Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 13.5s

Alternatives: 23

Speedup: 2.0×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 79.7% 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 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. add-sqr-sqrt 99.5%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

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

      5. sqrt-pow1 99.5%

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

      6. metadata-eval 99.5%

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

      7. pow1/2 99.5%

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

      8. sqrt-pow1 99.5%

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

      9. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around inf 99.6%

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

   8. Applied egg-rr 55.1%

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

      1. rem-log-exp 55.1%

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

   10. Simplified 55.1%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. add-sqr-sqrt 99.5%

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

      3. times-frac 99.1%

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

      4. pow1/2 99.1%

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

      5. sqrt-pow1 99.1%

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

      6. metadata-eval 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.1%

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

      9. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in th around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. unpow2 92.0%

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

      3. unpow2 92.0%

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

   8. Simplified 92.0%

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

4. Final simplification 77.1%

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

Alternative 3: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sqrt{2}}\\ \end{array} \end{array} \]

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 2.0)))))

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(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) :: 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(2.0d0)
    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(2.0);
	}
	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(2.0)
	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(2.0));
	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(2.0);
	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[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. add-sqr-sqrt 99.5%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

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

      5. sqrt-pow1 99.5%

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

      6. metadata-eval 99.5%

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

      7. pow1/2 99.5%

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

      8. sqrt-pow1 99.5%

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

      9. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around inf 99.6%

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

   8. Applied egg-rr 55.1%

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

      1. rem-log-exp 55.1%

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

   10. Simplified 55.1%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around inf 99.7%

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

      1. associate-/l* 99.7%

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

      2. unpow2 99.7%

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

      3. unpow2 99.7%

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

      4. fma-def 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in th around 0 92.1%

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

      1. unpow2 92.1%

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

      2. unpow2 92.1%

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

   9. Simplified 92.1%

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

4. Final simplification 77.2%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.5%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow1/2 99.3%

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

   8. sqrt-pow1 99.3%

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

   9. metadata-eval 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in th around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*r/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in th around inf 99.7%

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

   1. associate-/l* 99.6%

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

   2. unpow2 99.6%

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

   3. unpow2 99.6%

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

   4. fma-def 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in th around inf 99.7%

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

   1. associate-/l* 99.6%

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

   2. unpow2 99.6%

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

   3. unpow2 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 7: 69.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;a2 \leq 9 \cdot 10^{-46}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{elif}\;a2 \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a2 a2) (* a1 a1))))
   (if (<= a2 9e-46)
     (* a1 (* a1 (* (cos th) (sqrt 0.5))))
     (if (<= a2 1.35e+154) (/ t_1 (sqrt 2.0)) (* (cos th) t_1)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if (a2 <= 9e-46) {
		tmp = a1 * (a1 * (cos(th) * sqrt(0.5)));
	} else if (a2 <= 1.35e+154) {
		tmp = t_1 / sqrt(2.0);
	} else {
		tmp = cos(th) * t_1;
	}
	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 (a2 <= 9d-46) then
        tmp = a1 * (a1 * (cos(th) * sqrt(0.5d0)))
    else if (a2 <= 1.35d+154) then
        tmp = t_1 / sqrt(2.0d0)
    else
        tmp = cos(th) * t_1
    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 (a2 <= 9e-46) {
		tmp = a1 * (a1 * (Math.cos(th) * Math.sqrt(0.5)));
	} else if (a2 <= 1.35e+154) {
		tmp = t_1 / Math.sqrt(2.0);
	} else {
		tmp = Math.cos(th) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) + (a1 * a1)
	tmp = 0
	if a2 <= 9e-46:
		tmp = a1 * (a1 * (math.cos(th) * math.sqrt(0.5)))
	elif a2 <= 1.35e+154:
		tmp = t_1 / math.sqrt(2.0)
	else:
		tmp = math.cos(th) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) + Float64(a1 * a1))
	tmp = 0.0
	if (a2 <= 9e-46)
		tmp = Float64(a1 * Float64(a1 * Float64(cos(th) * sqrt(0.5))));
	elseif (a2 <= 1.35e+154)
		tmp = Float64(t_1 / sqrt(2.0));
	else
		tmp = Float64(cos(th) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) + (a1 * a1);
	tmp = 0.0;
	if (a2 <= 9e-46)
		tmp = a1 * (a1 * (cos(th) * sqrt(0.5)));
	elseif (a2 <= 1.35e+154)
		tmp = t_1 / sqrt(2.0);
	else
		tmp = cos(th) * t_1;
	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[a2, 9e-46], N[(a1 * N[(a1 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1.35e+154], N[(t$95$1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 9.00000000000000001e-46

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. add-sqr-sqrt 99.5%

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

      3. times-frac 99.2%

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

      4. pow1/2 99.2%

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

      5. sqrt-pow1 99.2%

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

      6. metadata-eval 99.2%

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

      7. pow1/2 99.2%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in a1 around inf 68.4%

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

      1. *-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. associate-*r* 68.4%

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

      4. associate-*r* 68.4%

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

   8. Simplified 68.4%

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

   if 9.00000000000000001e-46 < a2 < 1.35000000000000003e154

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around inf 99.6%

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

      1. associate-/l* 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. fma-def 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in th around 0 56.2%

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

      1. unpow2 56.2%

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

      2. unpow2 56.2%

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

   9. Simplified 56.2%

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

   if 1.35000000000000003e154 < a2

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. times-frac 100.0%

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

      4. pow1/2 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1/2 100.0%

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

      8. sqrt-pow1 100.0%

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

      9. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in th around inf 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. rem-log-exp 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 9 \cdot 10^{-46}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(\cos th \cdot \sqrt{0.5}\right)\right)\\ \mathbf{elif}\;a2 \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 8: 69.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;a2 \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a2 a2) (* a1 a1))))
   (if (<= a2 1.3e-45)
     (* a1 (* (sqrt 0.5) (* (cos th) a1)))
     (if (<= a2 1.15e+154) (/ t_1 (sqrt 2.0)) (* (cos th) t_1)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if (a2 <= 1.3e-45) {
		tmp = a1 * (sqrt(0.5) * (cos(th) * a1));
	} else if (a2 <= 1.15e+154) {
		tmp = t_1 / sqrt(2.0);
	} else {
		tmp = cos(th) * t_1;
	}
	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 (a2 <= 1.3d-45) then
        tmp = a1 * (sqrt(0.5d0) * (cos(th) * a1))
    else if (a2 <= 1.15d+154) then
        tmp = t_1 / sqrt(2.0d0)
    else
        tmp = cos(th) * t_1
    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 (a2 <= 1.3e-45) {
		tmp = a1 * (Math.sqrt(0.5) * (Math.cos(th) * a1));
	} else if (a2 <= 1.15e+154) {
		tmp = t_1 / Math.sqrt(2.0);
	} else {
		tmp = Math.cos(th) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) + (a1 * a1)
	tmp = 0
	if a2 <= 1.3e-45:
		tmp = a1 * (math.sqrt(0.5) * (math.cos(th) * a1))
	elif a2 <= 1.15e+154:
		tmp = t_1 / math.sqrt(2.0)
	else:
		tmp = math.cos(th) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) + Float64(a1 * a1))
	tmp = 0.0
	if (a2 <= 1.3e-45)
		tmp = Float64(a1 * Float64(sqrt(0.5) * Float64(cos(th) * a1)));
	elseif (a2 <= 1.15e+154)
		tmp = Float64(t_1 / sqrt(2.0));
	else
		tmp = Float64(cos(th) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) + (a1 * a1);
	tmp = 0.0;
	if (a2 <= 1.3e-45)
		tmp = a1 * (sqrt(0.5) * (cos(th) * a1));
	elseif (a2 <= 1.15e+154)
		tmp = t_1 / sqrt(2.0);
	else
		tmp = cos(th) * t_1;
	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[a2, 1.3e-45], N[(a1 * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a2, 1.15e+154], N[(t$95$1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 1.29999999999999993e-45

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around inf 68.4%

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

      1. unpow2 68.4%

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

   6. Simplified 68.4%

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

      1. expm1-log1p-u 66.7%

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

      2. expm1-udef 55.2%

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

      3. div-inv 55.2%

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

      4. add-sqr-sqrt 55.2%

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

      5. sqrt-unprod 55.2%

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

      6. frac-times 55.2%

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

      7. metadata-eval 55.2%

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

      8. add-sqr-sqrt 55.2%

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

      9. metadata-eval 55.2%

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

   8. Applied egg-rr 55.2%

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

      1. expm1-def 66.7%

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

      2. expm1-log1p 68.4%

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

      3. associate-*r* 68.3%

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

   10. Simplified 68.3%

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

   11. Taylor expanded in th around inf 68.4%

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

       1. unpow2 68.4%

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

       2. associate-*l* 68.4%

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

       3. associate-*r* 68.4%

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

       4. *-commutative 68.4%

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

       5. associate-*l* 68.4%

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

   13. Simplified 68.4%

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

   if 1.29999999999999993e-45 < a2 < 1.15e154

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around inf 99.6%

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

      1. associate-/l* 99.5%

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

      2. unpow2 99.5%

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

      3. unpow2 99.5%

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

      4. fma-def 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in th around 0 56.2%

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

      1. unpow2 56.2%

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

      2. unpow2 56.2%

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

   9. Simplified 56.2%

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

   if 1.15e154 < a2

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. times-frac 100.0%

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

      4. pow1/2 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1/2 100.0%

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

      8. sqrt-pow1 100.0%

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

      9. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in th around inf 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. rem-log-exp 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{elif}\;a2 \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 9: 69.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -1.28 \cdot 10^{-71}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -1.28e-71)
   (* a1 (* (sqrt 0.5) (* (cos th) a1)))
   (* a2 (/ a2 (/ (sqrt 2.0) (cos th))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.28e-71) {
		tmp = a1 * (sqrt(0.5) * (cos(th) * a1));
	} else {
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a1 <= (-1.28d-71)) then
        tmp = a1 * (sqrt(0.5d0) * (cos(th) * a1))
    else
        tmp = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.28e-71) {
		tmp = a1 * (Math.sqrt(0.5) * (Math.cos(th) * a1));
	} else {
		tmp = a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -1.28e-71:
		tmp = a1 * (math.sqrt(0.5) * (math.cos(th) * a1))
	else:
		tmp = a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -1.28e-71)
		tmp = Float64(a1 * Float64(sqrt(0.5) * Float64(cos(th) * a1)));
	else
		tmp = Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -1.28e-71)
		tmp = a1 * (sqrt(0.5) * (cos(th) * a1));
	else
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -1.28e-71], N[(a1 * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a1 < -1.28e-71

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around inf 69.4%

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

      1. unpow2 69.4%

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

   6. Simplified 69.4%

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

      1. expm1-log1p-u 67.3%

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

      2. expm1-udef 58.6%

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

      3. div-inv 58.6%

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

      4. add-sqr-sqrt 58.6%

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

      5. sqrt-unprod 58.6%

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

      6. frac-times 58.6%

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

      7. metadata-eval 58.6%

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

      8. add-sqr-sqrt 58.6%

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

      9. metadata-eval 58.6%

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

   8. Applied egg-rr 58.6%

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

      1. expm1-def 67.3%

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

      2. expm1-log1p 69.4%

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

      3. associate-*r* 69.4%

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

   10. Simplified 69.4%

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

   11. Taylor expanded in th around inf 69.4%

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

       1. unpow2 69.4%

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

       2. associate-*l* 69.4%

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

       3. associate-*r* 69.3%

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

       4. *-commutative 69.3%

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

       5. associate-*l* 69.3%

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

   13. Simplified 69.3%

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

   if -1.28e-71 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 64.0%

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

      1. unpow2 64.0%

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

      2. associate-*l* 64.0%

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

      3. associate-*r/ 64.0%

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

      4. associate-/l* 63.9%

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

   6. Simplified 63.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -1.28 \cdot 10^{-71}:\\ \;\;\;\;a1 \cdot \left(\sqrt{0.5} \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 10: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -9.5 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -9.5e-80)
   (* (cos th) (/ (* a1 a1) (sqrt 2.0)))
   (* a2 (/ a2 (/ (sqrt 2.0) (cos th))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -9.5e-80) {
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	} else {
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a1 <= (-9.5d-80)) then
        tmp = cos(th) * ((a1 * a1) / sqrt(2.0d0))
    else
        tmp = a2 * (a2 / (sqrt(2.0d0) / cos(th)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -9.5e-80) {
		tmp = Math.cos(th) * ((a1 * a1) / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 / (Math.sqrt(2.0) / Math.cos(th)));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -9.5e-80:
		tmp = math.cos(th) * ((a1 * a1) / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 / (math.sqrt(2.0) / math.cos(th)))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -9.5e-80)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) / sqrt(2.0)));
	else
		tmp = Float64(a2 * Float64(a2 / Float64(sqrt(2.0) / cos(th))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -9.5e-80)
		tmp = cos(th) * ((a1 * a1) / sqrt(2.0));
	else
		tmp = a2 * (a2 / (sqrt(2.0) / cos(th)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -9.5e-80], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a1 < -9.5000000000000003e-80

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around inf 68.9%

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

      1. unpow2 68.9%

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

   6. Simplified 68.9%

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

   if -9.5000000000000003e-80 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 64.1%

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

      1. unpow2 64.1%

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

      2. associate-*l* 64.1%

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

      3. associate-*r/ 64.1%

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

      4. associate-/l* 64.1%

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

   6. Simplified 64.1%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -9.5 \cdot 10^{-80}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 11: 53.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \cdot a2 \leq 10^{+277}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (* a2 a2) 4e-144)
   (* a1 (/ a1 (sqrt 2.0)))
   (if (<= (* a2 a2) 1e+277) (* a1 (* (cos th) a1)) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((a2 * a2) <= 4e-144) {
		tmp = a1 * (a1 / sqrt(2.0));
	} else if ((a2 * a2) <= 1e+277) {
		tmp = a1 * (cos(th) * a1);
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((a2 * a2) <= 4d-144) then
        tmp = a1 * (a1 / sqrt(2.0d0))
    else if ((a2 * a2) <= 1d+277) then
        tmp = a1 * (cos(th) * a1)
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((a2 * a2) <= 4e-144) {
		tmp = a1 * (a1 / Math.sqrt(2.0));
	} else if ((a2 * a2) <= 1e+277) {
		tmp = a1 * (Math.cos(th) * a1);
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (a2 * a2) <= 4e-144:
		tmp = a1 * (a1 / math.sqrt(2.0))
	elif (a2 * a2) <= 1e+277:
		tmp = a1 * (math.cos(th) * a1)
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a2 * a2) <= 4e-144)
		tmp = Float64(a1 * Float64(a1 / sqrt(2.0)));
	elseif (Float64(a2 * a2) <= 1e+277)
		tmp = Float64(a1 * Float64(cos(th) * a1));
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((a2 * a2) <= 4e-144)
		tmp = a1 * (a1 / sqrt(2.0));
	elseif ((a2 * a2) <= 1e+277)
		tmp = a1 * (cos(th) * a1);
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[(a2 * a2), $MachinePrecision], 4e-144], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a2 * a2), $MachinePrecision], 1e+277], N[(a1 * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a2 a2) < 3.9999999999999998e-144

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 68.8%

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

   5. Taylor expanded in a1 around inf 63.7%

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

      1. unpow2 63.7%

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

      2. associate-*r/ 63.7%

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

   7. Simplified 63.7%

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

   if 3.9999999999999998e-144 < (*.f64 a2 a2) < 1e277

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around inf 40.7%

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

      1. unpow2 40.7%

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

   6. Simplified 40.7%

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

      1. expm1-log1p-u 39.0%

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

      2. expm1-udef 38.1%

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

      3. div-inv 38.1%

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

      4. add-sqr-sqrt 38.1%

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

      5. sqrt-unprod 38.1%

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

      6. frac-times 38.1%

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

      7. metadata-eval 38.1%

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

      8. add-sqr-sqrt 38.1%

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

      9. metadata-eval 38.1%

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

   8. Applied egg-rr 38.1%

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

      1. expm1-def 39.0%

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

      2. expm1-log1p 40.7%

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

      3. associate-*r* 40.8%

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

   10. Simplified 40.8%

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

   11. Taylor expanded in th around inf 40.7%

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

       1. unpow2 40.7%

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

       2. associate-*l* 40.7%

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

       3. associate-*r* 40.7%

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

       4. *-commutative 40.7%

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

       5. associate-*l* 40.7%

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

   13. Simplified 40.7%

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

   15. Applied egg-rr 23.5%

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

       1. log-pow 23.5%

          \[\leadsto a1 \cdot \color{blue}{\left(\cos th \cdot \log \left(e^{a1}\right)\right)} \]

       2. rem-log-exp 26.6%

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

       3. *-commutative 26.6%

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

   17. Simplified 26.6%

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

   if 1e277 < (*.f64 a2 a2)

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in th around 0 76.4%

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

   5. Taylor expanded in a1 around 0 73.8%

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

      1. unpow2 73.8%

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

   7. Simplified 73.8%

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

   9. Applied egg-rr 70.3%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 4 \cdot 10^{-144}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \cdot a2 \leq 10^{+277}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]

Alternative 12: 63.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 2 \cdot 10^{-278}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (* a2 a2) 2e-278)
   (/ (* a1 a1) (sqrt 2.0))
   (* (cos th) (+ (* a2 a2) (* a1 a1)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((a2 * a2) <= 2e-278) {
		tmp = (a1 * a1) / sqrt(2.0);
	} else {
		tmp = cos(th) * ((a2 * a2) + (a1 * a1));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((a2 * a2) <= 2d-278) then
        tmp = (a1 * a1) / sqrt(2.0d0)
    else
        tmp = cos(th) * ((a2 * a2) + (a1 * a1))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((a2 * a2) <= 2e-278) {
		tmp = (a1 * a1) / Math.sqrt(2.0);
	} else {
		tmp = Math.cos(th) * ((a2 * a2) + (a1 * a1));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (a2 * a2) <= 2e-278:
		tmp = (a1 * a1) / math.sqrt(2.0)
	else:
		tmp = math.cos(th) * ((a2 * a2) + (a1 * a1))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a2 * a2) <= 2e-278)
		tmp = Float64(Float64(a1 * a1) / sqrt(2.0));
	else
		tmp = Float64(cos(th) * Float64(Float64(a2 * a2) + Float64(a1 * a1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((a2 * a2) <= 2e-278)
		tmp = (a1 * a1) / sqrt(2.0);
	else
		tmp = cos(th) * ((a2 * a2) + (a1 * a1));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[(a2 * a2), $MachinePrecision], 2e-278], N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a2 a2) < 1.99999999999999988e-278

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around inf 97.4%

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

      1. unpow2 97.4%

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

      2. associate-/l* 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in th around 0 72.8%

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

   if 1.99999999999999988e-278 < (*.f64 a2 a2)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. add-sqr-sqrt 99.5%

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

      3. times-frac 99.3%

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

      4. pow1/2 99.3%

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

      5. sqrt-pow1 99.3%

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

      6. metadata-eval 99.3%

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

      7. pow1/2 99.3%

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

      8. sqrt-pow1 99.3%

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

      9. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in th around inf 99.5%

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

   8. Applied egg-rr 58.4%

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

      1. rem-log-exp 58.4%

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

   10. Simplified 58.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 2 \cdot 10^{-278}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 13: 50.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -1.95 \cdot 10^{+151}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -4 \cdot 10^{+38}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -1.95e+151)
   (* a1 (* (cos th) a1))
   (if (<= a1 -4e+38) (* a1 (/ a1 (sqrt 2.0))) (* (* a2 a2) (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.95e+151) {
		tmp = a1 * (cos(th) * a1);
	} else if (a1 <= -4e+38) {
		tmp = a1 * (a1 / sqrt(2.0));
	} else {
		tmp = (a2 * a2) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a1 <= (-1.95d+151)) then
        tmp = a1 * (cos(th) * a1)
    else if (a1 <= (-4d+38)) then
        tmp = a1 * (a1 / sqrt(2.0d0))
    else
        tmp = (a2 * a2) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.95e+151) {
		tmp = a1 * (Math.cos(th) * a1);
	} else if (a1 <= -4e+38) {
		tmp = a1 * (a1 / Math.sqrt(2.0));
	} else {
		tmp = (a2 * a2) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -1.95e+151:
		tmp = a1 * (math.cos(th) * a1)
	elif a1 <= -4e+38:
		tmp = a1 * (a1 / math.sqrt(2.0))
	else:
		tmp = (a2 * a2) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -1.95e+151)
		tmp = Float64(a1 * Float64(cos(th) * a1));
	elseif (a1 <= -4e+38)
		tmp = Float64(a1 * Float64(a1 / sqrt(2.0)));
	else
		tmp = Float64(Float64(a2 * a2) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -1.95e+151)
		tmp = a1 * (cos(th) * a1);
	elseif (a1 <= -4e+38)
		tmp = a1 * (a1 / sqrt(2.0));
	else
		tmp = (a2 * a2) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -1.95e+151], N[(a1 * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision], If[LessEqual[a1, -4e+38], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a1 < -1.94999999999999988e151

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around inf 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. sqrt-unprod 100.0%

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

      6. frac-times 100.0%

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

      7. metadata-eval 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in th around inf 100.0%

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

       1. unpow2 100.0%

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

       2. associate-*l* 100.0%

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

       3. associate-*r* 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*l* 100.0%

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

   13. Simplified 100.0%

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

   15. Applied egg-rr 100.0%

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

       1. log-pow 100.0%

          \[\leadsto a1 \cdot \color{blue}{\left(\cos th \cdot \log \left(e^{a1}\right)\right)} \]

       2. rem-log-exp 100.0%

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

       3. *-commutative 100.0%

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

   17. Simplified 100.0%

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

   if -1.94999999999999988e151 < a1 < -3.99999999999999991e38

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 61.2%

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

   5. Taylor expanded in a1 around inf 31.6%

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

      1. unpow2 31.6%

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

      2. associate-*r/ 31.7%

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

   7. Simplified 31.7%

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

   if -3.99999999999999991e38 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. add-sqr-sqrt 99.5%

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

      3. times-frac 99.2%

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

      4. pow1/2 99.2%

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

      5. sqrt-pow1 99.2%

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

      6. metadata-eval 99.2%

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

      7. pow1/2 99.2%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in th around 0 63.6%

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

      1. *-commutative 63.6%

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

      2. unpow2 63.6%

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

      3. unpow2 63.6%

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

   8. Simplified 63.6%

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

   9. Taylor expanded in a2 around inf 41.9%

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

       1. *-commutative 41.9%

          \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{0.5}} \]

       2. unpow2 41.9%

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

   11. Simplified 41.9%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -1.95 \cdot 10^{+151}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -4 \cdot 10^{+38}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 14: 50.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -1.95 \cdot 10^{+151}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -1.55 \cdot 10^{+39}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -1.95e+151)
   (* a1 (* (cos th) a1))
   (if (<= a1 -1.55e+39) (* a1 (/ a1 (sqrt 2.0))) (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.95e+151) {
		tmp = a1 * (cos(th) * a1);
	} else if (a1 <= -1.55e+39) {
		tmp = a1 * (a1 / sqrt(2.0));
	} 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 (a1 <= (-1.95d+151)) then
        tmp = a1 * (cos(th) * a1)
    else if (a1 <= (-1.55d+39)) then
        tmp = a1 * (a1 / sqrt(2.0d0))
    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 (a1 <= -1.95e+151) {
		tmp = a1 * (Math.cos(th) * a1);
	} else if (a1 <= -1.55e+39) {
		tmp = a1 * (a1 / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -1.95e+151:
		tmp = a1 * (math.cos(th) * a1)
	elif a1 <= -1.55e+39:
		tmp = a1 * (a1 / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -1.95e+151)
		tmp = Float64(a1 * Float64(cos(th) * a1));
	elseif (a1 <= -1.55e+39)
		tmp = Float64(a1 * Float64(a1 / sqrt(2.0)));
	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 (a1 <= -1.95e+151)
		tmp = a1 * (cos(th) * a1);
	elseif (a1 <= -1.55e+39)
		tmp = a1 * (a1 / sqrt(2.0));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -1.95e+151], N[(a1 * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision], If[LessEqual[a1, -1.55e+39], N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a1 < -1.94999999999999988e151

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around inf 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. sqrt-unprod 100.0%

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

      6. frac-times 100.0%

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

      7. metadata-eval 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in th around inf 100.0%

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

       1. unpow2 100.0%

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

       2. associate-*l* 100.0%

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

       3. associate-*r* 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*l* 100.0%

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

   13. Simplified 100.0%

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

   15. Applied egg-rr 100.0%

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

       1. log-pow 100.0%

          \[\leadsto a1 \cdot \color{blue}{\left(\cos th \cdot \log \left(e^{a1}\right)\right)} \]

       2. rem-log-exp 100.0%

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

       3. *-commutative 100.0%

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

   17. Simplified 100.0%

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

   if -1.94999999999999988e151 < a1 < -1.5500000000000001e39

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in th around 0 61.2%

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

   5. Taylor expanded in a1 around inf 31.6%

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

      1. unpow2 31.6%

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

      2. associate-*r/ 31.7%

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

   7. Simplified 31.7%

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

   if -1.5500000000000001e39 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 63.6%

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

   5. Taylor expanded in a1 around 0 41.9%

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

      1. unpow2 41.9%

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

   7. Simplified 41.9%

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

   8. Taylor expanded in a2 around 0 41.9%

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

      1. unpow2 41.9%

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

      2. associate-/l* 41.9%

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

      3. associate-/r/ 41.9%

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

   10. Simplified 41.9%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -1.95 \cdot 10^{+151}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -1.55 \cdot 10^{+39}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 15: 50.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -1.95 \cdot 10^{+151}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -7 \cdot 10^{+37}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -1.95e+151)
   (* a1 (* (cos th) a1))
   (if (<= a1 -7e+37) (/ (* a1 a1) (sqrt 2.0)) (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -1.95e+151) {
		tmp = a1 * (cos(th) * a1);
	} else if (a1 <= -7e+37) {
		tmp = (a1 * a1) / sqrt(2.0);
	} 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 (a1 <= (-1.95d+151)) then
        tmp = a1 * (cos(th) * a1)
    else if (a1 <= (-7d+37)) then
        tmp = (a1 * a1) / sqrt(2.0d0)
    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 (a1 <= -1.95e+151) {
		tmp = a1 * (Math.cos(th) * a1);
	} else if (a1 <= -7e+37) {
		tmp = (a1 * a1) / Math.sqrt(2.0);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -1.95e+151:
		tmp = a1 * (math.cos(th) * a1)
	elif a1 <= -7e+37:
		tmp = (a1 * a1) / math.sqrt(2.0)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -1.95e+151)
		tmp = Float64(a1 * Float64(cos(th) * a1));
	elseif (a1 <= -7e+37)
		tmp = Float64(Float64(a1 * a1) / sqrt(2.0));
	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 (a1 <= -1.95e+151)
		tmp = a1 * (cos(th) * a1);
	elseif (a1 <= -7e+37)
		tmp = (a1 * a1) / sqrt(2.0);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -1.95e+151], N[(a1 * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision], If[LessEqual[a1, -7e+37], N[(N[(a1 * a1), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a1 < -1.94999999999999988e151

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around inf 100.0%

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

      1. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. sqrt-unprod 100.0%

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

      6. frac-times 100.0%

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

      7. metadata-eval 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in th around inf 100.0%

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

       1. unpow2 100.0%

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

       2. associate-*l* 100.0%

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

       3. associate-*r* 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*l* 100.0%

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

   13. Simplified 100.0%

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

   15. Applied egg-rr 100.0%

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

       1. log-pow 100.0%

          \[\leadsto a1 \cdot \color{blue}{\left(\cos th \cdot \log \left(e^{a1}\right)\right)} \]

       2. rem-log-exp 100.0%

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

       3. *-commutative 100.0%

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

   17. Simplified 100.0%

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

   if -1.94999999999999988e151 < a1 < -7e37

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around inf 63.4%

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

      1. unpow2 63.4%

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

      2. associate-/l* 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in th around 0 31.6%

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

   if -7e37 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 63.6%

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

   5. Taylor expanded in a1 around 0 41.9%

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

      1. unpow2 41.9%

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

   7. Simplified 41.9%

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

   8. Taylor expanded in a2 around 0 41.9%

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

      1. unpow2 41.9%

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

      2. associate-/l* 41.9%

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

      3. associate-/r/ 41.9%

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

   10. Simplified 41.9%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -1.95 \cdot 10^{+151}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{elif}\;a1 \leq -7 \cdot 10^{+37}:\\ \;\;\;\;\frac{a1 \cdot a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 16: 43.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 1.05 \cdot 10^{+117}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1.05e+117) (* a1 (* (cos th) a1)) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.05e+117) {
		tmp = a1 * (cos(th) * a1);
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 1.05d+117) then
        tmp = a1 * (cos(th) * a1)
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.05e+117) {
		tmp = a1 * (Math.cos(th) * a1);
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 1.05e+117:
		tmp = a1 * (math.cos(th) * a1)
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1.05e+117)
		tmp = Float64(a1 * Float64(cos(th) * a1));
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1.05e+117)
		tmp = a1 * (cos(th) * a1);
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 1.05e+117], N[(a1 * N[(N[Cos[th], $MachinePrecision] * a1), $MachinePrecision]), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 1.0500000000000001e117

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around inf 63.0%

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

      1. unpow2 63.0%

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

   6. Simplified 63.0%

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

      1. expm1-log1p-u 61.3%

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

      2. expm1-udef 51.7%

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

      3. div-inv 51.7%

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

      4. add-sqr-sqrt 51.7%

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

      5. sqrt-unprod 51.7%

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

      6. frac-times 51.7%

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

      7. metadata-eval 51.7%

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

      8. add-sqr-sqrt 51.7%

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

      9. metadata-eval 51.7%

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

   8. Applied egg-rr 51.7%

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

      1. expm1-def 61.3%

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

      2. expm1-log1p 62.9%

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

      3. associate-*r* 62.9%

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

   10. Simplified 62.9%

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

   11. Taylor expanded in th around inf 62.9%

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

       1. unpow2 62.9%

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

       2. associate-*l* 62.9%

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

       3. associate-*r* 62.9%

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

       4. *-commutative 62.9%

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

       5. associate-*l* 62.9%

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

   13. Simplified 62.9%

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

   15. Applied egg-rr 38.2%

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

       1. log-pow 38.2%

          \[\leadsto a1 \cdot \color{blue}{\left(\cos th \cdot \log \left(e^{a1}\right)\right)} \]

       2. rem-log-exp 39.4%

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

       3. *-commutative 39.4%

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

   17. Simplified 39.4%

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

   if 1.0500000000000001e117 < a2

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in th around 0 69.2%

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

   5. Taylor expanded in a1 around 0 66.9%

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

      1. unpow2 66.9%

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

   7. Simplified 66.9%

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

   9. Applied egg-rr 60.8%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.05 \cdot 10^{+117}:\\ \;\;\;\;a1 \cdot \left(\cos th \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]

Alternative 17: 36.1% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 2 \cdot 10^{+67}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(--2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 2e+67) (* a1 (* a1 (- -2.0))) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2e+67) {
		tmp = a1 * (a1 * -(-2.0));
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 2d+67) then
        tmp = a1 * (a1 * -(-2.0d0))
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 2e+67) {
		tmp = a1 * (a1 * -(-2.0));
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 2e+67:
		tmp = a1 * (a1 * -(-2.0))
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 2e+67)
		tmp = Float64(a1 * Float64(a1 * Float64(-(-2.0))));
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 2e+67)
		tmp = a1 * (a1 * -(-2.0));
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 2e+67], N[(a1 * N[(a1 * (--2.0)), $MachinePrecision]), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 1.99999999999999997e67

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 64.1%

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

   5. Taylor expanded in a1 around inf 43.9%

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

      1. unpow2 43.9%

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

      2. associate-*r/ 43.9%

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

   7. Simplified 43.9%

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

      1. frac-2neg 43.9%

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

      2. div-inv 43.9%

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

   9. Applied egg-rr 43.9%

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

   11. Applied egg-rr 32.5%

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

   if 1.99999999999999997e67 < a2

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 65.4%

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

   5. Taylor expanded in a1 around 0 62.2%

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

      1. unpow2 62.2%

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

   7. Simplified 62.2%

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

   9. Applied egg-rr 51.6%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 2 \cdot 10^{+67}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(--2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]

Alternative 18: 36.6% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 6.2 \cdot 10^{+66}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 6.2e+66) (* a1 (* a1 (- -0.5))) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 6.2e+66) {
		tmp = a1 * (a1 * -(-0.5));
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 6.2d+66) then
        tmp = a1 * (a1 * -(-0.5d0))
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 6.2e+66) {
		tmp = a1 * (a1 * -(-0.5));
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 6.2e+66:
		tmp = a1 * (a1 * -(-0.5))
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 6.2e+66)
		tmp = Float64(a1 * Float64(a1 * Float64(-(-0.5))));
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 6.2e+66)
		tmp = a1 * (a1 * -(-0.5));
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 6.2e+66], N[(a1 * N[(a1 * (--0.5)), $MachinePrecision]), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 6.20000000000000037e66

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 64.1%

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

   5. Taylor expanded in a1 around inf 43.9%

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

      1. unpow2 43.9%

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

      2. associate-*r/ 43.9%

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

   7. Simplified 43.9%

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

      1. frac-2neg 43.9%

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

      2. div-inv 43.9%

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

   9. Applied egg-rr 43.9%

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

   11. Applied egg-rr 33.0%

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

   if 6.20000000000000037e66 < a2

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 65.4%

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

   5. Taylor expanded in a1 around 0 62.2%

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

      1. unpow2 62.2%

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

   7. Simplified 62.2%

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

   9. Applied egg-rr 51.6%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 6.2 \cdot 10^{+66}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]

Alternative 19: 30.4% accurate, 58.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a1 \leq -2 \cdot 10^{+158}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a1 -2e+158) (* a1 (* a1 -0.5)) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -2e+158) {
		tmp = a1 * (a1 * -0.5);
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a1 <= (-2d+158)) then
        tmp = a1 * (a1 * (-0.5d0))
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a1 <= -2e+158) {
		tmp = a1 * (a1 * -0.5);
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a1 <= -2e+158:
		tmp = a1 * (a1 * -0.5)
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a1 <= -2e+158)
		tmp = Float64(a1 * Float64(a1 * -0.5));
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a1 <= -2e+158)
		tmp = a1 * (a1 * -0.5);
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a1, -2e+158], N[(a1 * N[(a1 * -0.5), $MachinePrecision]), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a1 < -1.99999999999999991e158

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. times-frac 100.0%

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

      4. pow1/2 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1/2 100.0%

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

      8. sqrt-pow1 100.0%

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

      9. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in th around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. unpow2 70.8%

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

      3. unpow2 70.8%

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

   8. Simplified 70.8%

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

   9. Taylor expanded in a2 around 0 70.8%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot {a1}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 70.8%

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

   11. Simplified 70.8%

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

   13. Applied egg-rr 29.2%

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

       1. *-rgt-identity 29.2%

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

       2. metadata-eval 29.2%

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

       3. associate-*r/ 29.2%

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

       4. associate-/r/ 29.2%

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

       5. metadata-eval 29.2%

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

       6. metadata-eval 29.2%

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

   15. Simplified 29.2%

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

   if -1.99999999999999991e158 < a1

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 63.7%

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

   5. Taylor expanded in a1 around 0 40.4%

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

      1. unpow2 40.4%

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

   7. Simplified 40.4%

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

   9. Applied egg-rr 30.2%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \leq -2 \cdot 10^{+158}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]

Alternative 20: 36.5% accurate, 82.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 6 \cdot 10^{+66}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (if (<= a2 6e+66) (* a1 a1) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 6e+66) {
		tmp = a1 * a1;
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (a2 <= 6d+66) then
        tmp = a1 * a1
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 6e+66) {
		tmp = a1 * a1;
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 6e+66:
		tmp = a1 * a1
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 6e+66)
		tmp = Float64(a1 * a1);
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 6e+66)
		tmp = a1 * a1;
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 6e+66], N[(a1 * a1), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 6.00000000000000005e66

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 64.1%

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

   5. Taylor expanded in a1 around inf 43.9%

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

      1. unpow2 43.9%

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

      2. associate-*r/ 43.9%

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

   7. Simplified 43.9%

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

      1. frac-2neg 43.9%

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

      2. div-inv 43.9%

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

   9. Applied egg-rr 43.9%

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

   11. Applied egg-rr 33.0%

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

   if 6.00000000000000005e66 < a2

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 65.4%

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

   5. Taylor expanded in a1 around 0 62.2%

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

      1. unpow2 62.2%

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

   7. Simplified 62.2%

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

   9. Applied egg-rr 51.6%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 6 \cdot 10^{+66}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]

Alternative 21: 29.9% 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 64.3%

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

5. Taylor expanded in a1 around 0 38.3%

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

   1. unpow2 38.3%

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

7. Simplified 38.3%

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

9. Applied egg-rr 29.1%

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

10. Final simplification 29.1%

    \[\leadsto a2 \cdot a2 \]

Alternative 22: 3.5% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := 1.0

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.5%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow1/2 99.3%

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

   8. sqrt-pow1 99.3%

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

   9. metadata-eval 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in th around inf 99.6%

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

8. Applied egg-rr 3.5%

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

   1. *-inverses 3.5%

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

10. Simplified 3.5%

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

11. Final simplification 3.5%

    \[\leadsto 1 \]

Alternative 23: 3.6% 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 64.3%

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

5. Taylor expanded in a1 around 0 38.3%

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

   1. unpow2 38.3%

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

7. Simplified 38.3%

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

9. Applied egg-rr 4.6%

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

    1. unpow1 4.6%

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

    2. metadata-eval 4.6%

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

    3. sqr-pow 2.3%

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

    4. fabs-sqr 2.3%

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

    5. sqr-pow 3.7%

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

    6. metadata-eval 3.7%

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

    7. unpow1 3.7%

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

11. Simplified 3.7%

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

12. Final simplification 3.7%

    \[\leadsto a2 \]

Reproduce

herbie shell --seed 2023195 
(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))))