Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.7s

Alternatives: 18

Speedup: 2.0×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 71.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* a2 (* (cos th) a2))
   (/ (+ (* a2 a2) (* a1 a1)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = ((a2 * a2) + (a1 * a1)) / sqrt(2.0);
	}
	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 = ((a2 * a2) + (a1 * a1)) / sqrt(2.0d0)
    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 = ((a2 * a2) + (a1 * a1)) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = ((a2 * a2) + (a1 * a1)) / math.sqrt(2.0)
	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(Float64(Float64(a2 * a2) + Float64(a1 * a1)) / sqrt(2.0));
	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 = ((a2 * a2) + (a1 * a1)) / sqrt(2.0);
	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[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 57.8%

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

      1. unpow2 57.8%

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

      2. associate-/l* 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in th around inf 57.7%

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

      1. associate-/l* 57.7%

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

      2. unpow2 57.7%

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

   9. Simplified 57.7%

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

   11. Applied egg-rr 36.4%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 90.3%

      \[\leadsto \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 90.3%

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

      2. unpow2 90.3%

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

      3. +-commutative 90.3%

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

   6. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-def 99.6%

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

   2. div-inv 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. associate-*l* 99.4%

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

   5. hypot-def 99.4%

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

   6. hypot-def 99.4%

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

   7. pow1/2 99.4%

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

   8. pow-flip 99.6%

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

   9. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left(\left({a2}^{2} + {a1}^{2}\right) \cdot \cos th\right)} \]
7. Step-by-step derivation

   1. +-commutative 99.6%

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

   2. unpow2 99.6%

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

   3. unpow2 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around inf 99.6%

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

   1. associate-/l* 99.6%

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

   2. unpow2 99.6%

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

   3. unpow2 99.6%

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

   4. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 46.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* a2 (* (cos th) a2)) (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	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 = a2 * (a2 / sqrt(2.0d0))
    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 = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	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(a2 * Float64(a2 / sqrt(2.0)));
	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 = a2 * (a2 / sqrt(2.0));
	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[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 57.8%

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

      1. unpow2 57.8%

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

      2. associate-/l* 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in th around inf 57.7%

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

      1. associate-/l* 57.7%

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

      2. unpow2 57.7%

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

   9. Simplified 57.7%

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

   11. Applied egg-rr 36.4%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-/l* 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in th around 0 46.5%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\frac{\sqrt{2}}{a2 \cdot a2}} \]
   8. Step-by-step derivation

      1. unpow2 46.5%

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

   9. Simplified 46.5%

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

   10. Taylor expanded in th around 0 53.5%

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

       1. unpow2 53.5%

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

       2. associate-*l/ 53.4%

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

       3. *-commutative 53.4%

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

   12. Simplified 53.4%

       \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 6: 46.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = (a2 * a2) * math.sqrt(0.5)
	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(Float64(a2 * a2) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = (a2 * a2) * sqrt(0.5);
	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[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 57.8%

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

      1. unpow2 57.8%

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

      2. associate-/l* 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in th around inf 57.7%

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

      1. associate-/l* 57.7%

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

      2. unpow2 57.7%

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

   9. Simplified 57.7%

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

   11. Applied egg-rr 36.4%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-/l* 58.3%

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

   6. Simplified 58.3%

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

      1. associate-/r/ 58.3%

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

      2. div-inv 58.3%

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

      3. pow1/2 58.3%

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

      4. pow-flip 58.4%

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

      5. metadata-eval 58.4%

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

      6. add-sqr-sqrt 58.0%

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

      7. sqrt-unprod 58.4%

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

      8. pow-prod-up 58.4%

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

      9. metadata-eval 58.4%

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

      10. metadata-eval 58.4%

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

   8. Applied egg-rr 58.4%

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

   9. Taylor expanded in th around 0 53.6%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot {a2}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 53.6%

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

   11. Simplified 53.6%

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

4. Final simplification 47.9%

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

Alternative 7: 46.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* a2 (* (cos th) a2)) (/ a2 (/ (sqrt 2.0) a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = a2 / (sqrt(2.0) / 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 = a2 / (sqrt(2.0d0) / 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 = a2 / (Math.sqrt(2.0) / 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 = a2 / (math.sqrt(2.0) / 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(a2 / Float64(sqrt(2.0) / 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 = a2 / (sqrt(2.0) / 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[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 57.8%

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

      1. unpow2 57.8%

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

      2. associate-/l* 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in th around inf 57.7%

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

      1. associate-/l* 57.7%

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

      2. unpow2 57.7%

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

   9. Simplified 57.7%

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

   11. Applied egg-rr 36.4%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. associate-/l* 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in th around inf 58.3%

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

      1. associate-/l* 58.3%

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

      2. unpow2 58.3%

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

   9. Simplified 58.3%

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

   10. Taylor expanded in th around 0 53.5%

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

       1. unpow2 53.5%

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

       2. associate-/l* 53.5%

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

   12. Simplified 53.5%

       \[\leadsto \color{blue}{\frac{a2}{\frac{\sqrt{2}}{a2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 8: 46.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* a2 (* (cos th) a2)) (/ (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = (a2 * a2) / sqrt(2.0);
	}
	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 = (a2 * a2) / sqrt(2.0d0)
    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 = (a2 * a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = (a2 * a2) / math.sqrt(2.0)
	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(Float64(a2 * a2) / sqrt(2.0));
	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 = (a2 * a2) / sqrt(2.0);
	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[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 57.8%

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

      1. unpow2 57.8%

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

      2. associate-/l* 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in th around inf 57.7%

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

      1. associate-/l* 57.7%

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

      2. unpow2 57.7%

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

   9. Simplified 57.7%

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

   11. Applied egg-rr 36.4%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 90.3%

      \[\leadsto \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\sqrt{2}}} \]
   5. Step-by-step derivation

      1. unpow2 90.3%

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

      2. unpow2 90.3%

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

      3. +-commutative 90.3%

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

   6. Simplified 90.3%

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

   7. Taylor expanded in a1 around 0 53.5%

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

      1. unpow2 53.5%

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

   9. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 9: 46.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	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(a2 * Float64(a2 * sqrt(0.5)));
	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 = a2 * (a2 * sqrt(0.5));
	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[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 57.8%

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

      1. unpow2 57.8%

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

      2. associate-/l* 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in th around inf 57.7%

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

      1. associate-/l* 57.7%

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

      2. unpow2 57.7%

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

   9. Simplified 57.7%

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

   11. Applied egg-rr 36.4%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. associate-/l* 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in th around 0 53.5%

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

      1. associate-/r/ 53.4%

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

      2. div-inv 53.4%

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

      3. pow1/2 53.4%

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

      4. pow-flip 53.5%

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

      5. metadata-eval 53.5%

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

      6. add-sqr-sqrt 53.3%

         \[\leadsto 1 \cdot \left(\left(a2 \cdot \color{blue}{\left(\sqrt{{2}^{-0.5}} \cdot \sqrt{{2}^{-0.5}}\right)}\right) \cdot a2\right) \]

      7. sqrt-unprod 53.5%

         \[\leadsto 1 \cdot \left(\left(a2 \cdot \color{blue}{\sqrt{{2}^{-0.5} \cdot {2}^{-0.5}}}\right) \cdot a2\right) \]

      8. pow-prod-up 53.5%

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

      9. metadata-eval 53.5%

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

      10. metadata-eval 53.5%

          \[\leadsto 1 \cdot \left(\left(a2 \cdot \sqrt{\color{blue}{0.5}}\right) \cdot a2\right) \]

   9. Applied egg-rr 53.5%

      \[\leadsto 1 \cdot \color{blue}{\left(\left(a2 \cdot \sqrt{0.5}\right) \cdot a2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

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

Alternative 10: 56.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.1%

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

   1. unpow2 58.1%

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

   2. associate-*l* 58.1%

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

   3. associate-*r/ 58.1%

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

   4. associate-/l* 58.1%

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

6. Simplified 58.1%

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

7. Final simplification 58.1%

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

Alternative 11: 56.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.1%

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

   1. unpow2 58.1%

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

   2. associate-/l* 58.1%

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

6. Simplified 58.1%

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

7. Final simplification 58.1%

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

Alternative 12: 56.6% accurate, 2.0× speedup

Math

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

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(Float64(a2 * 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[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.1%

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

   1. unpow2 58.1%

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

6. Simplified 58.1%

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

7. Final simplification 58.1%

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

Alternative 13: 36.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;a2 \cdot \left(-a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{th + -2}{-2 + \left(th + th\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.02)
   (* a2 (- a2))
   (* (* a2 a2) (/ (+ th -2.0) (+ -2.0 (+ th th))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.02) {
		tmp = a2 * -a2;
	} else {
		tmp = (a2 * a2) * ((th + -2.0) / (-2.0 + (th + th)));
	}
	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) <= (-0.02d0)) then
        tmp = a2 * -a2
    else
        tmp = (a2 * a2) * ((th + (-2.0d0)) / ((-2.0d0) + (th + th)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -0.02) {
		tmp = a2 * -a2;
	} else {
		tmp = (a2 * a2) * ((th + -2.0) / (-2.0 + (th + th)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -0.02:
		tmp = a2 * -a2
	else:
		tmp = (a2 * a2) * ((th + -2.0) / (-2.0 + (th + th)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.02)
		tmp = Float64(a2 * Float64(-a2));
	else
		tmp = Float64(Float64(a2 * a2) * Float64(Float64(th + -2.0) / Float64(-2.0 + Float64(th + th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -0.02)
		tmp = a2 * -a2;
	else
		tmp = (a2 * a2) * ((th + -2.0) / (-2.0 + (th + th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.02], N[(a2 * (-a2)), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[(N[(th + -2.0), $MachinePrecision] / N[(-2.0 + N[(th + th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -0.0200000000000000004

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 57.0%

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

      1. unpow2 57.0%

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

      2. *-commutative 57.0%

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

      3. associate-/l* 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in th around 0 34.8%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\frac{\sqrt{2}}{a2 \cdot a2}} \]
   8. Step-by-step derivation

      1. unpow2 34.8%

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

   9. Simplified 34.8%

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

   10. Taylor expanded in th around 0 10.9%

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

       1. unpow2 10.9%

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

       2. associate-*l/ 10.9%

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

       3. *-commutative 10.9%

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

   12. Simplified 10.9%

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

   14. Applied egg-rr 40.1%

       \[\leadsto a2 \cdot \color{blue}{\left(0 - a2\right)} \]
   15. Step-by-step derivation

       1. neg-sub0 40.1%

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

   16. Simplified 40.1%

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

   if -0.0200000000000000004 < (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%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 58.4%

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

      1. unpow2 58.4%

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

      2. *-commutative 58.4%

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

      3. associate-/l* 58.4%

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

   6. Simplified 58.4%

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

      1. associate-/r/ 58.4%

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

      2. div-inv 58.4%

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

      3. pow1/2 58.4%

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

      4. pow-flip 58.5%

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

      5. metadata-eval 58.5%

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

      6. add-sqr-sqrt 58.1%

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

      7. sqrt-unprod 58.5%

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

      8. pow-prod-up 58.5%

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

      9. metadata-eval 58.5%

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

      10. metadata-eval 58.5%

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

   8. Applied egg-rr 58.5%

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

   9. Taylor expanded in th around 0 39.2%

      \[\leadsto \left(\color{blue}{\left(1 + -0.5 \cdot {th}^{2}\right)} \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right) \]
   10. Step-by-step derivation

       1. unpow2 39.1%

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

   11. Simplified 39.2%

       \[\leadsto \left(\color{blue}{\left(1 + -0.5 \cdot \left(th \cdot th\right)\right)} \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right) \]

   13. Applied egg-rr 35.5%

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

       1. +-commutative 35.5%

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

   15. Simplified 35.5%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.02:\\ \;\;\;\;a2 \cdot \left(-a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{th + -2}{-2 + \left(th + th\right)}\\ \end{array} \]

Alternative 14: 37.0% 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%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.1%

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

   1. unpow2 58.1%

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

   2. associate-/l* 58.1%

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

6. Simplified 58.1%

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

7. Taylor expanded in th around inf 58.1%

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

   1. associate-/l* 58.1%

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

   2. unpow2 58.1%

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

9. Simplified 58.1%

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

11. Applied egg-rr 36.4%

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

12. Final simplification 36.4%

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

Alternative 15: 29.8% accurate, 68.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 4.6 \cdot 10^{+220}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(-a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 4.6e+220) (* a2 a2) (* a2 (- a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.6e+220) {
		tmp = a2 * a2;
	} else {
		tmp = 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 <= 4.6d+220) then
        tmp = a2 * a2
    else
        tmp = a2 * -a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.6e+220) {
		tmp = a2 * a2;
	} else {
		tmp = a2 * -a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 4.6e+220:
		tmp = a2 * a2
	else:
		tmp = a2 * -a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 4.6e+220)
		tmp = Float64(a2 * a2);
	else
		tmp = Float64(a2 * Float64(-a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 4.6e+220)
		tmp = a2 * a2;
	else
		tmp = a2 * -a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 4.6e+220], N[(a2 * a2), $MachinePrecision], N[(a2 * (-a2)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 4.59999999999999993e220

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 58.3%

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

      1. unpow2 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-/l* 58.2%

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

   6. Simplified 58.2%

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

   7. Taylor expanded in th around 0 38.9%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\frac{\sqrt{2}}{a2 \cdot a2}} \]
   8. Step-by-step derivation

      1. unpow2 38.9%

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

   9. Simplified 38.9%

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

   10. Taylor expanded in th around 0 43.7%

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

       1. unpow2 43.7%

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

       2. associate-*l/ 43.7%

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

       3. *-commutative 43.7%

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

   12. Simplified 43.7%

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

   14. Applied egg-rr 20.7%

       \[\leadsto a2 \cdot \color{blue}{\left|a2\right|} \]
   15. Step-by-step derivation

       1. unpow1 20.7%

          \[\leadsto a2 \cdot \left|\color{blue}{{a2}^{1}}\right| \]

       2. sqr-pow 16.1%

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

       3. fabs-sqr 16.1%

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

       4. sqr-pow 31.3%

          \[\leadsto a2 \cdot \color{blue}{{a2}^{1}} \]

       5. unpow1 31.3%

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

   16. Simplified 31.3%

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

   if 4.59999999999999993e220 < 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%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.8%

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

      4. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a1 around 0 56.6%

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

      1. unpow2 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-/l* 56.5%

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

   6. Simplified 56.5%

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

   7. Taylor expanded in th around 0 31.0%

      \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\frac{\sqrt{2}}{a2 \cdot a2}} \]
   8. Step-by-step derivation

      1. unpow2 31.0%

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

   9. Simplified 31.0%

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

   10. Taylor expanded in th around 0 18.2%

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

       1. unpow2 18.2%

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

       2. associate-*l/ 18.2%

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

       3. *-commutative 18.2%

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

   12. Simplified 18.2%

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

   14. Applied egg-rr 33.4%

       \[\leadsto a2 \cdot \color{blue}{\left(0 - a2\right)} \]
   15. Step-by-step derivation

       1. neg-sub0 33.4%

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

   16. Simplified 33.4%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 4.6 \cdot 10^{+220}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(-a2\right)\\ \end{array} \]

Alternative 16: 29.6% accurate, 138.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.1%

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

   1. unpow2 58.1%

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

   2. *-commutative 58.1%

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

   3. associate-/l* 58.1%

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

6. Simplified 58.1%

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

7. Taylor expanded in th around 0 38.3%

   \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\frac{\sqrt{2}}{a2 \cdot a2}} \]
8. Step-by-step derivation

   1. unpow2 38.3%

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

9. Simplified 38.3%

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

10. Taylor expanded in th around 0 41.7%

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

    1. unpow2 41.7%

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

    2. associate-*l/ 41.7%

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

    3. *-commutative 41.7%

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

12. Simplified 41.7%

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

14. Applied egg-rr 20.5%

    \[\leadsto a2 \cdot \color{blue}{\left|a2\right|} \]
15. Step-by-step derivation

    1. unpow1 20.5%

       \[\leadsto a2 \cdot \left|\color{blue}{{a2}^{1}}\right| \]

    2. sqr-pow 15.8%

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

    3. fabs-sqr 15.8%

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

    4. sqr-pow 30.3%

       \[\leadsto a2 \cdot \color{blue}{{a2}^{1}} \]

    5. unpow1 30.3%

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

16. Simplified 30.3%

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

17. Final simplification 30.3%

    \[\leadsto a2 \cdot a2 \]

Alternative 17: 3.5% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := 1.0

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 58.1%

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

   1. unpow2 58.1%

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

   2. *-commutative 58.1%

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

   3. associate-/l* 58.1%

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

6. Simplified 58.1%

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

7. Taylor expanded in th around 0 38.3%

   \[\leadsto \frac{\color{blue}{1 + -0.5 \cdot {th}^{2}}}{\frac{\sqrt{2}}{a2 \cdot a2}} \]
8. Step-by-step derivation

   1. unpow2 38.3%

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

9. Simplified 38.3%

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

10. Taylor expanded in th around 0 41.7%

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

    1. unpow2 41.7%

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

    2. associate-*l/ 41.7%

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

    3. *-commutative 41.7%

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

12. Simplified 41.7%

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

14. Applied egg-rr 3.8%

    \[\leadsto \color{blue}{\frac{a2}{a2}} \]
15. Step-by-step derivation

    1. *-inverses 3.8%

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

16. Simplified 3.8%

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

17. Final simplification 3.8%

    \[\leadsto 1 \]

Alternative 18: 3.6% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a1

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 68.2%

   \[\leadsto \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 68.2%

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

   2. unpow2 68.2%

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

   3. +-commutative 68.2%

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

6. Simplified 68.2%

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

7. Taylor expanded in a1 around inf 37.9%

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} \]
8. Step-by-step derivation

   1. unpow2 37.9%

      \[\leadsto \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}} \]

9. Simplified 37.9%

   \[\leadsto \color{blue}{\frac{a1 \cdot a1}{\sqrt{2}}} \]
10. Step-by-step derivation

    1. frac-2neg 37.9%

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

    2. div-inv 37.8%

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

    3. distribute-rgt-neg-in 37.8%

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

11. Applied egg-rr 37.8%

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

13. Applied egg-rr 3.3%

    \[\leadsto \color{blue}{0 + a1} \]
14. Step-by-step derivation

    1. +-lft-identity 3.3%

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

15. Simplified 3.3%

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

16. Final simplification 3.3%

    \[\leadsto a1 \]

Reproduce

herbie shell --seed 2023222 
(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))))