Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 9.6s

Alternatives: 14

Speedup: 2.0×

• 44.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot \left(\left(a1 - a2\right) \cdot \left(a1 - a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.68)
   (* (cos th) (* (- a1 a2) (- a1 a2)))
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.68) {
		tmp = cos(th) * ((a1 - a2) * (a1 - a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(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 (cos(th) <= 0.68d0) then
        tmp = cos(th) * ((a1 - a2) * (a1 - a2))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.68) {
		tmp = Math.cos(th) * ((a1 - a2) * (a1 - a2));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.68:
		tmp = math.cos(th) * ((a1 - a2) * (a1 - a2))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.68)
		tmp = Float64(cos(th) * Float64(Float64(a1 - a2) * Float64(a1 - a2)));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.68)
		tmp = cos(th) * ((a1 - a2) * (a1 - a2));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.68], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 - a2), $MachinePrecision] * N[(a1 - a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.680000000000000049

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. flip-+ 24.6%

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

      3. frac-times 24.6%

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

      4. *-un-lft-identity 24.6%

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

      5. pow2 24.6%

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

      6. pow2 24.6%

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

      7. pow-prod-up 24.6%

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

      8. metadata-eval 24.6%

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

      9. pow2 24.6%

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

      10. pow2 24.6%

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

      11. pow-prod-up 24.6%

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

      12. metadata-eval 24.6%

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

      13. pow2 24.6%

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

      14. pow2 24.6%

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

   5. Applied egg-rr 24.6%

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

   7. Applied egg-rr 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 60.3%

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

   9. Simplified 60.3%

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

   if 0.680000000000000049 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   4. 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/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.6%

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

      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 0 92.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot \left(\left(a1 - a2\right) \cdot \left(a1 - a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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)) * (cos(th) * sqrt(0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.6%

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

   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%

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

Alternative 4: 56.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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(0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.8%

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

   1. clear-num 58.4%

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

   2. pow2 58.4%

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

   3. associate-/r/ 58.8%

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

   4. pow1/2 58.8%

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

   5. pow-flip 58.8%

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

   6. metadata-eval 58.8%

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

   7. *-commutative 58.8%

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

   8. associate-*l* 58.8%

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

   9. associate-*r* 58.8%

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

   10. add-sqr-sqrt 58.5%

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

   11. sqrt-unprod 58.8%

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

   12. pow-prod-up 58.8%

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

   13. metadata-eval 58.8%

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

   14. metadata-eval 58.8%

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

6. Applied egg-rr 58.8%

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

7. Final simplification 58.8%

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

Alternative 5: 60.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. clear-num 99.6%

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

   2. flip-+ 28.6%

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

   3. frac-times 28.6%

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

   4. *-un-lft-identity 28.6%

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

   5. pow2 28.6%

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

   6. pow2 28.6%

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

   7. pow-prod-up 28.6%

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

   8. metadata-eval 28.6%

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

   9. pow2 28.6%

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

   10. pow2 28.6%

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

   11. pow-prod-up 28.6%

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

   12. metadata-eval 28.6%

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

   13. pow2 28.6%

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

   14. pow2 28.6%

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

5. Applied egg-rr 28.6%

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

7. Applied egg-rr 59.7%

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

   1. *-commutative 59.7%

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

   2. associate-*l* 59.7%

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

9. Simplified 59.7%

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

10. Final simplification 59.7%

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

Alternative 6: 37.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. flip-+ 28.6%

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

   3. frac-times 28.6%

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

   4. *-un-lft-identity 28.6%

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

   5. pow2 28.6%

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

   6. pow2 28.6%

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

   7. pow-prod-up 28.6%

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

   8. metadata-eval 28.6%

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

   9. pow2 28.6%

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

   10. pow2 28.6%

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

   11. pow-prod-up 28.6%

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

   12. metadata-eval 28.6%

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

   13. pow2 28.6%

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

   14. pow2 28.6%

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

5. Applied egg-rr 28.6%

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

7. Applied egg-rr 59.7%

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

8. Taylor expanded in a1 around 0 38.4%

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

   1. mul-1-neg 38.4%

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

   2. distribute-lft-neg-out 38.4%

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

   3. *-commutative 38.4%

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

10. Simplified 38.4%

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

11. Final simplification 38.4%

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

Alternative 7: 47.0% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\left(a1 - a2\right) \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 5e+68) (* (- a1 a2) (- a1 a2)) (* (+ (* a1 a1) (* a2 a2)) -0.5)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 5e+68) {
		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 <= 5d+68) 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 <= 5e+68) {
		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 <= 5e+68:
		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 <= 5e+68)
		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 <= 5e+68)
		tmp = (a1 - a2) * (a1 - a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 5e+68], 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 < 5.0000000000000004e68

   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. clear-num 99.6%

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

      2. flip-+ 28.3%

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

      3. frac-times 28.3%

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

      4. *-un-lft-identity 28.3%

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

      5. pow2 28.3%

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

      6. pow2 28.3%

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

      7. pow-prod-up 28.3%

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

      8. metadata-eval 28.3%

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

      9. pow2 28.3%

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

      10. pow2 28.3%

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

      11. pow-prod-up 28.3%

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

      12. metadata-eval 28.3%

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

      13. pow2 28.3%

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

      14. pow2 28.3%

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

   5. Applied egg-rr 28.3%

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

   7. Applied egg-rr 60.9%

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

   8. Taylor expanded in th around 0 50.3%

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

   if 5.0000000000000004e68 < th

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 25.7%

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

   6. Applied egg-rr 39.0%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\left(a1 - a2\right) \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 8: 47.2% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 5 \cdot 10^{+68}:\\ \;\;\;\;t_1 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 5e+68) (* t_1 0.5) (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 5e+68) {
		tmp = t_1 * 0.5;
	} 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 <= 5d+68) then
        tmp = t_1 * 0.5d0
    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 <= 5e+68) {
		tmp = t_1 * 0.5;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 5e+68:
		tmp = t_1 * 0.5
	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 <= 5e+68)
		tmp = Float64(t_1 * 0.5);
	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 <= 5e+68)
		tmp = t_1 * 0.5;
	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[LessEqual[th, 5e+68], N[(t$95$1 * 0.5), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 5.0000000000000004e68

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 75.1%

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

   6. Applied egg-rr 50.6%

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

   if 5.0000000000000004e68 < th

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 25.7%

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

   6. Applied egg-rr 39.0%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 9: 46.9% accurate, 41.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\left(a1 - a2\right) \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(-a1\right) - a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 5e+68) (* (- a1 a2) (- a1 a2)) (- (* a1 (- a1)) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 5e+68) {
		tmp = (a1 - a2) * (a1 - a2);
	} else {
		tmp = (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 (th <= 5d+68) then
        tmp = (a1 - a2) * (a1 - a2)
    else
        tmp = (a1 * -a1) - (a2 * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 5e+68) {
		tmp = (a1 - a2) * (a1 - a2);
	} else {
		tmp = (a1 * -a1) - (a2 * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 5e+68:
		tmp = (a1 - a2) * (a1 - a2)
	else:
		tmp = (a1 * -a1) - (a2 * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 5e+68)
		tmp = Float64(Float64(a1 - a2) * Float64(a1 - a2));
	else
		tmp = Float64(Float64(a1 * Float64(-a1)) - Float64(a2 * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 5e+68)
		tmp = (a1 - a2) * (a1 - a2);
	else
		tmp = (a1 * -a1) - (a2 * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 5e+68], N[(N[(a1 - a2), $MachinePrecision] * N[(a1 - a2), $MachinePrecision]), $MachinePrecision], N[(N[(a1 * (-a1)), $MachinePrecision] - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 5.0000000000000004e68

   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. clear-num 99.6%

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

      2. flip-+ 28.3%

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

      3. frac-times 28.3%

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

      4. *-un-lft-identity 28.3%

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

      5. pow2 28.3%

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

      6. pow2 28.3%

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

      7. pow-prod-up 28.3%

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

      8. metadata-eval 28.3%

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

      9. pow2 28.3%

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

      10. pow2 28.3%

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

      11. pow-prod-up 28.3%

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

      12. metadata-eval 28.3%

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

      13. pow2 28.3%

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

      14. pow2 28.3%

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

   5. Applied egg-rr 28.3%

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

   7. Applied egg-rr 60.9%

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

   8. Taylor expanded in th around 0 50.3%

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

   if 5.0000000000000004e68 < th

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 25.7%

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

   6. Applied egg-rr 38.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\left(a1 - a2\right) \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(-a1\right) - a2 \cdot a2\\ \end{array} \]

Alternative 10: 47.3% 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 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. clear-num 99.6%

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

   2. flip-+ 28.6%

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

   3. frac-times 28.6%

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

   4. *-un-lft-identity 28.6%

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

   5. pow2 28.6%

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

   6. pow2 28.6%

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

   7. pow-prod-up 28.6%

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

   8. metadata-eval 28.6%

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

   9. pow2 28.6%

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

   10. pow2 28.6%

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

   11. pow-prod-up 28.6%

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

   12. metadata-eval 28.6%

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

   13. pow2 28.6%

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

   14. pow2 28.6%

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

5. Applied egg-rr 28.6%

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

7. Applied egg-rr 59.7%

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

8. Taylor expanded in th around 0 46.5%

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

9. Final simplification 46.5%

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

Alternative 11: 4.1% accurate, 69.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-a2\right) - a1}{-2} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ (- (- a2) a1) -2.0))

C

double code(double a1, double a2, double th) {
	return (-a2 - a1) / -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 - a1) / (-2.0d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (-a2 - a1) / -2.0;
}

Python

def code(a1, a2, th):
	return (-a2 - a1) / -2.0

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(-a2) - a1) / -2.0)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (-a2 - a1) / -2.0;
end

Wolfram

code[a1_, a2_, th_] := N[(N[((-a2) - a1), $MachinePrecision] / -2.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 67.6%

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

6. Applied egg-rr 4.2%

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

7. Final simplification 4.2%

   \[\leadsto \frac{\left(-a2\right) - a1}{-2} \]

Alternative 12: 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 67.6%

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

6. Applied egg-rr 4.2%

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

7. Final simplification 4.2%

   \[\leadsto a1 + a2 \]

Alternative 13: 2.0% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 -2.0)

C

double code(double a1, double a2, double th) {
	return -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 = -2.0d0
end function

Java

public static double code(double a1, double a2, double th) {
	return -2.0;
}

Python

def code(a1, a2, th):
	return -2.0

Julia

function code(a1, a2, th)
	return -2.0
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = -2.0;
end

Wolfram

code[a1_, a2_, th_] := -2.0

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.2%

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

   4. pow1/2 99.2%

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

   5. sqrt-pow1 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. pow1/2 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. sqrt-pow1 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
6. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. *-lft-identity 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\cos th}{{2}^{0.25}}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

9. Applied egg-rr 1.9%

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

    1. associate-*l* 1.9%

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

    2. times-frac 1.9%

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

    3. *-inverses 1.9%

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

    4. *-commutative 1.9%

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

    5. associate-/l* 1.9%

       \[\leadsto 1 \cdot \color{blue}{\frac{-2}{\frac{a1 - a2}{a1 - a2}}} \]

    6. *-inverses 1.9%

       \[\leadsto 1 \cdot \frac{-2}{\color{blue}{1}} \]

    7. metadata-eval 1.9%

       \[\leadsto 1 \cdot \color{blue}{-2} \]

    8. metadata-eval 1.9%

       \[\leadsto \color{blue}{-2} \]

11. Simplified 1.9%

    \[\leadsto \color{blue}{-2} \]

12. Final simplification 1.9%

    \[\leadsto -2 \]

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 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.2%

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

   4. pow1/2 99.2%

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

   5. sqrt-pow1 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. pow1/2 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. sqrt-pow1 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
6. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. *-lft-identity 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\cos th}{{2}^{0.25}}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

9. Applied egg-rr 2.5%

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

    1. distribute-lft-neg-out 2.5%

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

    2. distribute-rgt-neg-out 2.5%

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

    3. *-commutative 2.5%

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

    4. *-inverses 3.8%

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

11. Simplified 3.8%

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

12. Final simplification 3.8%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023337 
(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))))