}\right) \end{array} \]

FPCore

a1_m = (fabs.f64 a1)
a2_m = (fabs.f64 a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1_m a2_m th) :precision binary64 (* a2_m (* a2_m (sqrt 0.5))))

C

a1_m = fabs(a1);
a2_m = fabs(a2);
assert(a1_m < a2_m && a2_m < th);
double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m * sqrt(0.5));
}

Fortran

a1_m = abs(a1)
a2_m = abs(a2)
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1_m, a2_m, th)
    real(8), intent (in) :: a1_m
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = a2_m * (a2_m * sqrt(0.5d0))
end function

Java

a1_m = Math.abs(a1);
a2_m = Math.abs(a2);
assert a1_m < a2_m && a2_m < th;
public static double code(double a1_m, double a2_m, double th) {
	return a2_m * (a2_m * Math.sqrt(0.5));
}

Python

a1_m = math.fabs(a1)
a2_m = math.fabs(a2)
[a1_m, a2_m, th] = sort([a1_m, a2_m, th])
def code(a1_m, a2_m, th):
	return a2_m * (a2_m * math.sqrt(0.5))

Julia

a1_m = abs(a1)
a2_m = abs(a2)
a1_m, a2_m, th = sort([a1_m, a2_m, th])
function code(a1_m, a2_m, th)
	return Float64(a2_m * Float64(a2_m * sqrt(0.5)))
end

MATLAB

a1_m = abs(a1);
a2_m = abs(a2);
a1_m, a2_m, th = num2cell(sort([a1_m, a2_m, th])){:}
function tmp = code(a1_m, a2_m, th)
	tmp = a2_m * (a2_m * sqrt(0.5));
end

Wolfram

a1_m = N[Abs[a1], $MachinePrecision]
a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1_m, a2_m, and th should be sorted in increasing order before calling this function.
code[a1$95$m_, a2$95$m_, th_] := N[(a2$95$m * N[(a2$95$m * 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%

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

   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. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 57.7%

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

   1. clear-num 57.6%

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

   2. pow2 57.6%

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

   3. associate-/r/ 57.7%

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

   4. *-commutative 57.7%

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

   5. associate-*l* 57.7%

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

   6. pow1/2 57.7%

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

   7. pow-flip 57.7%

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

   8. metadata-eval 57.7%

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

   9. associate-*r* 57.8%

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

7. Applied egg-rr 57.8%

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

8. Taylor expanded in th around 0 41.8%

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

9. Final simplification 41.8%

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

Reproduce

herbie shell --seed 2024058 
(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))))