Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 18.5s

Alternatives: 19

Speedup: 2.0×

• 43.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{0.25}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ (/ (* (cos th) (pow (hypot a2 a1) 2.0)) (pow 2.0 0.25)) (pow 2.0 0.25)))

C

double code(double a1, double a2, double th) {
	return ((cos(th) * pow(hypot(a2, a1), 2.0)) / pow(2.0, 0.25)) / pow(2.0, 0.25);
}

Java

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) * Math.pow(Math.hypot(a2, a1), 2.0)) / Math.pow(2.0, 0.25)) / Math.pow(2.0, 0.25);
}

Python

def code(a1, a2, th):
	return ((math.cos(th) * math.pow(math.hypot(a2, a1), 2.0)) / math.pow(2.0, 0.25)) / math.pow(2.0, 0.25)

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) * (hypot(a2, a1) ^ 2.0)) / (2.0 ^ 0.25)) / (2.0 ^ 0.25))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) * (hypot(a2, a1) ^ 2.0)) / (2.0 ^ 0.25)) / (2.0 ^ 0.25);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] * N[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $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. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. associate-*l/ 99.6%

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

   2. +-commutative 99.6%

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

   3. fma-udef 99.6%

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

   4. add-sqr-sqrt 99.5%

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

   5. associate-/r* 99.6%

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

   6. fma-udef 99.6%

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

   7. +-commutative 99.6%

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

   8. add-sqr-sqrt 99.6%

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

   9. pow2 99.6%

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

   10. hypot-def 99.6%

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

   11. pow1/2 99.6%

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

   12. sqrt-pow1 99.6%

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

   13. metadata-eval 99.6%

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

   14. pow1/2 99.6%

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

   15. sqrt-pow1 99.6%

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

   16. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{0.25}} \]

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (/ (/ (cos th) (pow 2.0 0.25)) (pow 2.0 0.25)) (+ (* a2 a2) (* a1 a1))))

C

double code(double a1, double a2, double th) {
	return ((cos(th) / pow(2.0, 0.25)) / pow(2.0, 0.25)) * ((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) / (2.0d0 ** 0.25d0)) / (2.0d0 ** 0.25d0)) * ((a2 * a2) + (a1 * a1))
end function

Java

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) / Math.pow(2.0, 0.25)) / Math.pow(2.0, 0.25)) * ((a2 * a2) + (a1 * a1));
}

Python

def code(a1, a2, th):
	return ((math.cos(th) / math.pow(2.0, 0.25)) / math.pow(2.0, 0.25)) * ((a2 * a2) + (a1 * a1))

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * Float64(Float64(a2 * a2) + Float64(a1 * a1)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * ((a2 * a2) + (a1 * a1));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.2%

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

   4. pow1/2 99.2%

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

   5. sqrt-pow1 99.2%

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

   6. metadata-eval 99.2%

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

   7. pow1/2 99.2%

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

   8. sqrt-pow1 99.2%

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

   9. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 55.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * ((a2 * a2) + (a1 * a1));
	} 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 = cos(th) * ((a2 * a2) + (a1 * a1))
    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 = Math.cos(th) * ((a2 * a2) + (a1 * a1));
	} 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 = math.cos(th) * ((a2 * a2) + (a1 * a1))
	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(cos(th) * Float64(Float64(a2 * a2) + Float64(a1 * a1)));
	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 = cos(th) * ((a2 * a2) + (a1 * a1));
	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[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around inf 99.6%

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

   8. Applied egg-rr 62.4%

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

      1. rem-log-exp 62.4%

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

   10. Simplified 62.4%

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

   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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 93.4%

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

   5. Taylor expanded in a2 around inf 58.2%

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

      1. unpow2 58.2%

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

      2. associate-/l* 58.2%

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

      3. associate-/r/ 58.2%

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

   7. Simplified 58.2%

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

4. Final simplification 60.0%

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

Alternative 4: 79.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a2 a2) (* a1 a1))))
   (if (<= (cos th) 0.7) (* (cos th) t_1) (* t_1 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * t_1;
	} else {
		tmp = t_1 * 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) :: t_1
    real(8) :: tmp
    t_1 = (a2 * a2) + (a1 * a1)
    if (cos(th) <= 0.7d0) then
        tmp = cos(th) * t_1
    else
        tmp = t_1 * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * t_1;
	} else {
		tmp = t_1 * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) + (a1 * a1)
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * t_1
	else:
		tmp = t_1 * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) + Float64(a1 * a1))
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * t_1);
	else
		tmp = Float64(t_1 * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) + (a1 * a1);
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * t_1;
	else
		tmp = t_1 * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around inf 99.6%

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

   8. Applied egg-rr 62.4%

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

      1. rem-log-exp 62.4%

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

   10. Simplified 62.4%

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

   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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 93.4%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
   5. Step-by-step derivation

      1. expm1-log1p-u 93.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      2. expm1-udef 93.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)} - 1\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      3. add-sqr-sqrt 93.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      4. sqrt-unprod 93.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      5. frac-times 93.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      6. metadata-eval 93.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      7. add-sqr-sqrt 92.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{2}}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      8. metadata-eval 92.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5}}\right)} - 1\right) \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

   6. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
   7. Step-by-step derivation

      1. expm1-def 92.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]

      2. expm1-log1p 93.5%

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

   8. Simplified 93.5%

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

4. Final simplification 80.4%

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

Alternative 5: 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 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 56.8% 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 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a2 around inf 56.7%

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

   1. unpow2 56.7%

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

   2. associate-*l/ 56.7%

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

   3. associate-/l* 56.7%

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

   4. associate-/r/ 56.7%

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

6. Simplified 56.7%

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

   1. associate-*l/ 56.7%

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

   2. *-commutative 56.7%

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

   3. associate-/l* 56.7%

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

   4. associate-*r/ 56.7%

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

8. Applied egg-rr 56.7%

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

   1. div-inv 56.6%

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

   2. *-commutative 56.6%

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

   3. clear-num 56.7%

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

   4. associate-*l* 56.7%

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

   5. clear-num 56.7%

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

   6. associate-/r/ 56.6%

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

   7. pow1/2 56.6%

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

   8. pow-flip 56.7%

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

   9. metadata-eval 56.7%

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

   10. add-sqr-sqrt 56.3%

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

   11. sqrt-unprod 56.7%

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

   12. pow-prod-up 56.7%

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

   13. metadata-eval 56.7%

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

   14. metadata-eval 56.7%

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

10. Applied egg-rr 56.7%

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

11. Final simplification 56.7%

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

Alternative 7: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * N[(N[Cos[th], $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. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

7. Taylor expanded in a2 around inf 56.7%

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

   1. unpow2 56.7%

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

   2. associate-*l* 56.7%

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

9. Simplified 56.7%

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

10. Final simplification 56.7%

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

Alternative 8: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in a2 around inf 56.7%

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

   1. unpow2 56.7%

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

   2. *-commutative 56.7%

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

8. Simplified 56.7%

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

9. Final simplification 56.7%

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

Alternative 9: 39.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8)
   (* (* a2 a2) (sqrt 0.5))
   (if (or (<= th 1.65e+209) (not (<= th 2.65e+230)))
     (* a2 (* a2 (- 0.5)))
     (* a2 (* a2 (- -0.25))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = (a2 * a2) * sqrt(0.5);
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 7.8d0) then
        tmp = (a2 * a2) * sqrt(0.5d0)
    else if ((th <= 1.65d+209) .or. (.not. (th <= 2.65d+230))) then
        tmp = a2 * (a2 * -0.5d0)
    else
        tmp = a2 * (a2 * -(-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8:
		tmp = (a2 * a2) * math.sqrt(0.5)
	elif (th <= 1.65e+209) or not (th <= 2.65e+230):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = a2 * (a2 * -(-0.25))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8)
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	elseif ((th <= 1.65e+209) || !(th <= 2.65e+230))
		tmp = Float64(a2 * Float64(a2 * Float64(-0.5)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8)
		tmp = (a2 * a2) * sqrt(0.5);
	elseif ((th <= 1.65e+209) || ~((th <= 2.65e+230)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = a2 * (a2 * -(-0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 1.65e+209], N[Not[LessEqual[th, 2.65e+230]], $MachinePrecision]], N[(a2 * N[(a2 * (-0.5)), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * (--0.25)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.79999999999999982

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 73.5%

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

   5. Taylor expanded in a2 around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-/l* 46.4%

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

      3. associate-/r/ 46.5%

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

   7. Simplified 46.5%

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

      1. associate-*l/ 46.4%

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

      2. expm1-log1p-u 45.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a2 \cdot a2}{\sqrt{2}}\right)\right)} \]

      3. expm1-udef 38.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a2 \cdot a2}{\sqrt{2}}\right)} - 1} \]

      4. *-un-lft-identity 38.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(a2 \cdot a2\right)}}{\sqrt{2}}\right)} - 1 \]

      5. associate-*l/ 38.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)}\right)} - 1 \]

      6. add-sqr-sqrt 38.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}\right)} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

      7. sqrt-unprod 38.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

      8. frac-times 38.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

      9. metadata-eval 38.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

      10. add-sqr-sqrt 38.6%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{2}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

      11. metadata-eval 38.6%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

   9. Applied egg-rr 38.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 45.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\right)} \]

       2. expm1-log1p 46.4%

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

   11. Simplified 46.4%

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

   if 7.79999999999999982 < th < 1.6499999999999999e209 or 2.65000000000000017e230 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 26.0%

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

   5. Taylor expanded in a2 around inf 11.6%

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

      1. unpow2 11.6%

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

      2. associate-/l* 11.6%

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

      3. associate-/r/ 11.6%

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

   7. Simplified 11.6%

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

      1. frac-2neg 11.6%

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

      2. div-inv 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative 11.6%

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

   11. Simplified 11.6%

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

   13. Applied egg-rr 21.4%

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

   if 1.6499999999999999e209 < th < 2.65000000000000017e230

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 80.0%

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

   5. Taylor expanded in a2 around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

      1. frac-2neg 80.0%

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

      2. div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

   11. Simplified 80.0%

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

   13. Applied egg-rr 80.0%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \]

Alternative 10: 39.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8)
   (* a2 (/ a2 (sqrt 2.0)))
   (if (or (<= th 1.65e+209) (not (<= th 2.65e+230)))
     (* a2 (* a2 (- 0.5)))
     (* a2 (* a2 (- -0.25))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * (a2 / sqrt(2.0));
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 7.8d0) then
        tmp = a2 * (a2 / sqrt(2.0d0))
    else if ((th <= 1.65d+209) .or. (.not. (th <= 2.65d+230))) then
        tmp = a2 * (a2 * -0.5d0)
    else
        tmp = a2 * (a2 * -(-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8:
		tmp = a2 * (a2 / math.sqrt(2.0))
	elif (th <= 1.65e+209) or not (th <= 2.65e+230):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = a2 * (a2 * -(-0.25))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8)
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	elseif ((th <= 1.65e+209) || !(th <= 2.65e+230))
		tmp = Float64(a2 * Float64(a2 * Float64(-0.5)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8)
		tmp = a2 * (a2 / sqrt(2.0));
	elseif ((th <= 1.65e+209) || ~((th <= 2.65e+230)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = a2 * (a2 * -(-0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 1.65e+209], N[Not[LessEqual[th, 2.65e+230]], $MachinePrecision]], N[(a2 * N[(a2 * (-0.5)), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * (--0.25)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.79999999999999982

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 73.5%

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

   5. Taylor expanded in a2 around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-/l* 46.4%

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

      3. associate-/r/ 46.5%

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

   7. Simplified 46.5%

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

   if 7.79999999999999982 < th < 1.6499999999999999e209 or 2.65000000000000017e230 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 26.0%

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

   5. Taylor expanded in a2 around inf 11.6%

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

      1. unpow2 11.6%

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

      2. associate-/l* 11.6%

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

      3. associate-/r/ 11.6%

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

   7. Simplified 11.6%

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

      1. frac-2neg 11.6%

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

      2. div-inv 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative 11.6%

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

   11. Simplified 11.6%

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

   13. Applied egg-rr 21.4%

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

   if 1.6499999999999999e209 < th < 2.65000000000000017e230

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 80.0%

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

   5. Taylor expanded in a2 around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

      1. frac-2neg 80.0%

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

      2. div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

   11. Simplified 80.0%

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

   13. Applied egg-rr 80.0%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \]

Alternative 11: 29.8% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.015625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8)
   (* a2 a2)
   (if (or (<= th 1.65e+209) (not (<= th 2.65e+230)))
     (* a2 (* a2 (- 0.015625)))
     (* a2 (* a2 (- -0.25))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.015625);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 7.8d0) then
        tmp = a2 * a2
    else if ((th <= 1.65d+209) .or. (.not. (th <= 2.65d+230))) then
        tmp = a2 * (a2 * -0.015625d0)
    else
        tmp = a2 * (a2 * -(-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.015625);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8:
		tmp = a2 * a2
	elif (th <= 1.65e+209) or not (th <= 2.65e+230):
		tmp = a2 * (a2 * -0.015625)
	else:
		tmp = a2 * (a2 * -(-0.25))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8)
		tmp = Float64(a2 * a2);
	elseif ((th <= 1.65e+209) || !(th <= 2.65e+230))
		tmp = Float64(a2 * Float64(a2 * Float64(-0.015625)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8)
		tmp = a2 * a2;
	elseif ((th <= 1.65e+209) || ~((th <= 2.65e+230)))
		tmp = a2 * (a2 * -0.015625);
	else
		tmp = a2 * (a2 * -(-0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 1.65e+209], N[Not[LessEqual[th, 2.65e+230]], $MachinePrecision]], N[(a2 * N[(a2 * (-0.015625)), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * (--0.25)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.79999999999999982

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 73.5%

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

   5. Taylor expanded in a2 around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-/l* 46.4%

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

      3. associate-/r/ 46.5%

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

   7. Simplified 46.5%

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

      1. frac-2neg 46.5%

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

      2. div-inv 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. *-commutative 46.4%

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

   11. Simplified 46.4%

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

   13. Applied egg-rr 28.2%

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

   if 7.79999999999999982 < th < 1.6499999999999999e209 or 2.65000000000000017e230 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 26.0%

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

   5. Taylor expanded in a2 around inf 11.6%

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

      1. unpow2 11.6%

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

      2. associate-/l* 11.6%

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

      3. associate-/r/ 11.6%

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

   7. Simplified 11.6%

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

      1. frac-2neg 11.6%

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

      2. div-inv 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative 11.6%

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

   11. Simplified 11.6%

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

   13. Applied egg-rr 20.7%

       \[\leadsto \left(\color{blue}{0.015625} \cdot \left(-a2\right)\right) \cdot a2 \]

   if 1.6499999999999999e209 < th < 2.65000000000000017e230

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 80.0%

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

   5. Taylor expanded in a2 around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

      1. frac-2neg 80.0%

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

      2. div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

   11. Simplified 80.0%

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

   13. Applied egg-rr 80.0%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.015625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \]

Alternative 12: 29.8% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8)
   (* a2 a2)
   (if (or (<= th 1.65e+209) (not (<= th 2.65e+230)))
     (* a2 (* a2 (- 0.0625)))
     (* a2 (* a2 (- -0.25))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.0625);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 7.8d0) then
        tmp = a2 * a2
    else if ((th <= 1.65d+209) .or. (.not. (th <= 2.65d+230))) then
        tmp = a2 * (a2 * -0.0625d0)
    else
        tmp = a2 * (a2 * -(-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.0625);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8:
		tmp = a2 * a2
	elif (th <= 1.65e+209) or not (th <= 2.65e+230):
		tmp = a2 * (a2 * -0.0625)
	else:
		tmp = a2 * (a2 * -(-0.25))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8)
		tmp = Float64(a2 * a2);
	elseif ((th <= 1.65e+209) || !(th <= 2.65e+230))
		tmp = Float64(a2 * Float64(a2 * Float64(-0.0625)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8)
		tmp = a2 * a2;
	elseif ((th <= 1.65e+209) || ~((th <= 2.65e+230)))
		tmp = a2 * (a2 * -0.0625);
	else
		tmp = a2 * (a2 * -(-0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 1.65e+209], N[Not[LessEqual[th, 2.65e+230]], $MachinePrecision]], N[(a2 * N[(a2 * (-0.0625)), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * (--0.25)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.79999999999999982

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 73.5%

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

   5. Taylor expanded in a2 around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-/l* 46.4%

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

      3. associate-/r/ 46.5%

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

   7. Simplified 46.5%

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

      1. frac-2neg 46.5%

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

      2. div-inv 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. *-commutative 46.4%

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

   11. Simplified 46.4%

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

   13. Applied egg-rr 28.2%

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

   if 7.79999999999999982 < th < 1.6499999999999999e209 or 2.65000000000000017e230 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 26.0%

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

   5. Taylor expanded in a2 around inf 11.6%

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

      1. unpow2 11.6%

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

      2. associate-/l* 11.6%

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

      3. associate-/r/ 11.6%

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

   7. Simplified 11.6%

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

      1. frac-2neg 11.6%

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

      2. div-inv 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative 11.6%

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

   11. Simplified 11.6%

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

   13. Applied egg-rr 20.9%

       \[\leadsto \left(\color{blue}{0.0625} \cdot \left(-a2\right)\right) \cdot a2 \]

   if 1.6499999999999999e209 < th < 2.65000000000000017e230

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 80.0%

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

   5. Taylor expanded in a2 around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

      1. frac-2neg 80.0%

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

      2. div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

   11. Simplified 80.0%

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

   13. Applied egg-rr 80.0%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \]

Alternative 13: 29.8% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;-a2 \cdot \left(a2 \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8)
   (* a2 a2)
   (if (or (<= th 1.65e+209) (not (<= th 2.65e+230)))
     (- (* a2 (* a2 0.125)))
     (* a2 (* a2 (- -0.25))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = -(a2 * (a2 * 0.125));
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 7.8d0) then
        tmp = a2 * a2
    else if ((th <= 1.65d+209) .or. (.not. (th <= 2.65d+230))) then
        tmp = -(a2 * (a2 * 0.125d0))
    else
        tmp = a2 * (a2 * -(-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = -(a2 * (a2 * 0.125));
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8:
		tmp = a2 * a2
	elif (th <= 1.65e+209) or not (th <= 2.65e+230):
		tmp = -(a2 * (a2 * 0.125))
	else:
		tmp = a2 * (a2 * -(-0.25))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8)
		tmp = Float64(a2 * a2);
	elseif ((th <= 1.65e+209) || !(th <= 2.65e+230))
		tmp = Float64(-Float64(a2 * Float64(a2 * 0.125)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8)
		tmp = a2 * a2;
	elseif ((th <= 1.65e+209) || ~((th <= 2.65e+230)))
		tmp = -(a2 * (a2 * 0.125));
	else
		tmp = a2 * (a2 * -(-0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 1.65e+209], N[Not[LessEqual[th, 2.65e+230]], $MachinePrecision]], (-N[(a2 * N[(a2 * 0.125), $MachinePrecision]), $MachinePrecision]), N[(a2 * N[(a2 * (--0.25)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.79999999999999982

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 73.5%

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

   5. Taylor expanded in a2 around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-/l* 46.4%

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

      3. associate-/r/ 46.5%

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

   7. Simplified 46.5%

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

      1. frac-2neg 46.5%

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

      2. div-inv 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. *-commutative 46.4%

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

   11. Simplified 46.4%

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

   13. Applied egg-rr 28.2%

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

   if 7.79999999999999982 < th < 1.6499999999999999e209 or 2.65000000000000017e230 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 26.0%

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

   5. Taylor expanded in a2 around inf 11.6%

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

      1. unpow2 11.6%

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

      2. associate-/l* 11.6%

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

      3. associate-/r/ 11.6%

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

   7. Simplified 11.6%

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

      1. frac-2neg 11.6%

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

      2. div-inv 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative 11.6%

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

   11. Simplified 11.6%

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

   13. Applied egg-rr 21.0%

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

   if 1.6499999999999999e209 < th < 2.65000000000000017e230

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 80.0%

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

   5. Taylor expanded in a2 around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

      1. frac-2neg 80.0%

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

      2. div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

   11. Simplified 80.0%

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

   13. Applied egg-rr 80.0%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;-a2 \cdot \left(a2 \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \]

Alternative 14: 29.9% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8)
   (* a2 a2)
   (if (or (<= th 1.65e+209) (not (<= th 2.65e+230)))
     (* a2 (* a2 (- 0.25)))
     (* a2 (* a2 (- -0.25))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.25);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 7.8d0) then
        tmp = a2 * a2
    else if ((th <= 1.65d+209) .or. (.not. (th <= 2.65d+230))) then
        tmp = a2 * (a2 * -0.25d0)
    else
        tmp = a2 * (a2 * -(-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.25);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8:
		tmp = a2 * a2
	elif (th <= 1.65e+209) or not (th <= 2.65e+230):
		tmp = a2 * (a2 * -0.25)
	else:
		tmp = a2 * (a2 * -(-0.25))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8)
		tmp = Float64(a2 * a2);
	elseif ((th <= 1.65e+209) || !(th <= 2.65e+230))
		tmp = Float64(a2 * Float64(a2 * Float64(-0.25)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8)
		tmp = a2 * a2;
	elseif ((th <= 1.65e+209) || ~((th <= 2.65e+230)))
		tmp = a2 * (a2 * -0.25);
	else
		tmp = a2 * (a2 * -(-0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 1.65e+209], N[Not[LessEqual[th, 2.65e+230]], $MachinePrecision]], N[(a2 * N[(a2 * (-0.25)), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * (--0.25)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.79999999999999982

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 73.5%

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

   5. Taylor expanded in a2 around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-/l* 46.4%

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

      3. associate-/r/ 46.5%

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

   7. Simplified 46.5%

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

      1. frac-2neg 46.5%

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

      2. div-inv 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. *-commutative 46.4%

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

   11. Simplified 46.4%

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

   13. Applied egg-rr 28.2%

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

   if 7.79999999999999982 < th < 1.6499999999999999e209 or 2.65000000000000017e230 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 26.0%

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

   5. Taylor expanded in a2 around inf 11.6%

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

      1. unpow2 11.6%

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

      2. associate-/l* 11.6%

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

      3. associate-/r/ 11.6%

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

   7. Simplified 11.6%

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

      1. frac-2neg 11.6%

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

      2. div-inv 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative 11.6%

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

   11. Simplified 11.6%

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

   13. Applied egg-rr 21.1%

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

   if 1.6499999999999999e209 < th < 2.65000000000000017e230

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 80.0%

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

   5. Taylor expanded in a2 around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

      1. frac-2neg 80.0%

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

      2. div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

   11. Simplified 80.0%

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

   13. Applied egg-rr 80.0%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \]

Alternative 15: 29.9% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8)
   (* a2 a2)
   (if (or (<= th 1.65e+209) (not (<= th 2.65e+230)))
     (* a2 (* a2 (- 0.5)))
     (* a2 (* a2 (- -0.25))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 7.8d0) then
        tmp = a2 * a2
    else if ((th <= 1.65d+209) .or. (.not. (th <= 2.65d+230))) then
        tmp = a2 * (a2 * -0.5d0)
    else
        tmp = a2 * (a2 * -(-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8) {
		tmp = a2 * a2;
	} else if ((th <= 1.65e+209) || !(th <= 2.65e+230)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 * (a2 * -(-0.25));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8:
		tmp = a2 * a2
	elif (th <= 1.65e+209) or not (th <= 2.65e+230):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = a2 * (a2 * -(-0.25))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8)
		tmp = Float64(a2 * a2);
	elseif ((th <= 1.65e+209) || !(th <= 2.65e+230))
		tmp = Float64(a2 * Float64(a2 * Float64(-0.5)));
	else
		tmp = Float64(a2 * Float64(a2 * Float64(-(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8)
		tmp = a2 * a2;
	elseif ((th <= 1.65e+209) || ~((th <= 2.65e+230)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = a2 * (a2 * -(-0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 1.65e+209], N[Not[LessEqual[th, 2.65e+230]], $MachinePrecision]], N[(a2 * N[(a2 * (-0.5)), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * (--0.25)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 7.79999999999999982

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 73.5%

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

   5. Taylor expanded in a2 around inf 46.4%

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

      1. unpow2 46.4%

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

      2. associate-/l* 46.4%

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

      3. associate-/r/ 46.5%

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

   7. Simplified 46.5%

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

      1. frac-2neg 46.5%

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

      2. div-inv 46.4%

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

   9. Applied egg-rr 46.4%

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

       1. *-commutative 46.4%

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

   11. Simplified 46.4%

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

   13. Applied egg-rr 28.2%

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

   if 7.79999999999999982 < th < 1.6499999999999999e209 or 2.65000000000000017e230 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 26.0%

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

   5. Taylor expanded in a2 around inf 11.6%

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

      1. unpow2 11.6%

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

      2. associate-/l* 11.6%

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

      3. associate-/r/ 11.6%

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

   7. Simplified 11.6%

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

      1. frac-2neg 11.6%

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

      2. div-inv 11.6%

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

   9. Applied egg-rr 11.6%

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

       1. *-commutative 11.6%

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

   11. Simplified 11.6%

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

   13. Applied egg-rr 21.4%

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

   if 1.6499999999999999e209 < th < 2.65000000000000017e230

   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. +-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 80.0%

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

   5. Taylor expanded in a2 around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-/l* 80.0%

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

      3. associate-/r/ 80.0%

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

   7. Simplified 80.0%

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

      1. frac-2neg 80.0%

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

      2. div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

       1. *-commutative 80.0%

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

   11. Simplified 80.0%

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

   13. Applied egg-rr 80.0%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.65 \cdot 10^{+209} \lor \neg \left(th \leq 2.65 \cdot 10^{+230}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(-0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \left(--0.25\right)\right)\\ \end{array} \]

Alternative 16: 29.7% accurate, 69.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around 0 63.1%

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

5. Taylor expanded in a2 around inf 39.3%

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

   1. unpow2 39.3%

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

   2. associate-/l* 39.3%

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

   3. associate-/r/ 39.3%

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

7. Simplified 39.3%

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

   1. frac-2neg 39.3%

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

   2. div-inv 39.3%

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

9. Applied egg-rr 39.3%

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

    1. *-commutative 39.3%

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

11. Simplified 39.3%

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

13. Applied egg-rr 25.6%

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

14. Final simplification 25.6%

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

Alternative 17: 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. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around 0 63.1%

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

5. Taylor expanded in a2 around inf 39.3%

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

   1. unpow2 39.3%

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

   2. associate-/l* 39.3%

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

   3. associate-/r/ 39.3%

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

7. Simplified 39.3%

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

   1. frac-2neg 39.3%

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

   2. div-inv 39.3%

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

9. Applied egg-rr 39.3%

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

    1. *-commutative 39.3%

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

11. Simplified 39.3%

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

13. Applied egg-rr 25.5%

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

14. Final simplification 25.5%

    \[\leadsto a2 \cdot a2 \]

Alternative 18: 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. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.4%

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

   3. pow1/2 99.4%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

8. Applied egg-rr 3.7%

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

   1. *-inverses 3.7%

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

10. Simplified 3.7%

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

11. Final simplification 3.7%

    \[\leadsto 1 \]

Alternative 19: 3.5% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a2

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. +-commutative 99.5%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around 0 63.1%

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

5. Taylor expanded in a2 around inf 39.3%

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

   1. unpow2 39.3%

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

   2. associate-/l* 39.3%

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

   3. associate-/r/ 39.3%

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

7. Simplified 39.3%

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

   1. associate-*l/ 39.3%

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

   2. expm1-log1p-u 38.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a2 \cdot a2}{\sqrt{2}}\right)\right)} \]

   3. expm1-udef 33.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a2 \cdot a2}{\sqrt{2}}\right)} - 1} \]

   4. *-un-lft-identity 33.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(a2 \cdot a2\right)}}{\sqrt{2}}\right)} - 1 \]

   5. associate-*l/ 33.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)}\right)} - 1 \]

   6. add-sqr-sqrt 33.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}\right)} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

   7. sqrt-unprod 33.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

   8. frac-times 33.2%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

   9. metadata-eval 33.2%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

   10. add-sqr-sqrt 33.2%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{2}}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

   11. metadata-eval 33.2%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5}} \cdot \left(a2 \cdot a2\right)\right)} - 1 \]

9. Applied egg-rr 33.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 38.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\right)} \]

    2. expm1-log1p 39.3%

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

11. Simplified 39.3%

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

13. Applied egg-rr 3.8%

    \[\leadsto \color{blue}{0 + a2} \]
14. Step-by-step derivation

    1. +-lft-identity 3.8%

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

15. Simplified 3.8%

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

16. Final simplification 3.8%

    \[\leadsto a2 \]

Reproduce

herbie shell --seed 2023275 
(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))))