Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.8s

Alternatives: 17

Speedup: 2.0×

• 43.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*l/ 99.6%

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

   6. fma-def 99.6%

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

   7. cos-neg 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 71.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(a2 * Float64(a2 * cos(th)));
	else
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * 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 * (a2 * cos(th));
	else
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.6%

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

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

      3. associate-/r/ 99.4%

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

      4. cos-neg 99.4%

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

      5. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a1 around 0 61.0%

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

      1. pow2 61.0%

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

      2. associate-/r/ 61.1%

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

      3. pow1/2 61.1%

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

      4. metadata-eval 61.1%

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

      5. pow-prod-up 61.0%

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

      6. associate-/r* 61.0%

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

      7. div-inv 60.9%

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

      8. *-commutative 60.9%

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

      9. associate-*r* 60.9%

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

   6. Applied egg-rr 61.0%

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

   8. Applied egg-rr 38.4%

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

      1. log-pow 38.4%

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

      2. rem-log-exp 39.8%

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

   10. Simplified 39.8%

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

   if 0.69999999999999996 < (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 99.6%

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

      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 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 89.5%

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

4. Final simplification 69.3%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 4: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 47.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (a2 * cos(th));
	} 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 * (a2 * cos(th))
    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 * (a2 * Math.cos(th));
	} 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 * (a2 * math.cos(th))
	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(a2 * cos(th)));
	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 * (a2 * cos(th));
	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[(a2 * N[Cos[th], $MachinePrecision]), $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 99.6%

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

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

      3. associate-/r/ 99.4%

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

      4. cos-neg 99.4%

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

      5. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a1 around 0 61.0%

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

      1. pow2 61.0%

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

      2. associate-/r/ 61.1%

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

      3. pow1/2 61.1%

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

      4. metadata-eval 61.1%

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

      5. pow-prod-up 61.0%

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

      6. associate-/r* 61.0%

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

      7. div-inv 60.9%

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

      8. *-commutative 60.9%

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

      9. associate-*r* 60.9%

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

   6. Applied egg-rr 61.0%

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

   8. Applied egg-rr 38.4%

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

      1. log-pow 38.4%

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

      2. rem-log-exp 39.8%

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

   10. Simplified 39.8%

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

   if 0.69999999999999996 < (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%

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

      2. cos-neg 99.6%

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

      3. associate-/r/ 99.3%

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

      4. cos-neg 99.3%

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

      5. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a1 around 0 53.1%

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

      1. pow2 53.1%

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

      2. associate-/r/ 53.0%

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

      3. pow1/2 53.0%

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

      4. metadata-eval 53.0%

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

      5. pow-prod-up 53.1%

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

      6. associate-/r* 53.1%

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

      7. div-inv 52.9%

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

      8. *-commutative 52.9%

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

      9. associate-*r* 52.9%

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

   6. Applied egg-rr 53.1%

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

   7. Taylor expanded in th around 0 49.9%

      \[\leadsto \color{blue}{\left(a2 \cdot \sqrt{0.5}\right)} \cdot a2 \]
   8. Step-by-step derivation

      1. *-commutative 49.9%

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

   9. Simplified 49.9%

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

4. Final simplification 45.8%

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

Alternative 6: 47.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (a2 * cos(th));
	} 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 * (a2 * cos(th))
    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 * (a2 * Math.cos(th));
	} 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 * (a2 * math.cos(th))
	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(a2 * cos(th)));
	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 * (a2 * cos(th));
	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[(a2 * N[Cos[th], $MachinePrecision]), $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 99.6%

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

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

      3. associate-/r/ 99.4%

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

      4. cos-neg 99.4%

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

      5. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a1 around 0 61.0%

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

      1. pow2 61.0%

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

      2. associate-/r/ 61.1%

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

      3. pow1/2 61.1%

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

      4. metadata-eval 61.1%

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

      5. pow-prod-up 61.0%

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

      6. associate-/r* 61.0%

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

      7. div-inv 60.9%

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

      8. *-commutative 60.9%

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

      9. associate-*r* 60.9%

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

   6. Applied egg-rr 61.0%

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

   8. Applied egg-rr 38.4%

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

      1. log-pow 38.4%

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

      2. rem-log-exp 39.8%

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

   10. Simplified 39.8%

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

   if 0.69999999999999996 < (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%

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

      2. cos-neg 99.6%

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

      3. associate-/r/ 99.3%

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

      4. cos-neg 99.3%

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

      5. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a1 around 0 53.1%

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

   5. Taylor expanded in th around 0 49.9%

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

      1. clear-num 49.9%

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

      2. unpow2 49.9%

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

      3. *-un-lft-identity 49.9%

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

      4. times-frac 49.9%

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

   7. Applied egg-rr 49.9%

      \[\leadsto \color{blue}{\frac{a2}{1} \cdot \frac{a2}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

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

Alternative 7: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. cos-neg 99.6%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 56.3%

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

   1. pow2 56.3%

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

   2. associate-/r/ 56.3%

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

   3. pow1/2 56.3%

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

   4. metadata-eval 56.3%

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

   5. pow-prod-up 56.3%

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

   6. associate-/r* 56.3%

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

   7. div-inv 56.1%

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

   8. *-commutative 56.1%

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

   9. associate-*r* 56.1%

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

6. Applied egg-rr 56.3%

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

7. Taylor expanded in th around inf 56.3%

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

   1. associate-*r* 56.3%

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

   2. *-commutative 56.3%

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

9. Simplified 56.3%

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

10. Final simplification 56.3%

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

Alternative 8: 57.6% 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%

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

   2. cos-neg 99.6%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 56.3%

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

   1. pow2 56.3%

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

   2. associate-/r/ 56.3%

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

   3. pow1/2 56.3%

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

   4. metadata-eval 56.3%

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

   5. pow-prod-up 56.3%

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

   6. associate-/r* 56.3%

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

   7. div-inv 56.1%

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

   8. *-commutative 56.1%

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

   9. associate-*r* 56.1%

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

6. Applied egg-rr 56.3%

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

7. Final simplification 56.3%

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

Alternative 9: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. cos-neg 99.6%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 56.3%

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

   1. pow2 56.3%

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

   2. associate-/r/ 56.3%

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

   3. pow1/2 56.3%

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

   4. metadata-eval 56.3%

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

   5. pow-prod-up 56.3%

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

   6. associate-/r* 56.3%

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

   7. div-inv 56.1%

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

   8. *-commutative 56.1%

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

   9. associate-*r* 56.1%

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

6. Applied egg-rr 56.3%

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

7. Taylor expanded in th around inf 56.3%

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

   1. associate-*r* 56.3%

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

   2. *-commutative 56.3%

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

9. Simplified 56.3%

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

    1. add-log-exp 39.7%

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

    2. *-un-lft-identity 39.7%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{\left(\cos th \cdot a2\right) \cdot \sqrt{0.5}}\right)} \cdot a2 \]

    3. log-prod 39.7%

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

    4. metadata-eval 39.7%

       \[\leadsto \left(\color{blue}{0} + \log \left(e^{\left(\cos th \cdot a2\right) \cdot \sqrt{0.5}}\right)\right) \cdot a2 \]

    5. add-log-exp 56.3%

       \[\leadsto \left(0 + \color{blue}{\left(\cos th \cdot a2\right) \cdot \sqrt{0.5}}\right) \cdot a2 \]

    6. associate-*l* 56.3%

       \[\leadsto \left(0 + \color{blue}{\cos th \cdot \left(a2 \cdot \sqrt{0.5}\right)}\right) \cdot a2 \]

11. Applied egg-rr 56.3%

    \[\leadsto \color{blue}{\left(0 + \cos th \cdot \left(a2 \cdot \sqrt{0.5}\right)\right)} \cdot a2 \]

12. Final simplification 56.3%

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

Alternative 10: 37.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. cos-neg 99.6%

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

   3. associate-/r/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 56.3%

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

   1. pow2 56.3%

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

   2. associate-/r/ 56.3%

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

   3. pow1/2 56.3%

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

   4. metadata-eval 56.3%

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

   5. pow-prod-up 56.3%

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

   6. associate-/r* 56.3%

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

   7. div-inv 56.1%

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

   8. *-commutative 56.1%

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

   9. associate-*r* 56.1%

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

6. Applied egg-rr 56.3%

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

8. Applied egg-rr 37.8%

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

   1. log-pow 37.8%

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

   2. rem-log-exp 37.3%

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

10. Simplified 37.3%

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

11. Final simplification 37.3%

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

Alternative 11: 43.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.4 \cdot 10^{+92} \lor \neg \left(th \leq 2.5 \cdot 10^{+254}\right) \land th \leq 1.85 \cdot 10^{+267}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 2.4e+92) (and (not (<= th 2.5e+254)) (<= th 1.85e+267)))
   (* (+ a1 a2) (+ a1 a2))
   (- a1 (* a2 a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 2.4e+92) || (!(th <= 2.5e+254) && (th <= 1.85e+267))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = 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 <= 2.4d+92) .or. (.not. (th <= 2.5d+254)) .and. (th <= 1.85d+267)) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = a1 - (a2 * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 2.4e+92) || (!(th <= 2.5e+254) && (th <= 1.85e+267))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = a1 - (a2 * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 2.4e+92) or (not (th <= 2.5e+254) and (th <= 1.85e+267)):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = a1 - (a2 * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 2.4e+92) || (!(th <= 2.5e+254) && (th <= 1.85e+267)))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(a1 - Float64(a2 * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 2.4e+92) || (~((th <= 2.5e+254)) && (th <= 1.85e+267)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = a1 - (a2 * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 2.4e+92], And[N[Not[LessEqual[th, 2.5e+254]], $MachinePrecision], LessEqual[th, 1.85e+267]]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(a1 - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 2.40000000000000005e92 or 2.49999999999999997e254 < th < 1.85e267

   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/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 73.4%

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

   8. Applied egg-rr 46.3%

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

      1. +-commutative 46.3%

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

      2. distribute-lft-out 52.1%

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

   10. Simplified 52.1%

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

   if 2.40000000000000005e92 < th < 2.49999999999999997e254 or 1.85e267 < th

   1. Initial program 99.8%

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

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

      2. associate-/r/ 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around 0 16.3%

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

   8. Applied egg-rr 29.6%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2.4 \cdot 10^{+92} \lor \neg \left(th \leq 2.5 \cdot 10^{+254}\right) \land th \leq 1.85 \cdot 10^{+267}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \]

Alternative 12: 43.5% accurate, 31.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.4 \cdot 10^{+92}:\\ \;\;\;\;a2 \cdot a2 + a1 \cdot a1\\ \mathbf{elif}\;th \leq 2.5 \cdot 10^{+254} \lor \neg \left(th \leq 1.85 \cdot 10^{+267}\right):\\ \;\;\;\;a1 - a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 2.4e+92)
   (+ (* a2 a2) (* a1 a1))
   (if (or (<= th 2.5e+254) (not (<= th 1.85e+267)))
     (- a1 (* a2 a2))
     (* (+ a1 a2) (+ a1 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.4e+92) {
		tmp = (a2 * a2) + (a1 * a1);
	} else if ((th <= 2.5e+254) || !(th <= 1.85e+267)) {
		tmp = a1 - (a2 * a2);
	} else {
		tmp = (a1 + a2) * (a1 + 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 <= 2.4d+92) then
        tmp = (a2 * a2) + (a1 * a1)
    else if ((th <= 2.5d+254) .or. (.not. (th <= 1.85d+267))) then
        tmp = a1 - (a2 * a2)
    else
        tmp = (a1 + a2) * (a1 + a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 2.4e+92) {
		tmp = (a2 * a2) + (a1 * a1);
	} else if ((th <= 2.5e+254) || !(th <= 1.85e+267)) {
		tmp = a1 - (a2 * a2);
	} else {
		tmp = (a1 + a2) * (a1 + a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 2.4e+92:
		tmp = (a2 * a2) + (a1 * a1)
	elif (th <= 2.5e+254) or not (th <= 1.85e+267):
		tmp = a1 - (a2 * a2)
	else:
		tmp = (a1 + a2) * (a1 + a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 2.4e+92)
		tmp = Float64(Float64(a2 * a2) + Float64(a1 * a1));
	elseif ((th <= 2.5e+254) || !(th <= 1.85e+267))
		tmp = Float64(a1 - Float64(a2 * a2));
	else
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 2.4e+92)
		tmp = (a2 * a2) + (a1 * a1);
	elseif ((th <= 2.5e+254) || ~((th <= 1.85e+267)))
		tmp = a1 - (a2 * a2);
	else
		tmp = (a1 + a2) * (a1 + a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 2.4e+92], N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 2.5e+254], N[Not[LessEqual[th, 1.85e+267]], $MachinePrecision]], N[(a1 - N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.40000000000000005e92

   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/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around inf 99.6%

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

      1. *-commutative 99.6%

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

   8. Simplified 99.6%

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

   10. Applied egg-rr 51.7%

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

       1. *-inverses 51.7%

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

   12. Simplified 51.7%

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

   if 2.40000000000000005e92 < th < 2.49999999999999997e254 or 1.85e267 < th

   1. Initial program 99.8%

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

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

      2. associate-/r/ 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around 0 16.3%

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

   8. Applied egg-rr 29.6%

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

   if 2.49999999999999997e254 < th < 1.85e267

   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 100.0%

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

      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 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in th around 0 75.1%

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

   8. Applied egg-rr 50.1%

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

      1. +-commutative 50.1%

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

      2. distribute-lft-out 75.1%

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

   10. Simplified 75.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2.4 \cdot 10^{+92}:\\ \;\;\;\;a2 \cdot a2 + a1 \cdot a1\\ \mathbf{elif}\;th \leq 2.5 \cdot 10^{+254} \lor \neg \left(th \leq 1.85 \cdot 10^{+267}\right):\\ \;\;\;\;a1 - a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \end{array} \]

Alternative 13: 15.5% accurate, 58.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 9.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{a1}{\frac{-2}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 9.2e-128) (/ a1 (/ -2.0 a1)) (- a1 (* a2 a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 9.2e-128) {
		tmp = a1 / (-2.0 / a1);
	} else {
		tmp = 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 (a2 <= 9.2d-128) then
        tmp = a1 / ((-2.0d0) / a1)
    else
        tmp = a1 - (a2 * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 9.2e-128) {
		tmp = a1 / (-2.0 / a1);
	} else {
		tmp = a1 - (a2 * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 9.2e-128:
		tmp = a1 / (-2.0 / a1)
	else:
		tmp = a1 - (a2 * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 9.2e-128)
		tmp = Float64(a1 / Float64(-2.0 / a1));
	else
		tmp = Float64(a1 - Float64(a2 * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 9.2e-128)
		tmp = a1 / (-2.0 / a1);
	else
		tmp = a1 - (a2 * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 9.2e-128], N[(a1 / N[(-2.0 / a1), $MachinePrecision]), $MachinePrecision], N[(a1 - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 9.2000000000000003e-128

   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 60.3%

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

   5. Taylor expanded in a1 around inf 42.3%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{{a1}^{2}} \]

   7. Applied egg-rr 20.4%

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

   if 9.2000000000000003e-128 < a2

   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. 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 66.4%

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

   8. Applied egg-rr 12.4%

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

4. Final simplification 17.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 9.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{a1}{\frac{-2}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \]

Alternative 14: 14.6% accurate, 83.0× speedup

Math

\[\begin{array}{l} \\ \frac{a1}{\frac{-2}{a1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 / N[(-2.0 / 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. Taylor expanded in th around 0 62.7%

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

5. Taylor expanded in a1 around inf 36.3%

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{{a1}^{2}} \]

7. Applied egg-rr 15.3%

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

8. Final simplification 15.3%

   \[\leadsto \frac{a1}{\frac{-2}{a1}} \]

Alternative 15: 3.6% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 \cdot 0.5 \end{array} \]

FPCore

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

C

double code(double a1, double a2, double th) {
	return a1 * 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 * 0.5d0
end function

Java

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

Python

def code(a1, a2, th):
	return a1 * 0.5

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 * 0.5), $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 62.7%

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

5. Taylor expanded in a1 around inf 36.3%

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{{a1}^{2}} \]

7. Applied egg-rr 3.3%

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

   1. neg-mul-1 3.3%

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

   2. associate-/l* 3.3%

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

   3. associate-/r/ 3.3%

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

   4. metadata-eval 3.3%

      \[\leadsto \color{blue}{0.5} \cdot a1 \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{0.5 \cdot a1} \]

10. Final simplification 3.3%

    \[\leadsto a1 \cdot 0.5 \]

Alternative 16: 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. 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 0 62.7%

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

8. Applied egg-rr 4.0%

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

9. Final simplification 4.0%

   \[\leadsto a1 + a2 \]

Alternative 17: 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 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 62.7%

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

5. Taylor expanded in a1 around inf 36.3%

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{{a1}^{2}} \]

7. Applied egg-rr 3.3%

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

   1. associate-/r/ 3.3%

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

   2. metadata-eval 3.3%

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

   3. *-lft-identity 3.3%

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

9. Simplified 3.3%

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

10. Final simplification 3.3%

    \[\leadsto a1 \]

Reproduce

herbie shell --seed 2023320 
(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))))