Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 10.4s

Alternatives: 11

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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 inf 99.7%

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

   1. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 71.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* a2 (* (cos th) 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 = a2 * (cos(th) * 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 = a2 * (cos(th) * 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 = a2 * (Math.cos(th) * 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 = a2 * (math.cos(th) * 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(a2 * Float64(cos(th) * 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 = a2 * (cos(th) * 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[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $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 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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 a1 around 0 56.7%

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

      1. associate-/l* 56.7%

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

   6. Simplified 56.7%

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

   8. Applied egg-rr 38.5%

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

   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/ 99.4%

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

      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 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in th around 0 90.0%

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

4. Final simplification 70.1%

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

Alternative 3: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 55.7%

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

   1. associate-/l* 55.8%

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

   2. associate-/r/ 55.8%

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

8. Simplified 55.8%

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

   1. pow2 55.8%

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

   2. div-inv 55.7%

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

   3. pow1/2 55.7%

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

   4. pow-flip 55.8%

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

   5. metadata-eval 55.8%

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

   6. associate-*l* 55.7%

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

   7. add-sqr-sqrt 55.5%

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

   8. sqrt-unprod 55.7%

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

   9. pow-prod-up 55.7%

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

   10. metadata-eval 55.7%

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

   11. metadata-eval 55.7%

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

10. Applied egg-rr 55.7%

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

11. Final simplification 55.7%

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

Alternative 4: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return cos(th) * (a2 * (a2 / 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 = cos(th) * (a2 * (a2 / sqrt(2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 / 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 55.7%

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

   1. associate-/l* 55.8%

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

   2. associate-/r/ 55.8%

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

8. Simplified 55.8%

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

   1. pow2 55.8%

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

   2. *-un-lft-identity 55.8%

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

   3. times-frac 55.7%

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

10. Applied egg-rr 55.7%

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

11. Final simplification 55.7%

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

Alternative 5: 37.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Cos[th], $MachinePrecision] * 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 a1 around 0 55.7%

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

   1. associate-/l* 55.8%

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

6. Simplified 55.8%

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

8. Applied egg-rr 39.0%

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

9. Final simplification 39.0%

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

Alternative 6: 46.2% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+151} \lor \neg \left(th \leq 3.3 \cdot 10^{+220}\right) \land th \leq 3.3 \cdot 10^{+243}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 9.2e+151) (and (not (<= th 3.3e+220)) (<= th 3.3e+243)))
   (* (+ a1 a2) (+ a1 a2))
   (* (+ (* a1 a1) (* a2 a2)) -0.5)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	}
	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 ((th <= 9.2d+151) .or. (.not. (th <= 3.3d+220)) .and. (th <= 3.3d+243)) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 9.2e+151) or (not (th <= 3.3e+220) and (th <= 3.3e+243)):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243)))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 9.2e+151) || (~((th <= 3.3e+220)) && (th <= 3.3e+243)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 9.2e+151], And[N[Not[LessEqual[th, 3.3e+220]], $MachinePrecision], LessEqual[th, 3.3e+243]]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.2000000000000003e151 or 3.30000000000000021e220 < th < 3.29999999999999994e243

   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 69.4%

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

   6. Applied egg-rr 47.0%

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

      1. +-commutative 47.0%

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

      2. distribute-lft-in 51.7%

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

   8. Simplified 51.7%

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

   if 9.2000000000000003e151 < th < 3.30000000000000021e220 or 3.29999999999999994e243 < 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 15.1%

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

   6. Applied egg-rr 46.7%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+151} \lor \neg \left(th \leq 3.3 \cdot 10^{+220}\right) \land th \leq 3.3 \cdot 10^{+243}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 7: 46.4% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.2 \cdot 10^{+151} \lor \neg \left(th \leq 3.3 \cdot 10^{+220}\right) \land th \leq 3.3 \cdot 10^{+243}:\\ \;\;\;\;0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 9.2e+151) (and (not (<= th 3.3e+220)) (<= th 3.3e+243)))
     (* 0.5 t_1)
     (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243))) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.5;
	}
	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 <= 9.2d+151) .or. (.not. (th <= 3.3d+220)) .and. (th <= 3.3d+243)) then
        tmp = 0.5d0 * t_1
    else
        tmp = t_1 * (-0.5d0)
    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 <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243))) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if (th <= 9.2e+151) or (not (th <= 3.3e+220) and (th <= 3.3e+243)):
		tmp = 0.5 * t_1
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243)))
		tmp = Float64(0.5 * t_1);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if ((th <= 9.2e+151) || (~((th <= 3.3e+220)) && (th <= 3.3e+243)))
		tmp = 0.5 * t_1;
	else
		tmp = t_1 * -0.5;
	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, 9.2e+151], And[N[Not[LessEqual[th, 3.3e+220]], $MachinePrecision], LessEqual[th, 3.3e+243]]], N[(0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 9.2000000000000003e151 or 3.30000000000000021e220 < th < 3.29999999999999994e243

   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 69.4%

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

   6. Applied egg-rr 52.0%

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

   if 9.2000000000000003e151 < th < 3.30000000000000021e220 or 3.29999999999999994e243 < 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 15.1%

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

   6. Applied egg-rr 46.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+151} \lor \neg \left(th \leq 3.3 \cdot 10^{+220}\right) \land th \leq 3.3 \cdot 10^{+243}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 8: 46.1% accurate, 29.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+151} \lor \neg \left(th \leq 3.3 \cdot 10^{+220}\right) \land th \leq 3.3 \cdot 10^{+243}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-a2 \cdot a2\right) - a1 \cdot a1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 9.2e+151) (and (not (<= th 3.3e+220)) (<= th 3.3e+243)))
   (* (+ a1 a2) (+ a1 a2))
   (- (- (* a2 a2)) (* a1 a1))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = -(a2 * a2) - (a1 * a1);
	}
	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 ((th <= 9.2d+151) .or. (.not. (th <= 3.3d+220)) .and. (th <= 3.3d+243)) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = -(a2 * a2) - (a1 * a1)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = -(a2 * a2) - (a1 * a1);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 9.2e+151) or (not (th <= 3.3e+220) and (th <= 3.3e+243)):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = -(a2 * a2) - (a1 * a1)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 9.2e+151) || (!(th <= 3.3e+220) && (th <= 3.3e+243)))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(-Float64(a2 * a2)) - Float64(a1 * a1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 9.2e+151) || (~((th <= 3.3e+220)) && (th <= 3.3e+243)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = -(a2 * a2) - (a1 * a1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 9.2e+151], And[N[Not[LessEqual[th, 3.3e+220]], $MachinePrecision], LessEqual[th, 3.3e+243]]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[((-N[(a2 * a2), $MachinePrecision]) - N[(a1 * a1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.2000000000000003e151 or 3.30000000000000021e220 < th < 3.29999999999999994e243

   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 69.4%

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

   6. Applied egg-rr 47.0%

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

      1. +-commutative 47.0%

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

      2. distribute-lft-in 51.7%

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

   8. Simplified 51.7%

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

   if 9.2000000000000003e151 < th < 3.30000000000000021e220 or 3.29999999999999994e243 < 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 15.1%

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

   6. Applied egg-rr 45.7%

      \[\leadsto \color{blue}{-1} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+151} \lor \neg \left(th \leq 3.3 \cdot 10^{+220}\right) \land th \leq 3.3 \cdot 10^{+243}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-a2 \cdot a2\right) - a1 \cdot a1\\ \end{array} \]

Alternative 9: 46.2% accurate, 59.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + 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 64.7%

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

6. Applied egg-rr 44.3%

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

   1. +-commutative 44.3%

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

   2. distribute-lft-in 48.6%

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

8. Simplified 48.6%

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

9. Final simplification 48.6%

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

Alternative 10: 4.1% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 + a2 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 + a2), $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 64.7%

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

6. Applied egg-rr 4.1%

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

7. Final simplification 4.1%

   \[\leadsto a1 + a2 \]

Alternative 11: 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%

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

3. Simplified 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. *-commutative 99.5%

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

   4. div-inv 99.5%

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

   5. associate-*r* 99.5%

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

   6. fma-def 99.5%

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

   7. pow2 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-flip 99.5%

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

   10. metadata-eval 99.5%

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

   11. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

7. Applied egg-rr 3.2%

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

   1. *-inverses 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023314 
(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))))