Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.4s

Alternatives: 9

Speedup: 2.0×

• 42.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{\cos th}\\ \mathsf{fma}\left(\frac{a2_m}{t_1}, a2_m, \frac{{a1}^{2}}{t_1}\right) \end{array} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (let* ((t_1 (/ (sqrt 2.0) (cos th))))
   (fma (/ a2_m t_1) a2_m (/ (pow a1 2.0) t_1))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	double t_1 = sqrt(2.0) / cos(th);
	return fma((a2_m / t_1), a2_m, (pow(a1, 2.0) / t_1));
}

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	t_1 = Float64(sqrt(2.0) / cos(th))
	return fma(Float64(a2_m / t_1), a2_m, Float64((a1 ^ 2.0) / t_1))
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]}, N[(N[(a2$95$m / t$95$1), $MachinePrecision] * a2$95$m + N[(N[Power[a1, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. associate-*r* 99.6%

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

   4. fma-def 99.6%

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

   5. clear-num 99.5%

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

   6. associate-*l/ 99.6%

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

   7. *-un-lft-identity 99.6%

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

   8. *-commutative 99.6%

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

   9. clear-num 99.6%

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

   10. un-div-inv 99.6%

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

   11. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[N[Sqrt[a1 ^ 2 + a2$95$m ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 79.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)\right)} \]

   2. expm1-udef 58.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)} - 1} \]

   3. add-sqr-sqrt 58.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\cos th}{\sqrt{2}} \cdot \color{blue}{\left(\sqrt{a1 \cdot a1 + a2 \cdot a2} \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}\right)}\right)} - 1 \]

   4. pow2 58.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\cos th}{\sqrt{2}} \cdot \color{blue}{{\left(\sqrt{a1 \cdot a1 + a2 \cdot a2}\right)}^{2}}\right)} - 1 \]

   5. hypot-def 58.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\cos th}{\sqrt{2}} \cdot {\color{blue}{\left(\mathsf{hypot}\left(a1, a2\right)\right)}}^{2}\right)} - 1 \]

5. Applied egg-rr 58.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\cos th}{\sqrt{2}} \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 79.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cos th}{\sqrt{2}} \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}\right)\right)} \]

   2. expm1-log1p 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r/ 99.6%

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

   5. associate-*l/ 99.6%

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

   6. *-commutative 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \left(\cos th \cdot {2}^{-0.5}\right) \cdot \left(a1 \cdot a1 + a2_m \cdot a2_m\right) \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (* (* (cos th) (pow 2.0 -0.5)) (+ (* a1 a1) (* a2_m a2_m))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (cos(th) * pow(2.0, -0.5)) * ((a1 * a1) + (a2_m * a2_m));
}

Fortran

a2_m = abs(a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2_m, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (cos(th) * (2.0d0 ** (-0.5d0))) * ((a1 * a1) + (a2_m * a2_m))
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return (Math.cos(th) * Math.pow(2.0, -0.5)) * ((a1 * a1) + (a2_m * a2_m));
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return (math.cos(th) * math.pow(2.0, -0.5)) * ((a1 * a1) + (a2_m * a2_m))

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(cos(th) * (2.0 ^ -0.5)) * Float64(Float64(a1 * a1) + Float64(a2_m * a2_m)))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (cos(th) * (2.0 ^ -0.5)) * ((a1 * a1) + (a2_m * a2_m));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   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 99.5%

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

   5. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 5: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{a2_m \cdot \sqrt{0.5}}{\frac{1}{a2_m \cdot \cos th}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th)
 :precision binary64
 (/ (* a2_m (sqrt 0.5)) (/ 1.0 (* a2_m (cos th)))))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (a2_m * sqrt(0.5)) / (1.0 / (a2_m * cos(th)));
}

Fortran

a2_m = abs(a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2_m, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (a2_m * sqrt(0.5d0)) / (1.0d0 / (a2_m * cos(th)))
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return (a2_m * Math.sqrt(0.5)) / (1.0 / (a2_m * Math.cos(th)));
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return (a2_m * math.sqrt(0.5)) / (1.0 / (a2_m * math.cos(th)))

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(a2_m * sqrt(0.5)) / Float64(1.0 / Float64(a2_m * cos(th))))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (a2_m * sqrt(0.5)) / (1.0 / (a2_m * cos(th)));
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(a2$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(a2$95$m * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 57.1%

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

   1. pow2 57.1%

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

   2. associate-/l* 57.1%

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

   3. associate-*l/ 57.1%

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

   4. associate-/r/ 57.1%

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

   5. associate-*l* 57.1%

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

   6. div-inv 57.0%

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

   7. add-sqr-sqrt 57.0%

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

   8. sqrt-unprod 57.0%

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

   9. frac-times 57.0%

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

   10. metadata-eval 57.0%

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

   11. rem-square-sqrt 57.1%

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

   12. metadata-eval 57.1%

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

6. Applied egg-rr 57.1%

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

   1. *-commutative 57.1%

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

   2. /-rgt-identity 57.1%

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

   3. associate-/r/ 57.0%

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

   4. clear-num 57.0%

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

   5. un-div-inv 57.0%

      \[\leadsto \color{blue}{\frac{a2 \cdot \sqrt{0.5}}{\frac{\frac{1}{\cos th}}{a2}}} \]

   6. associate-/l/ 57.0%

      \[\leadsto \frac{a2 \cdot \sqrt{0.5}}{\color{blue}{\frac{1}{a2 \cdot \cos th}}} \]

   7. *-commutative 57.0%

      \[\leadsto \frac{a2 \cdot \sqrt{0.5}}{\frac{1}{\color{blue}{\cos th \cdot a2}}} \]

8. Applied egg-rr 57.0%

   \[\leadsto \color{blue}{\frac{a2 \cdot \sqrt{0.5}}{\frac{1}{\cos th \cdot a2}}} \]

9. Final simplification 57.0%

   \[\leadsto \frac{a2 \cdot \sqrt{0.5}}{\frac{1}{a2 \cdot \cos th}} \]

Alternative 6: 77.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(a2$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2$95$m * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 57.1%

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

   1. pow2 57.1%

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

   2. associate-/l* 57.1%

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

   3. associate-*l/ 57.1%

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

   4. associate-/r/ 57.1%

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

   5. associate-*l* 57.1%

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

   6. div-inv 57.0%

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

   7. add-sqr-sqrt 57.0%

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

   8. sqrt-unprod 57.0%

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

   9. frac-times 57.0%

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

   10. metadata-eval 57.0%

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

   11. rem-square-sqrt 57.1%

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

   12. metadata-eval 57.1%

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

6. Applied egg-rr 57.1%

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

7. Final simplification 57.1%

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

Alternative 7: 66.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2$95$m * a2$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around 0 67.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
5. Step-by-step derivation

   1. expm1-log1p-u 67.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. expm1-udef 67.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. add-sqr-sqrt 67.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. sqrt-unprod 67.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   5. frac-times 67.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. metadata-eval 67.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. rem-square-sqrt 66.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{2}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. metadata-eval 66.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 66.7%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
7. Step-by-step derivation

   1. expm1-def 66.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. expm1-log1p 67.1%

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

8. Simplified 67.1%

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

9. Final simplification 67.1%

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

Alternative 8: 52.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} a2_m = \left|a2\right| \\ [a1, a2_m, th] = \mathsf{sort}([a1, a2_m, th])\\ \\ \frac{a2_m \cdot \sqrt{0.5}}{\frac{1}{a2_m}} \end{array} \]

FPCore

a2_m = (fabs.f64 a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
(FPCore (a1 a2_m th) :precision binary64 (/ (* a2_m (sqrt 0.5)) (/ 1.0 a2_m)))

C

a2_m = fabs(a2);
assert(a1 < a2_m && a2_m < th);
double code(double a1, double a2_m, double th) {
	return (a2_m * sqrt(0.5)) / (1.0 / a2_m);
}

Fortran

a2_m = abs(a2)
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
real(8) function code(a1, a2_m, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2_m
    real(8), intent (in) :: th
    code = (a2_m * sqrt(0.5d0)) / (1.0d0 / a2_m)
end function

Java

a2_m = Math.abs(a2);
assert a1 < a2_m && a2_m < th;
public static double code(double a1, double a2_m, double th) {
	return (a2_m * Math.sqrt(0.5)) / (1.0 / a2_m);
}

Python

a2_m = math.fabs(a2)
[a1, a2_m, th] = sort([a1, a2_m, th])
def code(a1, a2_m, th):
	return (a2_m * math.sqrt(0.5)) / (1.0 / a2_m)

Julia

a2_m = abs(a2)
a1, a2_m, th = sort([a1, a2_m, th])
function code(a1, a2_m, th)
	return Float64(Float64(a2_m * sqrt(0.5)) / Float64(1.0 / a2_m))
end

MATLAB

a2_m = abs(a2);
a1, a2_m, th = num2cell(sort([a1, a2_m, th])){:}
function tmp = code(a1, a2_m, th)
	tmp = (a2_m * sqrt(0.5)) / (1.0 / a2_m);
end

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(N[(a2$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 / a2$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 57.1%

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

   1. pow2 57.1%

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

   2. associate-/l* 57.1%

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

   3. associate-*l/ 57.1%

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

   4. associate-/r/ 57.1%

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

   5. associate-*l* 57.1%

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

   6. div-inv 57.0%

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

   7. add-sqr-sqrt 57.0%

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

   8. sqrt-unprod 57.0%

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

   9. frac-times 57.0%

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

   10. metadata-eval 57.0%

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

   11. rem-square-sqrt 57.1%

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

   12. metadata-eval 57.1%

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

6. Applied egg-rr 57.1%

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

   1. *-commutative 57.1%

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

   2. /-rgt-identity 57.1%

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

   3. associate-/r/ 57.0%

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

   4. clear-num 57.0%

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

   5. un-div-inv 57.0%

      \[\leadsto \color{blue}{\frac{a2 \cdot \sqrt{0.5}}{\frac{\frac{1}{\cos th}}{a2}}} \]

   6. associate-/l/ 57.0%

      \[\leadsto \frac{a2 \cdot \sqrt{0.5}}{\color{blue}{\frac{1}{a2 \cdot \cos th}}} \]

   7. *-commutative 57.0%

      \[\leadsto \frac{a2 \cdot \sqrt{0.5}}{\frac{1}{\color{blue}{\cos th \cdot a2}}} \]

8. Applied egg-rr 57.0%

   \[\leadsto \color{blue}{\frac{a2 \cdot \sqrt{0.5}}{\frac{1}{\cos th \cdot a2}}} \]

9. Taylor expanded in th around 0 40.9%

   \[\leadsto \frac{a2 \cdot \sqrt{0.5}}{\color{blue}{\frac{1}{a2}}} \]

10. Final simplification 40.9%

    \[\leadsto \frac{a2 \cdot \sqrt{0.5}}{\frac{1}{a2}} \]

Alternative 9: 52.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a2_m = N[Abs[a2], $MachinePrecision]
NOTE: a1, a2_m, and th should be sorted in increasing order before calling this function.
code[a1_, a2$95$m_, th_] := N[(a2$95$m * N[(a2$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 57.1%

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

   1. pow2 57.1%

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

   2. associate-/l* 57.1%

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

   3. associate-*l/ 57.1%

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

   4. associate-/r/ 57.1%

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

   5. associate-*l* 57.1%

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

   6. div-inv 57.0%

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

   7. add-sqr-sqrt 57.0%

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

   8. sqrt-unprod 57.0%

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

   9. frac-times 57.0%

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

   10. metadata-eval 57.0%

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

   11. rem-square-sqrt 57.1%

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

   12. metadata-eval 57.1%

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

6. Applied egg-rr 57.1%

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

7. Taylor expanded in th around 0 40.9%

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

8. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2023322 
(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))))