Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.7s

Alternatives: 13

Speedup: 2.0×

• 43.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (pow 2.0 -0.5) (* (cos th) (pow (hypot a1 a2) 2.0))))

C

double code(double a1, double a2, double th) {
	return pow(2.0, -0.5) * (cos(th) * pow(hypot(a1, a2), 2.0));
}

Java

public static double code(double a1, double a2, double th) {
	return Math.pow(2.0, -0.5) * (Math.cos(th) * Math.pow(Math.hypot(a1, a2), 2.0));
}

Python

def code(a1, a2, th):
	return math.pow(2.0, -0.5) * (math.cos(th) * math.pow(math.hypot(a1, a2), 2.0))

Julia

function code(a1, a2, th)
	return Float64((2.0 ^ -0.5) * Float64(cos(th) * (hypot(a1, a2) ^ 2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (2.0 ^ -0.5) * (cos(th) * (hypot(a1, a2) ^ 2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Power[2.0, -0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 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-in 99.5%

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

   2. div-inv 99.4%

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

   3. associate-*l* 99.5%

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

   4. pow1/2 99.5%

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

   5. pow-flip 99.6%

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

   6. metadata-eval 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. pow2 99.6%

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

   9. hypot-def 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l* 99.6%

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

   3. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(\left(a1 + a2\right) \cdot \left(a1 + a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* (cos th) (* (+ a1 a2) (+ a1 a2)))
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * ((a1 + a2) * (a1 + a2));
	} 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) <= 0.7d0) then
        tmp = cos(th) * ((a1 + a2) * (a1 + a2))
    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) <= 0.7) {
		tmp = Math.cos(th) * ((a1 + a2) * (a1 + a2));
	} else {
		tmp = Math.sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * ((a1 + a2) * (a1 + a2))
	else:
		tmp = math.sqrt(0.5) * ((a1 * a1) + (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(Float64(a1 + a2) * Float64(a1 + a2)));
	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) <= 0.7)
		tmp = cos(th) * ((a1 + a2) * (a1 + a2));
	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], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $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) < 0.69999999999999996

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

      1. expm1-log1p-u 42.1%

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

      2. expm1-udef 28.7%

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

      3. add-sqr-sqrt 28.7%

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

      4. pow2 28.7%

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

      5. hypot-def 28.7%

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

   5. Applied egg-rr 28.7%

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

      1. expm1-def 42.1%

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

      2. expm1-log1p 99.4%

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

      3. associate-/r/ 99.3%

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

   7. Simplified 99.3%

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

   9. Applied egg-rr 57.1%

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

       1. *-commutative 57.1%

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

       2. associate-*l* 57.1%

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

   11. Simplified 57.1%

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

   if 0.69999999999999996 < (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. 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 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 92.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(\left(a1 + a2\right) \cdot \left(a1 + a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 3: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.7) (* (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) <= 0.7) {
		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) <= 0.7d0) 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) <= 0.7) {
		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) <= 0.7:
		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) <= 0.7)
		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) <= 0.7)
		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], 0.7], 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) < 0.69999999999999996

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\left(0 + \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 0.69999999999999996 < (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. 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 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 92.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $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/ 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 59.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $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 75.8%

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

   2. expm1-udef 61.3%

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

   3. add-sqr-sqrt 61.3%

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

   4. pow2 61.3%

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

   5. hypot-def 61.3%

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

5. Applied egg-rr 61.3%

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

   1. expm1-def 75.8%

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

   2. expm1-log1p 99.5%

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

   3. associate-/r/ 99.2%

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

7. Simplified 99.2%

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

9. Applied egg-rr 58.6%

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

    1. *-commutative 58.6%

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

    2. associate-*l* 58.6%

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

11. Simplified 58.6%

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

12. Final simplification 58.6%

    \[\leadsto \cos th \cdot \left(\left(a1 + a2\right) \cdot \left(a1 + a2\right)\right) \]

Alternative 6: 46.4% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := 0.5 \cdot t_1\\ \mathbf{if}\;th \leq 1.06 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 2.15 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot t_1\\ \mathbf{elif}\;th \leq 5.9 \cdot 10^{+209} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))) (t_2 (* 0.5 t_1)))
   (if (<= th 1.06e+29)
     t_2
     (if (<= th 2.15e+135)
       (* -0.5 t_1)
       (if (or (<= th 5.9e+209) (not (<= th 1.05e+228))) t_2 (* t_1 -0.25))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double t_2 = 0.5 * t_1;
	double tmp;
	if (th <= 1.06e+29) {
		tmp = t_2;
	} else if (th <= 2.15e+135) {
		tmp = -0.5 * t_1;
	} else if ((th <= 5.9e+209) || !(th <= 1.05e+228)) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.25;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    t_2 = 0.5d0 * t_1
    if (th <= 1.06d+29) then
        tmp = t_2
    else if (th <= 2.15d+135) then
        tmp = (-0.5d0) * t_1
    else if ((th <= 5.9d+209) .or. (.not. (th <= 1.05d+228))) then
        tmp = t_2
    else
        tmp = t_1 * (-0.25d0)
    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 t_2 = 0.5 * t_1;
	double tmp;
	if (th <= 1.06e+29) {
		tmp = t_2;
	} else if (th <= 2.15e+135) {
		tmp = -0.5 * t_1;
	} else if ((th <= 5.9e+209) || !(th <= 1.05e+228)) {
		tmp = t_2;
	} else {
		tmp = t_1 * -0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	t_2 = 0.5 * t_1
	tmp = 0
	if th <= 1.06e+29:
		tmp = t_2
	elif th <= 2.15e+135:
		tmp = -0.5 * t_1
	elif (th <= 5.9e+209) or not (th <= 1.05e+228):
		tmp = t_2
	else:
		tmp = t_1 * -0.25
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	t_2 = Float64(0.5 * t_1)
	tmp = 0.0
	if (th <= 1.06e+29)
		tmp = t_2;
	elseif (th <= 2.15e+135)
		tmp = Float64(-0.5 * t_1);
	elseif ((th <= 5.9e+209) || !(th <= 1.05e+228))
		tmp = t_2;
	else
		tmp = Float64(t_1 * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	t_2 = 0.5 * t_1;
	tmp = 0.0;
	if (th <= 1.06e+29)
		tmp = t_2;
	elseif (th <= 2.15e+135)
		tmp = -0.5 * t_1;
	elseif ((th <= 5.9e+209) || ~((th <= 1.05e+228)))
		tmp = t_2;
	else
		tmp = t_1 * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * t$95$1), $MachinePrecision]}, If[LessEqual[th, 1.06e+29], t$95$2, If[LessEqual[th, 2.15e+135], N[(-0.5 * t$95$1), $MachinePrecision], If[Or[LessEqual[th, 5.9e+209], N[Not[LessEqual[th, 1.05e+228]], $MachinePrecision]], t$95$2, N[(t$95$1 * -0.25), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.0600000000000001e29 or 2.14999999999999986e135 < th < 5.8999999999999998e209 or 1.04999999999999997e228 < 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. Taylor expanded in th around 0 70.4%

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

   6. Applied egg-rr 47.9%

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

   if 1.0600000000000001e29 < th < 2.14999999999999986e135

   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. Taylor expanded in th around 0 18.7%

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

   6. Applied egg-rr 69.9%

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

   if 5.8999999999999998e209 < th < 1.04999999999999997e228

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 20.2%

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

   6. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.06 \cdot 10^{+29}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 2.15 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 5.9 \cdot 10^{+209} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]

Alternative 7: 44.9% accurate, 31.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 1.06 \cdot 10^{+29} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;t_1 \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 1.06e+29) (not (<= th 1.05e+228)))
     (* t_1 0.0625)
     (* -0.5 t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 1.06e+29) || !(th <= 1.05e+228)) {
		tmp = t_1 * 0.0625;
	} else {
		tmp = -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 ((th <= 1.06d+29) .or. (.not. (th <= 1.05d+228))) then
        tmp = t_1 * 0.0625d0
    else
        tmp = (-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 ((th <= 1.06e+29) || !(th <= 1.05e+228)) {
		tmp = t_1 * 0.0625;
	} else {
		tmp = -0.5 * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if (th <= 1.06e+29) or not (th <= 1.05e+228):
		tmp = t_1 * 0.0625
	else:
		tmp = -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 ((th <= 1.06e+29) || !(th <= 1.05e+228))
		tmp = Float64(t_1 * 0.0625);
	else
		tmp = Float64(-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 ((th <= 1.06e+29) || ~((th <= 1.05e+228)))
		tmp = t_1 * 0.0625;
	else
		tmp = -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[Or[LessEqual[th, 1.06e+29], N[Not[LessEqual[th, 1.05e+228]], $MachinePrecision]], N[(t$95$1 * 0.0625), $MachinePrecision], N[(-0.5 * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 1.0600000000000001e29 or 1.04999999999999997e228 < 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. Taylor expanded in th around 0 72.5%

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

   6. Applied egg-rr 46.6%

      \[\leadsto \color{blue}{0.0625} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 1.0600000000000001e29 < th < 1.04999999999999997e228

   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. Taylor expanded in th around 0 26.4%

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

   6. Applied egg-rr 52.4%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.06 \cdot 10^{+29} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 8: 45.1% accurate, 31.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 1.06 \cdot 10^{+29} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;t_1 \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 1.06e+29) (not (<= th 1.05e+228)))
     (* t_1 0.125)
     (* -0.5 t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 1.06e+29) || !(th <= 1.05e+228)) {
		tmp = t_1 * 0.125;
	} else {
		tmp = -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 ((th <= 1.06d+29) .or. (.not. (th <= 1.05d+228))) then
        tmp = t_1 * 0.125d0
    else
        tmp = (-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 ((th <= 1.06e+29) || !(th <= 1.05e+228)) {
		tmp = t_1 * 0.125;
	} else {
		tmp = -0.5 * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if (th <= 1.06e+29) or not (th <= 1.05e+228):
		tmp = t_1 * 0.125
	else:
		tmp = -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 ((th <= 1.06e+29) || !(th <= 1.05e+228))
		tmp = Float64(t_1 * 0.125);
	else
		tmp = Float64(-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 ((th <= 1.06e+29) || ~((th <= 1.05e+228)))
		tmp = t_1 * 0.125;
	else
		tmp = -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[Or[LessEqual[th, 1.06e+29], N[Not[LessEqual[th, 1.05e+228]], $MachinePrecision]], N[(t$95$1 * 0.125), $MachinePrecision], N[(-0.5 * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 1.0600000000000001e29 or 1.04999999999999997e228 < 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. Taylor expanded in th around 0 72.5%

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

   6. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{0.125} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 1.0600000000000001e29 < th < 1.04999999999999997e228

   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. Taylor expanded in th around 0 26.4%

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

   6. Applied egg-rr 52.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.06 \cdot 10^{+29} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 9: 45.5% accurate, 31.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 1.06 \cdot 10^{+29} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;t_1 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 1.06e+29) (not (<= th 1.05e+228)))
     (* t_1 0.25)
     (* -0.5 t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 1.06e+29) || !(th <= 1.05e+228)) {
		tmp = t_1 * 0.25;
	} else {
		tmp = -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 ((th <= 1.06d+29) .or. (.not. (th <= 1.05d+228))) then
        tmp = t_1 * 0.25d0
    else
        tmp = (-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 ((th <= 1.06e+29) || !(th <= 1.05e+228)) {
		tmp = t_1 * 0.25;
	} else {
		tmp = -0.5 * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if (th <= 1.06e+29) or not (th <= 1.05e+228):
		tmp = t_1 * 0.25
	else:
		tmp = -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 ((th <= 1.06e+29) || !(th <= 1.05e+228))
		tmp = Float64(t_1 * 0.25);
	else
		tmp = Float64(-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 ((th <= 1.06e+29) || ~((th <= 1.05e+228)))
		tmp = t_1 * 0.25;
	else
		tmp = -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[Or[LessEqual[th, 1.06e+29], N[Not[LessEqual[th, 1.05e+228]], $MachinePrecision]], N[(t$95$1 * 0.25), $MachinePrecision], N[(-0.5 * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 1.0600000000000001e29 or 1.04999999999999997e228 < 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. Taylor expanded in th around 0 72.5%

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

   6. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 1.0600000000000001e29 < th < 1.04999999999999997e228

   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. Taylor expanded in th around 0 26.4%

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

   6. Applied egg-rr 52.4%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.06 \cdot 10^{+29} \lor \neg \left(th \leq 1.05 \cdot 10^{+228}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 10: 20.6% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 64.6%

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

6. Applied egg-rr 22.0%

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

7. Final simplification 22.0%

   \[\leadsto -0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

Alternative 11: 20.4% accurate, 51.9× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return (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 = (a1 * -a1) - (a2 * a2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * (-a1)), $MachinePrecision] - N[(a2 * a2), $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 64.6%

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

6. Applied egg-rr 21.6%

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

7. Final simplification 21.6%

   \[\leadsto a1 \cdot \left(-a1\right) - a2 \cdot a2 \]

Alternative 12: 2.5% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a1} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ 1.0 a1))

C

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

Java

public static double code(double a1, double a2, double th) {
	return 1.0 / a1;
}

Python

def code(a1, a2, th):
	return 1.0 / a1

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(1.0 / a1), $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-in 99.5%

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

   2. div-inv 99.4%

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

   3. associate-*l* 99.5%

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

   4. pow1/2 99.5%

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

   5. pow-flip 99.6%

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

   6. metadata-eval 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. pow2 99.6%

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

   9. hypot-def 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l* 99.6%

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

   3. *-commutative 99.6%

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

5. Simplified 99.6%

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

7. Applied egg-rr 2.0%

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

   1. associate-/r* 2.0%

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

   2. *-inverses 2.0%

      \[\leadsto \frac{\color{blue}{1}}{a1 + a2} \]

9. Simplified 2.0%

   \[\leadsto \color{blue}{\frac{1}{a1 + a2}} \]

10. Taylor expanded in a1 around inf 2.4%

    \[\leadsto \color{blue}{\frac{1}{a1}} \]

11. Final simplification 2.4%

    \[\leadsto \frac{1}{a1} \]

Alternative 13: 3.5% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 1.0)

C

double code(double a1, double a2, double th) {
	return 1.0;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 1.0d0
end function

Java

public static double code(double a1, double a2, double th) {
	return 1.0;
}

Python

def code(a1, a2, th):
	return 1.0

Julia

function code(a1, a2, th)
	return 1.0
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 1.0;
end

Wolfram

code[a1_, a2_, th_] := 1.0

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-in 99.5%

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

   2. div-inv 99.4%

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

   3. associate-*l* 99.5%

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

   4. pow1/2 99.5%

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

   5. pow-flip 99.6%

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

   6. metadata-eval 99.6%

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

   7. add-sqr-sqrt 99.6%

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

   8. pow2 99.6%

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

   9. hypot-def 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l* 99.6%

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

   3. *-commutative 99.6%

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

5. Simplified 99.6%

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

7. Applied egg-rr 3.4%

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

   1. *-inverses 3.4%

      \[\leadsto \color{blue}{1} \]

9. Simplified 3.4%

   \[\leadsto \color{blue}{1} \]

10. Final simplification 3.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023319 
(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))))