Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.4s

Alternatives: 14

Speedup: 2.0×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-/r/ 99.5%

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

   4. cos-neg 99.5%

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

   5. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. associate-/l* 99.6%

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

   2. div-inv 99.5%

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

   3. fma-def 99.5%

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

   4. add-sqr-sqrt 99.5%

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

   5. pow2 99.5%

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

   6. hypot-def 99.5%

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

   7. pow1/2 99.5%

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

   8. pow-flip 99.6%

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

   9. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 71.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.72:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.72)
   (* (cos th) (pow a2 2.0))
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.72) {
		tmp = cos(th) * pow(a2, 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) <= 0.72d0) then
        tmp = cos(th) * (a2 ** 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) <= 0.72) {
		tmp = Math.cos(th) * Math.pow(a2, 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) <= 0.72:
		tmp = math.cos(th) * math.pow(a2, 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) <= 0.72)
		tmp = Float64(cos(th) * (a2 ^ 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) <= 0.72)
		tmp = cos(th) * (a2 ^ 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], 0.72], N[(N[Cos[th], $MachinePrecision] * N[Power[a2, 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) < 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. Step-by-step derivation

      1. frac-2neg 99.6%

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

      2. div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   7. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{\frac{-4}{\frac{-4}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. Taylor expanded in a1 around 0 31.4%

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

   if 0.71999999999999997 < (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 91.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.72:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 3: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.675:\\ \;\;\;\;\cos th \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.675) (* (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) <= 0.675) {
		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) <= 0.675d0) 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) <= 0.675) {
		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) <= 0.675:
		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) <= 0.675)
		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) <= 0.675)
		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], 0.675], 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) < 0.67500000000000004

   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. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   7. Applied egg-rr 57.0%

      \[\leadsto \color{blue}{\frac{-4}{\frac{-4}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. Taylor expanded in th around inf 57.0%

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

   if 0.67500000000000004 < (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 91.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.675:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

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/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

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: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. frac-2neg 99.5%

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

   2. div-inv 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in a1 around 0 59.1%

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

   1. associate-/l* 59.1%

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

8. Simplified 59.1%

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

   1. pow2 59.1%

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

   2. div-inv 59.1%

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

   3. clear-num 59.1%

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

   4. associate-*l* 59.1%

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

10. Applied egg-rr 59.1%

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

11. Final simplification 59.1%

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

Alternative 6: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}} \end{array} \]

FPCore

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

C

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

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 / (sqrt(2.0d0) / cos(th)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. frac-2neg 99.5%

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

   2. div-inv 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in a1 around 0 59.1%

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

   1. associate-/l* 59.1%

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

8. Simplified 59.1%

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

   1. pow2 59.1%

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

   2. *-un-lft-identity 59.1%

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

   3. times-frac 59.1%

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

10. Applied egg-rr 59.1%

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

11. Final simplification 59.1%

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

Alternative 7: 59.7% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. frac-2neg 99.5%

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

   2. div-inv 99.5%

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

5. Applied egg-rr 99.5%

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

7. Applied egg-rr 58.0%

   \[\leadsto \color{blue}{\frac{-4}{\frac{-4}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Taylor expanded in th around inf 58.0%

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

9. Final simplification 58.0%

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

Alternative 8: 45.0% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 130000000 \lor \neg \left(th \leq 2.6 \cdot 10^{+124} \lor \neg \left(th \leq 2.8 \cdot 10^{+181}\right) \land th \leq 1.15 \cdot 10^{+228}\right):\\ \;\;\;\;t_1 \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 130000000.0)
           (not
            (or (<= th 2.6e+124)
                (and (not (<= th 2.8e+181)) (<= th 1.15e+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 <= 130000000.0) || !((th <= 2.6e+124) || (!(th <= 2.8e+181) && (th <= 1.15e+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 <= 130000000.0d0) .or. (.not. (th <= 2.6d+124) .or. (.not. (th <= 2.8d+181)) .and. (th <= 1.15d+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 <= 130000000.0) || !((th <= 2.6e+124) || (!(th <= 2.8e+181) && (th <= 1.15e+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 <= 130000000.0) or not ((th <= 2.6e+124) or (not (th <= 2.8e+181) and (th <= 1.15e+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 <= 130000000.0) || !((th <= 2.6e+124) || (!(th <= 2.8e+181) && (th <= 1.15e+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 <= 130000000.0) || ~(((th <= 2.6e+124) || (~((th <= 2.8e+181)) && (th <= 1.15e+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, 130000000.0], N[Not[Or[LessEqual[th, 2.6e+124], And[N[Not[LessEqual[th, 2.8e+181]], $MachinePrecision], LessEqual[th, 1.15e+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 < 1.3e8 or 2.6e124 < th < 2.79999999999999984e181 or 1.15000000000000006e228 < 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 67.3%

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

   6. Applied egg-rr 44.8%

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

   if 1.3e8 < th < 2.6e124 or 2.79999999999999984e181 < th < 1.15000000000000006e228

   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 19.6%

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

   6. Applied egg-rr 45.5%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 130000000 \lor \neg \left(th \leq 2.6 \cdot 10^{+124} \lor \neg \left(th \leq 2.8 \cdot 10^{+181}\right) \land th \leq 1.15 \cdot 10^{+228}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 9: 45.4% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 130000000 \lor \neg \left(th \leq 2.6 \cdot 10^{+124} \lor \neg \left(th \leq 2.8 \cdot 10^{+181}\right) \land th \leq 1.15 \cdot 10^{+228}\right):\\ \;\;\;\;t_1 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 130000000.0)
           (not
            (or (<= th 2.6e+124)
                (and (not (<= th 2.8e+181)) (<= th 1.15e+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 <= 130000000.0) || !((th <= 2.6e+124) || (!(th <= 2.8e+181) && (th <= 1.15e+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 <= 130000000.0d0) .or. (.not. (th <= 2.6d+124) .or. (.not. (th <= 2.8d+181)) .and. (th <= 1.15d+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 <= 130000000.0) || !((th <= 2.6e+124) || (!(th <= 2.8e+181) && (th <= 1.15e+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 <= 130000000.0) or not ((th <= 2.6e+124) or (not (th <= 2.8e+181) and (th <= 1.15e+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 <= 130000000.0) || !((th <= 2.6e+124) || (!(th <= 2.8e+181) && (th <= 1.15e+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 <= 130000000.0) || ~(((th <= 2.6e+124) || (~((th <= 2.8e+181)) && (th <= 1.15e+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, 130000000.0], N[Not[Or[LessEqual[th, 2.6e+124], And[N[Not[LessEqual[th, 2.8e+181]], $MachinePrecision], LessEqual[th, 1.15e+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 < 1.3e8 or 2.6e124 < th < 2.79999999999999984e181 or 1.15000000000000006e228 < 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 67.3%

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

   6. Applied egg-rr 45.3%

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

   if 1.3e8 < th < 2.6e124 or 2.79999999999999984e181 < th < 1.15000000000000006e228

   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 19.6%

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

   6. Applied egg-rr 45.5%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 130000000 \lor \neg \left(th \leq 2.6 \cdot 10^{+124} \lor \neg \left(th \leq 2.8 \cdot 10^{+181}\right) \land th \leq 1.15 \cdot 10^{+228}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 10: 46.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := -0.5 \cdot t_1\\ t_3 := 0.5 \cdot t_1\\ \mathbf{if}\;th \leq 130000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 2.8 \cdot 10^{+181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;th \leq 1.15 \cdot 10^{+228}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))) (t_2 (* -0.5 t_1)) (t_3 (* 0.5 t_1)))
   (if (<= th 130000000.0)
     t_3
     (if (<= th 2.6e+124)
       t_2
       (if (<= th 2.8e+181) t_3 (if (<= th 1.15e+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 t_3 = 0.5 * t_1;
	double tmp;
	if (th <= 130000000.0) {
		tmp = t_3;
	} else if (th <= 2.6e+124) {
		tmp = t_2;
	} else if (th <= 2.8e+181) {
		tmp = t_3;
	} else if (th <= 1.15e+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) :: t_3
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    t_2 = (-0.5d0) * t_1
    t_3 = 0.5d0 * t_1
    if (th <= 130000000.0d0) then
        tmp = t_3
    else if (th <= 2.6d+124) then
        tmp = t_2
    else if (th <= 2.8d+181) then
        tmp = t_3
    else if (th <= 1.15d+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 t_3 = 0.5 * t_1;
	double tmp;
	if (th <= 130000000.0) {
		tmp = t_3;
	} else if (th <= 2.6e+124) {
		tmp = t_2;
	} else if (th <= 2.8e+181) {
		tmp = t_3;
	} else if (th <= 1.15e+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
	t_3 = 0.5 * t_1
	tmp = 0
	if th <= 130000000.0:
		tmp = t_3
	elif th <= 2.6e+124:
		tmp = t_2
	elif th <= 2.8e+181:
		tmp = t_3
	elif th <= 1.15e+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)
	t_3 = Float64(0.5 * t_1)
	tmp = 0.0
	if (th <= 130000000.0)
		tmp = t_3;
	elseif (th <= 2.6e+124)
		tmp = t_2;
	elseif (th <= 2.8e+181)
		tmp = t_3;
	elseif (th <= 1.15e+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;
	t_3 = 0.5 * t_1;
	tmp = 0.0;
	if (th <= 130000000.0)
		tmp = t_3;
	elseif (th <= 2.6e+124)
		tmp = t_2;
	elseif (th <= 2.8e+181)
		tmp = t_3;
	elseif (th <= 1.15e+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]}, Block[{t$95$3 = N[(0.5 * t$95$1), $MachinePrecision]}, If[LessEqual[th, 130000000.0], t$95$3, If[LessEqual[th, 2.6e+124], t$95$2, If[LessEqual[th, 2.8e+181], t$95$3, If[LessEqual[th, 1.15e+228], t$95$2, N[(t$95$1 * 0.25), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.3e8 or 2.6e124 < th < 2.79999999999999984e181

   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 69.7%

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

   6. Applied egg-rr 46.9%

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

   if 1.3e8 < th < 2.6e124 or 2.79999999999999984e181 < th < 1.15000000000000006e228

   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 19.6%

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

   6. Applied egg-rr 45.5%

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

   if 1.15000000000000006e228 < th

   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 35.1%

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

   6. Applied egg-rr 33.9%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 130000000:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 2.8 \cdot 10^{+181}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 1.15 \cdot 10^{+228}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \end{array} \]

Alternative 11: 46.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := -0.5 \cdot t_1\\ \mathbf{if}\;th \leq 130000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 2.8 \cdot 10^{+181}:\\ \;\;\;\;0.5 \cdot t_1\\ \mathbf{elif}\;th \leq 1.15 \cdot 10^{+228}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))) (t_2 (* -0.5 t_1)))
   (if (<= th 130000000.0)
     t_1
     (if (<= th 2.6e+124)
       t_2
       (if (<= th 2.8e+181)
         (* 0.5 t_1)
         (if (<= th 1.15e+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 <= 130000000.0) {
		tmp = t_1;
	} else if (th <= 2.6e+124) {
		tmp = t_2;
	} else if (th <= 2.8e+181) {
		tmp = 0.5 * t_1;
	} else if (th <= 1.15e+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 <= 130000000.0d0) then
        tmp = t_1
    else if (th <= 2.6d+124) then
        tmp = t_2
    else if (th <= 2.8d+181) then
        tmp = 0.5d0 * t_1
    else if (th <= 1.15d+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 <= 130000000.0) {
		tmp = t_1;
	} else if (th <= 2.6e+124) {
		tmp = t_2;
	} else if (th <= 2.8e+181) {
		tmp = 0.5 * t_1;
	} else if (th <= 1.15e+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 <= 130000000.0:
		tmp = t_1
	elif th <= 2.6e+124:
		tmp = t_2
	elif th <= 2.8e+181:
		tmp = 0.5 * t_1
	elif th <= 1.15e+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 <= 130000000.0)
		tmp = t_1;
	elseif (th <= 2.6e+124)
		tmp = t_2;
	elseif (th <= 2.8e+181)
		tmp = Float64(0.5 * t_1);
	elseif (th <= 1.15e+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 <= 130000000.0)
		tmp = t_1;
	elseif (th <= 2.6e+124)
		tmp = t_2;
	elseif (th <= 2.8e+181)
		tmp = 0.5 * t_1;
	elseif (th <= 1.15e+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, 130000000.0], t$95$1, If[LessEqual[th, 2.6e+124], t$95$2, If[LessEqual[th, 2.8e+181], N[(0.5 * t$95$1), $MachinePrecision], If[LessEqual[th, 1.15e+228], t$95$2, N[(t$95$1 * 0.25), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if th < 1.3e8

   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. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   7. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\frac{-4}{\frac{-4}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. Taylor expanded in th around 0 46.6%

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

   if 1.3e8 < th < 2.6e124 or 2.79999999999999984e181 < th < 1.15000000000000006e228

   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 19.6%

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

   6. Applied egg-rr 45.5%

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

   if 2.6e124 < th < 2.79999999999999984e181

   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 48.4%

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

   6. Applied egg-rr 49.3%

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

   if 1.15000000000000006e228 < th

   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 35.1%

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

   6. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 130000000:\\ \;\;\;\;a1 \cdot a1 + a2 \cdot a2\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+124}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 2.8 \cdot 10^{+181}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{elif}\;th \leq 1.15 \cdot 10^{+228}:\\ \;\;\;\;-0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \end{array} \]

Alternative 12: 20.1% 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 63.2%

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

6. Applied egg-rr 21.3%

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

7. Final simplification 21.3%

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

Alternative 13: 19.8% 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 63.2%

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

6. Applied egg-rr 21.0%

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

7. Final simplification 21.0%

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

Alternative 14: 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-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-/r/ 99.5%

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

   4. cos-neg 99.5%

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

   5. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. associate-/l* 99.6%

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

   2. div-inv 99.5%

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

   3. fma-def 99.5%

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

   4. add-sqr-sqrt 99.5%

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

   5. pow2 99.5%

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

   6. hypot-def 99.5%

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

   7. pow1/2 99.5%

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

   8. pow-flip 99.6%

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

   9. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

7. Applied egg-rr 3.6%

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

   1. *-inverses 3.6%

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

9. Simplified 3.6%

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

10. Final simplification 3.6%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023312 
(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))))