Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.1s

Alternatives: 10

Speedup: 2.0×

• 43.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% 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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (sqrt(0.5d0) * cos(th)) * ((a1 * a1) + (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

9. Simplified 99.7%

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

Alternative 2: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{{a2}^{2}}{-\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -5e-310)
   (/ (pow a2 2.0) (- (sqrt 2.0)))
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -5e-310) {
		tmp = pow(a2, 2.0) / -sqrt(2.0);
	} else {
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= (-5d-310)) then
        tmp = (a2 ** 2.0d0) / -sqrt(2.0d0)
    else
        tmp = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -5e-310) {
		tmp = Math.pow(a2, 2.0) / -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -5e-310:
		tmp = math.pow(a2, 2.0) / -math.sqrt(2.0)
	else:
		tmp = math.sqrt(0.5) * ((a1 * a1) + (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -5e-310)
		tmp = Float64((a2 ^ 2.0) / Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -5e-310)
		tmp = (a2 ^ 2.0) / -sqrt(2.0);
	else
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -5e-310], N[(N[Power[a2, 2.0], $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -4.999999999999985e-310

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. cos-neg 99.7%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 80.4%

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

      3. frac-times 80.3%

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

      4. pow2 80.3%

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

      5. add-sqr-sqrt 80.3%

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

      6. pow2 80.3%

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

      7. fma-undefine 80.3%

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

      8. hypot-define 80.3%

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

      9. rem-square-sqrt 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. unpow2 80.4%

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

      2. pow-sqr 80.5%

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

      3. hypot-undefine 80.5%

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

      4. unpow2 80.5%

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

      5. unpow2 80.5%

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

      6. +-commutative 80.5%

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

      7. unpow2 80.5%

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

      8. unpow2 80.5%

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

      9. hypot-define 80.5%

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

      10. metadata-eval 80.5%

          \[\leadsto \cos th \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{\color{blue}{4}}}{2}} \]

   8. Simplified 80.5%

      \[\leadsto \cos th \cdot \color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{4}}{2}}} \]

   9. Taylor expanded in a1 around 0 49.5%

      \[\leadsto \cos th \cdot \sqrt{\frac{\color{blue}{{a2}^{4}}}{2}} \]

   10. Taylor expanded in th around 0 16.9%

       \[\leadsto \color{blue}{1} \cdot \sqrt{\frac{{a2}^{4}}{2}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 16.9%

          \[\leadsto \color{blue}{\sqrt{\frac{{a2}^{4}}{2}}} \]

       2. sqrt-div 16.9%

          \[\leadsto \color{blue}{\frac{\sqrt{{a2}^{4}}}{\sqrt{2}}} \]

       3. sqrt-pow1 16.8%

          \[\leadsto \frac{\color{blue}{{a2}^{\left(\frac{4}{2}\right)}}}{\sqrt{2}} \]

       4. metadata-eval 16.8%

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

       5. pow2 16.8%

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

       6. frac-2neg 16.8%

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

       7. distribute-lft-neg-out 16.8%

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

       8. add-sqr-sqrt 8.0%

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

       9. sqrt-unprod 25.8%

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

       10. sqr-neg 25.8%

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

       11. sqrt-prod 17.7%

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

       12. add-sqr-sqrt 42.7%

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

       13. pow2 42.7%

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

   12. Applied egg-rr 42.7%

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

   if -4.999999999999985e-310 < (cos.f64 th)

   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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 85.8%

      \[\leadsto \color{blue}{\sqrt{0.5}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.72:\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.72) (* t_1 (* 0.5 (cos th))) (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.72) {
		tmp = t_1 * (0.5 * cos(th));
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.72d0) then
        tmp = t_1 * (0.5d0 * cos(th))
    else
        tmp = sqrt(0.5d0) * t_1
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (Math.cos(th) <= 0.72) {
		tmp = t_1 * (0.5 * Math.cos(th));
	} else {
		tmp = Math.sqrt(0.5) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.72:
		tmp = t_1 * (0.5 * math.cos(th))
	else:
		tmp = math.sqrt(0.5) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.72)
		tmp = Float64(t_1 * Float64(0.5 * cos(th)));
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.72)
		tmp = t_1 * (0.5 * cos(th));
	else
		tmp = sqrt(0.5) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], 0.72], N[(t$95$1 * N[(0.5 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.71999999999999997

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   8. Applied egg-rr 63.4%

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

   if 0.71999999999999997 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 90.1%

      \[\leadsto \color{blue}{\sqrt{0.5}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.72:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(0.5 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.72:\\ \;\;\;\;\cos th \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.72) (* (cos th) t_1) (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.72) {
		tmp = cos(th) * t_1;
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.72d0) then
        tmp = cos(th) * t_1
    else
        tmp = sqrt(0.5d0) * t_1
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (Math.cos(th) <= 0.72) {
		tmp = Math.cos(th) * t_1;
	} else {
		tmp = Math.sqrt(0.5) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.72:
		tmp = math.cos(th) * t_1
	else:
		tmp = math.sqrt(0.5) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.72)
		tmp = Float64(cos(th) * t_1);
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.72)
		tmp = cos(th) * t_1;
	else
		tmp = sqrt(0.5) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], 0.72], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.71999999999999997

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   8. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\left(0 + \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   9. Step-by-step derivation

      1. +-lft-identity 63.4%

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

   10. Simplified 63.4%

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

   if 0.71999999999999997 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 90.1%

      \[\leadsto \color{blue}{\sqrt{0.5}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 58.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * (a2 * (sqrt(0.5d0) * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. 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. Step-by-step derivation

   1. div-inv 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. associate-*l* 99.5%

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

   4. fma-undefine 99.5%

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

   5. hypot-define 99.5%

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

   6. fma-undefine 99.5%

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

   7. hypot-define 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-flip 99.7%

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

   10. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in a2 around inf 35.9%

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

   1. *-commutative 35.9%

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

9. Simplified 35.9%

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

10. Taylor expanded in a2 around inf 56.2%

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

Alternative 6: 52.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -5e-310)
   (* (+ a1 a2) (- a1 a2))
   (* 0.5 (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -5e-310) {
		tmp = (a1 + a2) * (a1 - a2);
	} else {
		tmp = 0.5 * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= (-5d-310)) then
        tmp = (a1 + a2) * (a1 - a2)
    else
        tmp = 0.5d0 * ((a1 * a1) + (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -5e-310) {
		tmp = (a1 + a2) * (a1 - a2);
	} else {
		tmp = 0.5 * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -5e-310:
		tmp = (a1 + a2) * (a1 - a2)
	else:
		tmp = 0.5 * ((a1 * a1) + (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -5e-310)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 - a2));
	else
		tmp = Float64(0.5 * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -5e-310)
		tmp = (a1 + a2) * (a1 - a2);
	else
		tmp = 0.5 * ((a1 * a1) + (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -5e-310], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 - a2), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -4.999999999999985e-310

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   8. Applied egg-rr 16.3%

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

      1. *-inverses 16.3%

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

   10. Simplified 16.3%

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

       1. add-sqr-sqrt 16.3%

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

       2. sqrt-prod 16.4%

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

       3. sqr-neg 16.4%

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

       4. swap-sqr 16.4%

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

       5. sqrt-unprod 15.8%

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

       6. add-sqr-sqrt 39.5%

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

       7. cancel-sign-sub-inv 39.5%

          \[\leadsto 1 \cdot \color{blue}{\left(a1 \cdot a1 - a2 \cdot a2\right)} \]

       8. difference-of-squares 41.1%

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

   12. Applied egg-rr 41.1%

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

   if -4.999999999999985e-310 < (cos.f64 th)

   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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   8. Applied egg-rr 62.7%

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

   9. Taylor expanded in th around 0 62.8%

      \[\leadsto \color{blue}{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return cos(th) * ((a1 * a1) + (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((a1 * a1) + (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

8. Applied egg-rr 63.4%

   \[\leadsto \color{blue}{\left(0 + \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
9. Step-by-step derivation

   1. +-lft-identity 63.4%

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

10. Simplified 63.4%

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

Alternative 8: 46.5% accurate, 46.1× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 0.5 * ((a1 * a1) + (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 0.5d0 * ((a1 * a1) + (a2 * a2))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(0.5 * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

8. Applied egg-rr 63.3%

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

9. Taylor expanded in th around 0 50.8%

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

Alternative 9: 30.5% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a2 \cdot a2 \end{array} \]

FPCore

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

C

double code(double a1, double a2, double th) {
	return a2 * a2;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * a2
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * a2;
}

Python

def code(a1, a2, th):
	return a2 * a2

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * a2), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

8. Applied egg-rr 50.5%

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

   1. *-inverses 50.5%

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

10. Simplified 50.5%

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

11. Taylor expanded in a1 around 0 31.1%

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

    1. pow2 31.1%

       \[\leadsto \color{blue}{a2 \cdot a2} \]

13. Applied egg-rr 31.1%

    \[\leadsto \color{blue}{a2 \cdot a2} \]
14. Add Preprocessing

Alternative 10: 3.6% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ a1 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 a1)

C

double code(double a1, double a2, double th) {
	return a1;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a1
end function

Java

public static double code(double a1, double a2, double th) {
	return a1;
}

Python

def code(a1, a2, th):
	return a1

Julia

function code(a1, a2, th)
	return a1
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a1

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. Step-by-step derivation

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 80.4%

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

   3. frac-times 80.2%

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

   4. pow2 80.2%

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

   5. add-sqr-sqrt 80.2%

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

   6. pow2 80.2%

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

   7. fma-undefine 80.2%

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

   8. hypot-define 80.2%

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

   9. rem-square-sqrt 80.5%

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

6. Applied egg-rr 80.5%

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

   1. unpow2 80.5%

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

   2. pow-sqr 80.5%

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

   3. hypot-undefine 80.5%

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

   4. unpow2 80.5%

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

   5. unpow2 80.5%

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

   6. +-commutative 80.5%

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

   7. unpow2 80.5%

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

   8. unpow2 80.5%

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

   9. hypot-define 80.5%

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

   10. metadata-eval 80.5%

       \[\leadsto \cos th \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{\color{blue}{4}}}{2}} \]

8. Simplified 80.5%

   \[\leadsto \cos th \cdot \color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{4}}{2}}} \]

10. Applied egg-rr 2.6%

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

11. Taylor expanded in th around 0 9.1%

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

12. Taylor expanded in a1 around inf 3.7%

    \[\leadsto \color{blue}{a1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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))))