Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 15.6s

Alternatives: 18

Speedup: 2.0×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. cos-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

   1. fma-udef 99.3%

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

   2. +-commutative 99.3%

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

5. Applied egg-rr 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(\left(a2 + a1\right) \cdot \left(a2 + a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.65)
   (* (cos th) (* (+ a2 a1) (+ a2 a1)))
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.65) {
		tmp = cos(th) * ((a2 + a1) * (a2 + a1));
	} else {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.65d0) then
        tmp = cos(th) * ((a2 + a1) * (a2 + a1))
    else
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.65) {
		tmp = Math.cos(th) * ((a2 + a1) * (a2 + a1));
	} else {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.65:
		tmp = math.cos(th) * ((a2 + a1) * (a2 + a1))
	else:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.65)
		tmp = Float64(cos(th) * Float64(Float64(a2 + a1) * Float64(a2 + a1)));
	else
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.65)
		tmp = cos(th) * ((a2 + a1) * (a2 + a1));
	else
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.65], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 + a1), $MachinePrecision] * N[(a2 + a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.650000000000000022

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-lft-out 98.8%

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

   3. Simplified 98.8%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.6%

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

      3. pow1/2 98.6%

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

      4. pow-flip 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in th around inf 98.8%

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

      1. *-commutative 98.8%

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

      2. unpow2 98.8%

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

      3. unpow2 98.8%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 49.1%

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

       1. +-commutative 49.1%

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

       2. distribute-lft-in 57.7%

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

   12. Simplified 57.7%

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

   if 0.650000000000000022 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 93.8%

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

      1. expm1-log1p-u 93.8%

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

      2. expm1-udef 93.8%

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

      3. add-sqr-sqrt 93.8%

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

      4. sqrt-unprod 93.8%

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

      5. frac-times 93.8%

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

      6. metadata-eval 93.8%

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

      7. add-sqr-sqrt 93.3%

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

      8. metadata-eval 93.3%

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

   6. Applied egg-rr 93.3%

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

      1. expm1-def 93.3%

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

      2. expm1-log1p 93.8%

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

   8. Simplified 93.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(\left(a2 + a1\right) \cdot \left(a2 + a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 3: 80.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(\left(a2 + a1\right) \cdot \left(a2 + a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.65)
   (* (cos th) (* (+ a2 a1) (+ a2 a1)))
   (/ (+ (* a2 a2) (* a1 a1)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.65) {
		tmp = cos(th) * ((a2 + a1) * (a2 + a1));
	} else {
		tmp = ((a2 * a2) + (a1 * a1)) / sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.65d0) then
        tmp = cos(th) * ((a2 + a1) * (a2 + a1))
    else
        tmp = ((a2 * a2) + (a1 * a1)) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.65) {
		tmp = Math.cos(th) * ((a2 + a1) * (a2 + a1));
	} else {
		tmp = ((a2 * a2) + (a1 * a1)) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.65:
		tmp = math.cos(th) * ((a2 + a1) * (a2 + a1))
	else:
		tmp = ((a2 * a2) + (a1 * a1)) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.65)
		tmp = Float64(cos(th) * Float64(Float64(a2 + a1) * Float64(a2 + a1)));
	else
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.65)
		tmp = cos(th) * ((a2 + a1) * (a2 + a1));
	else
		tmp = ((a2 * a2) + (a1 * a1)) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.65], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 + a1), $MachinePrecision] * N[(a2 + a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.650000000000000022

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-lft-out 98.8%

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

   3. Simplified 98.8%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.6%

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

      3. pow1/2 98.6%

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

      4. pow-flip 98.9%

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

      5. metadata-eval 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in th around inf 98.8%

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

      1. *-commutative 98.8%

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

      2. unpow2 98.8%

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

      3. unpow2 98.8%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 49.1%

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

       1. +-commutative 49.1%

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

       2. distribute-lft-in 57.7%

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

   12. Simplified 57.7%

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

   if 0.650000000000000022 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. cos-neg 99.6%

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

      3. associate-*l/ 99.8%

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

      4. cos-neg 99.8%

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

      5. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in th around 0 93.9%

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

      1. unpow2 93.9%

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

      2. unpow2 93.9%

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

      3. +-commutative 93.9%

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

   8. Simplified 93.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(\left(a2 + a1\right) \cdot \left(a2 + a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.2%

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

   2. associate-/r/ 99.2%

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

   3. pow1/2 99.2%

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

   4. pow-flip 99.3%

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

   5. metadata-eval 99.3%

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

5. Applied egg-rr 99.3%

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

6. Taylor expanded in th around inf 99.3%

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

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

   3. unpow2 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 5: 57.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a2 around inf 57.5%

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

   1. unpow2 57.5%

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

   2. associate-*r/ 57.5%

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

   3. associate-*r* 57.8%

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

6. Simplified 57.8%

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

7. Final simplification 57.8%

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

Alternative 6: 57.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. cos-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. cos-neg 99.3%

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

   5. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 57.5%

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

   1. unpow2 57.5%

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

   2. *-commutative 57.5%

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

6. Simplified 57.5%

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

   1. div-inv 57.4%

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

   2. associate-*r* 57.4%

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

   3. associate-*l* 57.7%

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

   4. add-sqr-sqrt 57.7%

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

   5. sqrt-unprod 57.7%

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

   6. frac-times 57.7%

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

   7. metadata-eval 57.7%

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

   8. add-sqr-sqrt 57.8%

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

   9. metadata-eval 57.8%

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

8. Applied egg-rr 57.8%

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

9. Final simplification 57.8%

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

Alternative 7: 39.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (/ (* a2 a2) (sqrt 2.0))
   (if (or (<= th 7.5e+18) (not (<= th 1.25e+76)))
     (* a2 (* a2 -0.5))
     (* (* a2 a2) (/ 1.0 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a2 * a2) / sqrt(2.0);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = (a2 * a2) * (1.0 / 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 (th <= 1.55d0) then
        tmp = (a2 * a2) / sqrt(2.0d0)
    else if ((th <= 7.5d+18) .or. (.not. (th <= 1.25d+76))) then
        tmp = a2 * (a2 * (-0.5d0))
    else
        tmp = (a2 * a2) * (1.0d0 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = (a2 * a2) * (1.0 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = (a2 * a2) / math.sqrt(2.0)
	elif (th <= 7.5e+18) or not (th <= 1.25e+76):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = (a2 * a2) * (1.0 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	elseif ((th <= 7.5e+18) || !(th <= 1.25e+76))
		tmp = Float64(a2 * Float64(a2 * -0.5));
	else
		tmp = Float64(Float64(a2 * a2) * Float64(1.0 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = (a2 * a2) / sqrt(2.0);
	elseif ((th <= 7.5e+18) || ~((th <= 1.25e+76)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = (a2 * a2) * (1.0 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 7.5e+18], N[Not[LessEqual[th, 1.25e+76]], $MachinePrecision]], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a2 around inf 46.6%

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

      1. unpow2 46.6%

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

   7. Simplified 46.6%

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

   if 1.55000000000000004 < th < 7.5e18 or 1.24999999999999998e76 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 20.6%

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

   5. Taylor expanded in a2 around inf 10.5%

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

      1. unpow2 10.5%

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

   7. Simplified 10.5%

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

      1. div-inv 10.5%

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

      2. add-sqr-sqrt 10.5%

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

      3. sqrt-unprod 10.5%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. add-sqr-sqrt 10.5%

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

      7. metadata-eval 10.5%

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

   9. Applied egg-rr 10.5%

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

   11. Applied egg-rr 20.7%

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

       1. *-lft-identity 20.7%

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

       2. *-inverses 20.7%

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

       3. associate-*l/ 20.7%

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

       4. associate-/r/ 20.7%

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

       5. *-inverses 20.7%

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

       6. metadata-eval 20.7%

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

       7. *-commutative 20.7%

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

       8. *-commutative 20.7%

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

   13. Simplified 20.7%

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

   if 7.5e18 < th < 1.24999999999999998e76

   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. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 68.5%

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

   5. Taylor expanded in a2 around inf 40.9%

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

      1. unpow2 40.9%

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

   7. Simplified 40.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \]

Alternative 8: 39.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (/ (* a2 a2) (sqrt 2.0))
   (if (or (<= th 7.5e+18) (not (<= th 1.25e+76)))
     (* a2 (* a2 -0.5))
     (/ 1.0 (/ (sqrt 2.0) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a2 * a2) / sqrt(2.0);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.55d0) then
        tmp = (a2 * a2) / sqrt(2.0d0)
    else if ((th <= 7.5d+18) .or. (.not. (th <= 1.25d+76))) then
        tmp = a2 * (a2 * (-0.5d0))
    else
        tmp = 1.0d0 / (sqrt(2.0d0) / (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = 1.0 / (Math.sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = (a2 * a2) / math.sqrt(2.0)
	elif (th <= 7.5e+18) or not (th <= 1.25e+76):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = 1.0 / (math.sqrt(2.0) / (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	elseif ((th <= 7.5e+18) || !(th <= 1.25e+76))
		tmp = Float64(a2 * Float64(a2 * -0.5));
	else
		tmp = Float64(1.0 / Float64(sqrt(2.0) / Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = (a2 * a2) / sqrt(2.0);
	elseif ((th <= 7.5e+18) || ~((th <= 1.25e+76)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 7.5e+18], N[Not[LessEqual[th, 1.25e+76]], $MachinePrecision]], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a2 around inf 46.6%

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

      1. unpow2 46.6%

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

   7. Simplified 46.6%

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

   if 1.55000000000000004 < th < 7.5e18 or 1.24999999999999998e76 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 20.6%

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

   5. Taylor expanded in a2 around inf 10.5%

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

      1. unpow2 10.5%

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

   7. Simplified 10.5%

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

      1. div-inv 10.5%

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

      2. add-sqr-sqrt 10.5%

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

      3. sqrt-unprod 10.5%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. add-sqr-sqrt 10.5%

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

      7. metadata-eval 10.5%

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

   9. Applied egg-rr 10.5%

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

   11. Applied egg-rr 20.7%

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

       1. *-lft-identity 20.7%

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

       2. *-inverses 20.7%

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

       3. associate-*l/ 20.7%

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

       4. associate-/r/ 20.7%

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

       5. *-inverses 20.7%

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

       6. metadata-eval 20.7%

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

       7. *-commutative 20.7%

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

       8. *-commutative 20.7%

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

   13. Simplified 20.7%

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

   if 7.5e18 < th < 1.24999999999999998e76

   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. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 68.5%

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

   5. Taylor expanded in a2 around inf 40.9%

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

      1. unpow2 40.9%

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

      2. associate-*r/ 40.9%

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

   7. Simplified 40.9%

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

      1. associate-*r/ 40.9%

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

      2. clear-num 40.9%

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

   9. Applied egg-rr 40.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]

Alternative 9: 63.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))
   (if (or (<= th 7.5e+18) (not (<= th 1.25e+76)))
     (* a2 (* a2 -0.5))
     (/ 1.0 (/ (sqrt 2.0) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.55d0) then
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    else if ((th <= 7.5d+18) .or. (.not. (th <= 1.25d+76))) then
        tmp = a2 * (a2 * (-0.5d0))
    else
        tmp = 1.0d0 / (sqrt(2.0d0) / (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = 1.0 / (Math.sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	elif (th <= 7.5e+18) or not (th <= 1.25e+76):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = 1.0 / (math.sqrt(2.0) / (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	elseif ((th <= 7.5e+18) || !(th <= 1.25e+76))
		tmp = Float64(a2 * Float64(a2 * -0.5));
	else
		tmp = Float64(1.0 / Float64(sqrt(2.0) / Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	elseif ((th <= 7.5e+18) || ~((th <= 1.25e+76)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 7.5e+18], N[Not[LessEqual[th, 1.25e+76]], $MachinePrecision]], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.4%

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

      1. expm1-log1p-u 73.4%

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

      2. expm1-udef 73.4%

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

      3. add-sqr-sqrt 73.4%

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

      4. sqrt-unprod 73.4%

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

      5. frac-times 73.4%

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

      6. metadata-eval 73.4%

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

      7. add-sqr-sqrt 73.0%

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

      8. metadata-eval 73.0%

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

   6. Applied egg-rr 73.0%

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

      1. expm1-def 73.0%

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

      2. expm1-log1p 73.4%

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

   8. Simplified 73.4%

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

   if 1.55000000000000004 < th < 7.5e18 or 1.24999999999999998e76 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 20.6%

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

   5. Taylor expanded in a2 around inf 10.5%

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

      1. unpow2 10.5%

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

   7. Simplified 10.5%

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

      1. div-inv 10.5%

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

      2. add-sqr-sqrt 10.5%

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

      3. sqrt-unprod 10.5%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. add-sqr-sqrt 10.5%

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

      7. metadata-eval 10.5%

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

   9. Applied egg-rr 10.5%

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

   11. Applied egg-rr 20.7%

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

       1. *-lft-identity 20.7%

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

       2. *-inverses 20.7%

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

       3. associate-*l/ 20.7%

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

       4. associate-/r/ 20.7%

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

       5. *-inverses 20.7%

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

       6. metadata-eval 20.7%

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

       7. *-commutative 20.7%

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

       8. *-commutative 20.7%

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

   13. Simplified 20.7%

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

   if 7.5e18 < th < 1.24999999999999998e76

   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. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 68.5%

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

   5. Taylor expanded in a2 around inf 40.9%

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

      1. unpow2 40.9%

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

      2. associate-*r/ 40.9%

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

   7. Simplified 40.9%

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

      1. associate-*r/ 40.9%

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

      2. clear-num 40.9%

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

   9. Applied egg-rr 40.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]

Alternative 10: 39.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55 \lor \neg \left(th \leq 7.5 \cdot 10^{+18}\right) \land th \leq 1.25 \cdot 10^{+76}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 1.55) (and (not (<= th 7.5e+18)) (<= th 1.25e+76)))
   (* a2 (/ a2 (sqrt 2.0)))
   (* a2 (* a2 -0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.55) || (!(th <= 7.5e+18) && (th <= 1.25e+76))) {
		tmp = a2 * (a2 / sqrt(2.0));
	} else {
		tmp = a2 * (a2 * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((th <= 1.55d0) .or. (.not. (th <= 7.5d+18)) .and. (th <= 1.25d+76)) then
        tmp = a2 * (a2 / sqrt(2.0d0))
    else
        tmp = a2 * (a2 * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.55) || (!(th <= 7.5e+18) && (th <= 1.25e+76))) {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	} else {
		tmp = a2 * (a2 * -0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 1.55) or (not (th <= 7.5e+18) and (th <= 1.25e+76)):
		tmp = a2 * (a2 / math.sqrt(2.0))
	else:
		tmp = a2 * (a2 * -0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 1.55) || (!(th <= 7.5e+18) && (th <= 1.25e+76)))
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	else
		tmp = Float64(a2 * Float64(a2 * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 1.55) || (~((th <= 7.5e+18)) && (th <= 1.25e+76)))
		tmp = a2 * (a2 / sqrt(2.0));
	else
		tmp = a2 * (a2 * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 1.55], And[N[Not[LessEqual[th, 7.5e+18]], $MachinePrecision], LessEqual[th, 1.25e+76]]], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.55000000000000004 or 7.5e18 < th < 1.24999999999999998e76

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.2%

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

   5. Taylor expanded in a2 around inf 46.3%

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

      1. unpow2 46.3%

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

      2. associate-*r/ 46.3%

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

   7. Simplified 46.3%

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

   if 1.55000000000000004 < th < 7.5e18 or 1.24999999999999998e76 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 20.6%

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

   5. Taylor expanded in a2 around inf 10.5%

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

      1. unpow2 10.5%

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

   7. Simplified 10.5%

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

      1. div-inv 10.5%

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

      2. add-sqr-sqrt 10.5%

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

      3. sqrt-unprod 10.5%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. add-sqr-sqrt 10.5%

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

      7. metadata-eval 10.5%

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

   9. Applied egg-rr 10.5%

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

   11. Applied egg-rr 20.7%

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

       1. *-lft-identity 20.7%

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

       2. *-inverses 20.7%

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

       3. associate-*l/ 20.7%

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

       4. associate-/r/ 20.7%

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

       5. *-inverses 20.7%

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

       6. metadata-eval 20.7%

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

       7. *-commutative 20.7%

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

       8. *-commutative 20.7%

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

   13. Simplified 20.7%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55 \lor \neg \left(th \leq 7.5 \cdot 10^{+18}\right) \land th \leq 1.25 \cdot 10^{+76}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \end{array} \]

Alternative 11: 39.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (* a2 (/ a2 (sqrt 2.0)))
   (if (or (<= th 7.5e+18) (not (<= th 1.25e+76)))
     (* a2 (* a2 -0.5))
     (* a2 (* a2 (sqrt 0.5))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = a2 * (a2 / sqrt(2.0));
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} 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 (th <= 1.55d0) then
        tmp = a2 * (a2 / sqrt(2.0d0))
    else if ((th <= 7.5d+18) .or. (.not. (th <= 1.25d+76))) then
        tmp = a2 * (a2 * (-0.5d0))
    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 (th <= 1.55) {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = a2 * (a2 / math.sqrt(2.0))
	elif (th <= 7.5e+18) or not (th <= 1.25e+76):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	elseif ((th <= 7.5e+18) || !(th <= 1.25e+76))
		tmp = Float64(a2 * Float64(a2 * -0.5));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = a2 * (a2 / sqrt(2.0));
	elseif ((th <= 7.5e+18) || ~((th <= 1.25e+76)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 7.5e+18], N[Not[LessEqual[th, 1.25e+76]], $MachinePrecision]], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a2 around inf 46.6%

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

      1. unpow2 46.6%

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

      2. associate-*r/ 46.5%

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

   7. Simplified 46.5%

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

   if 1.55000000000000004 < th < 7.5e18 or 1.24999999999999998e76 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 20.6%

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

   5. Taylor expanded in a2 around inf 10.5%

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

      1. unpow2 10.5%

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

   7. Simplified 10.5%

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

      1. div-inv 10.5%

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

      2. add-sqr-sqrt 10.5%

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

      3. sqrt-unprod 10.5%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. add-sqr-sqrt 10.5%

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

      7. metadata-eval 10.5%

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

   9. Applied egg-rr 10.5%

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

   11. Applied egg-rr 20.7%

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

       1. *-lft-identity 20.7%

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

       2. *-inverses 20.7%

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

       3. associate-*l/ 20.7%

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

       4. associate-/r/ 20.7%

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

       5. *-inverses 20.7%

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

       6. metadata-eval 20.7%

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

       7. *-commutative 20.7%

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

       8. *-commutative 20.7%

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

   13. Simplified 20.7%

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

   if 7.5e18 < th < 1.24999999999999998e76

   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. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 68.5%

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

   5. Taylor expanded in a2 around inf 40.9%

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

      1. unpow2 40.9%

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

   7. Simplified 40.9%

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

      1. *-un-lft-identity 40.9%

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

      2. associate-*l/ 40.9%

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

      3. associate-*r* 40.9%

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

      4. add-sqr-sqrt 40.9%

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

      5. sqrt-unprod 40.9%

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

      6. frac-times 40.9%

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

      7. metadata-eval 40.9%

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

      8. add-sqr-sqrt 40.9%

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

      9. metadata-eval 40.9%

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

   9. Applied egg-rr 40.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 12: 39.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (* a2 (/ a2 (sqrt 2.0)))
   (if (or (<= th 7.5e+18) (not (<= th 1.25e+76)))
     (* a2 (* a2 -0.5))
     (/ a2 (/ (sqrt 2.0) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = a2 * (a2 / sqrt(2.0));
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 / (sqrt(2.0) / a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.55d0) then
        tmp = a2 * (a2 / sqrt(2.0d0))
    else if ((th <= 7.5d+18) .or. (.not. (th <= 1.25d+76))) then
        tmp = a2 * (a2 * (-0.5d0))
    else
        tmp = a2 / (sqrt(2.0d0) / a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = a2 * (a2 / math.sqrt(2.0))
	elif (th <= 7.5e+18) or not (th <= 1.25e+76):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = a2 / (math.sqrt(2.0) / a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	elseif ((th <= 7.5e+18) || !(th <= 1.25e+76))
		tmp = Float64(a2 * Float64(a2 * -0.5));
	else
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = a2 * (a2 / sqrt(2.0));
	elseif ((th <= 7.5e+18) || ~((th <= 1.25e+76)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = a2 / (sqrt(2.0) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 7.5e+18], N[Not[LessEqual[th, 1.25e+76]], $MachinePrecision]], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a2 around inf 46.6%

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

      1. unpow2 46.6%

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

      2. associate-*r/ 46.5%

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

   7. Simplified 46.5%

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

   if 1.55000000000000004 < th < 7.5e18 or 1.24999999999999998e76 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 20.6%

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

   5. Taylor expanded in a2 around inf 10.5%

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

      1. unpow2 10.5%

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

   7. Simplified 10.5%

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

      1. div-inv 10.5%

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

      2. add-sqr-sqrt 10.5%

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

      3. sqrt-unprod 10.5%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. add-sqr-sqrt 10.5%

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

      7. metadata-eval 10.5%

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

   9. Applied egg-rr 10.5%

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

   11. Applied egg-rr 20.7%

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

       1. *-lft-identity 20.7%

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

       2. *-inverses 20.7%

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

       3. associate-*l/ 20.7%

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

       4. associate-/r/ 20.7%

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

       5. *-inverses 20.7%

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

       6. metadata-eval 20.7%

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

       7. *-commutative 20.7%

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

       8. *-commutative 20.7%

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

   13. Simplified 20.7%

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

   if 7.5e18 < th < 1.24999999999999998e76

   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. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 68.5%

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

   5. Taylor expanded in a2 around inf 40.9%

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

      1. unpow2 40.9%

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

      2. associate-*r/ 40.9%

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

   7. Simplified 40.9%

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

      1. clear-num 40.9%

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

      2. un-div-inv 40.9%

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

   9. Applied egg-rr 40.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 13: 39.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (/ (* a2 a2) (sqrt 2.0))
   (if (or (<= th 7.5e+18) (not (<= th 1.25e+76)))
     (* a2 (* a2 -0.5))
     (/ a2 (/ (sqrt 2.0) a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a2 * a2) / sqrt(2.0);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 / (sqrt(2.0) / a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.55d0) then
        tmp = (a2 * a2) / sqrt(2.0d0)
    else if ((th <= 7.5d+18) .or. (.not. (th <= 1.25d+76))) then
        tmp = a2 * (a2 * (-0.5d0))
    else
        tmp = a2 / (sqrt(2.0d0) / a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	} else if ((th <= 7.5e+18) || !(th <= 1.25e+76)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = (a2 * a2) / math.sqrt(2.0)
	elif (th <= 7.5e+18) or not (th <= 1.25e+76):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = a2 / (math.sqrt(2.0) / a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	elseif ((th <= 7.5e+18) || !(th <= 1.25e+76))
		tmp = Float64(a2 * Float64(a2 * -0.5));
	else
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = (a2 * a2) / sqrt(2.0);
	elseif ((th <= 7.5e+18) || ~((th <= 1.25e+76)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = a2 / (sqrt(2.0) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 7.5e+18], N[Not[LessEqual[th, 1.25e+76]], $MachinePrecision]], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a2 around inf 46.6%

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

      1. unpow2 46.6%

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

   7. Simplified 46.6%

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

   if 1.55000000000000004 < th < 7.5e18 or 1.24999999999999998e76 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 20.6%

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

   5. Taylor expanded in a2 around inf 10.5%

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

      1. unpow2 10.5%

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

   7. Simplified 10.5%

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

      1. div-inv 10.5%

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

      2. add-sqr-sqrt 10.5%

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

      3. sqrt-unprod 10.5%

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

      4. frac-times 10.5%

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

      5. metadata-eval 10.5%

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

      6. add-sqr-sqrt 10.5%

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

      7. metadata-eval 10.5%

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

   9. Applied egg-rr 10.5%

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

   11. Applied egg-rr 20.7%

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

       1. *-lft-identity 20.7%

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

       2. *-inverses 20.7%

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

       3. associate-*l/ 20.7%

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

       4. associate-/r/ 20.7%

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

       5. *-inverses 20.7%

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

       6. metadata-eval 20.7%

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

       7. *-commutative 20.7%

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

       8. *-commutative 20.7%

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

   13. Simplified 20.7%

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

   if 7.5e18 < th < 1.24999999999999998e76

   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. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 68.5%

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

   5. Taylor expanded in a2 around inf 40.9%

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

      1. unpow2 40.9%

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

      2. associate-*r/ 40.9%

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

   7. Simplified 40.9%

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

      1. clear-num 40.9%

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

      2. un-div-inv 40.9%

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

   9. Applied egg-rr 40.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 14: 30.3% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 4.4 \cdot 10^{+73}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot -0.5\right) \cdot \left(-a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55)
   (* a2 a2)
   (if (or (<= th 7.5e+18) (not (<= th 4.4e+73)))
     (* a2 (* a2 -0.5))
     (* (* a2 -0.5) (- a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = a2 * a2;
	} else if ((th <= 7.5e+18) || !(th <= 4.4e+73)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = (a2 * -0.5) * -a2;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.55d0) then
        tmp = a2 * a2
    else if ((th <= 7.5d+18) .or. (.not. (th <= 4.4d+73))) then
        tmp = a2 * (a2 * (-0.5d0))
    else
        tmp = (a2 * (-0.5d0)) * -a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = a2 * a2;
	} else if ((th <= 7.5e+18) || !(th <= 4.4e+73)) {
		tmp = a2 * (a2 * -0.5);
	} else {
		tmp = (a2 * -0.5) * -a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = a2 * a2
	elif (th <= 7.5e+18) or not (th <= 4.4e+73):
		tmp = a2 * (a2 * -0.5)
	else:
		tmp = (a2 * -0.5) * -a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(a2 * a2);
	elseif ((th <= 7.5e+18) || !(th <= 4.4e+73))
		tmp = Float64(a2 * Float64(a2 * -0.5));
	else
		tmp = Float64(Float64(a2 * -0.5) * Float64(-a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = a2 * a2;
	elseif ((th <= 7.5e+18) || ~((th <= 4.4e+73)))
		tmp = a2 * (a2 * -0.5);
	else
		tmp = (a2 * -0.5) * -a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 7.5e+18], N[Not[LessEqual[th, 4.4e+73]], $MachinePrecision]], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * -0.5), $MachinePrecision] * (-a2)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.55000000000000004

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a2 around inf 46.6%

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

      1. unpow2 46.6%

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

      2. associate-*r/ 46.5%

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

   7. Simplified 46.5%

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

      1. frac-2neg 46.5%

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

      2. div-inv 46.5%

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

   9. Applied egg-rr 46.5%

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

   11. Applied egg-rr 33.9%

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

   if 1.55000000000000004 < th < 7.5e18 or 4.4e73 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 22.2%

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

   5. Taylor expanded in a2 around inf 12.3%

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

      1. unpow2 12.3%

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

   7. Simplified 12.3%

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

      1. div-inv 12.3%

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

      2. add-sqr-sqrt 12.3%

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

      3. sqrt-unprod 12.3%

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

      4. frac-times 12.3%

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

      5. metadata-eval 12.3%

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

      6. add-sqr-sqrt 12.3%

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

      7. metadata-eval 12.3%

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

   9. Applied egg-rr 12.3%

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

   11. Applied egg-rr 20.3%

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

       1. *-lft-identity 20.3%

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

       2. *-inverses 20.3%

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

       3. associate-*l/ 20.3%

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

       4. associate-/r/ 20.3%

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

       5. *-inverses 20.3%

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

       6. metadata-eval 20.3%

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

       7. *-commutative 20.3%

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

       8. *-commutative 20.3%

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

   13. Simplified 20.3%

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

   if 7.5e18 < th < 4.4e73

   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. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in th around 0 64.0%

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

   5. Taylor expanded in a2 around inf 32.5%

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

      1. unpow2 32.5%

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

      2. associate-*r/ 32.5%

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

   7. Simplified 32.5%

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

      1. frac-2neg 32.5%

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

      2. div-inv 32.5%

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

   9. Applied egg-rr 32.5%

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

   11. Applied egg-rr 32.5%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 7.5 \cdot 10^{+18} \lor \neg \left(th \leq 4.4 \cdot 10^{+73}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot -0.5\right) \cdot \left(-a2\right)\\ \end{array} \]

Alternative 15: 30.3% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55 \lor \neg \left(th \leq 7.5 \cdot 10^{+18}\right) \land th \leq 4.4 \cdot 10^{+73}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 1.55) (and (not (<= th 7.5e+18)) (<= th 4.4e+73)))
   (* a2 a2)
   (* a2 (* a2 -0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.55) || (!(th <= 7.5e+18) && (th <= 4.4e+73))) {
		tmp = a2 * a2;
	} else {
		tmp = a2 * (a2 * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((th <= 1.55d0) .or. (.not. (th <= 7.5d+18)) .and. (th <= 4.4d+73)) then
        tmp = a2 * a2
    else
        tmp = a2 * (a2 * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.55) || (!(th <= 7.5e+18) && (th <= 4.4e+73))) {
		tmp = a2 * a2;
	} else {
		tmp = a2 * (a2 * -0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 1.55) or (not (th <= 7.5e+18) and (th <= 4.4e+73)):
		tmp = a2 * a2
	else:
		tmp = a2 * (a2 * -0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 1.55) || (!(th <= 7.5e+18) && (th <= 4.4e+73)))
		tmp = Float64(a2 * a2);
	else
		tmp = Float64(a2 * Float64(a2 * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 1.55) || (~((th <= 7.5e+18)) && (th <= 4.4e+73)))
		tmp = a2 * a2;
	else
		tmp = a2 * (a2 * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 1.55], And[N[Not[LessEqual[th, 7.5e+18]], $MachinePrecision], LessEqual[th, 4.4e+73]]], N[(a2 * a2), $MachinePrecision], N[(a2 * N[(a2 * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.55000000000000004 or 7.5e18 < th < 4.4e73

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 73.1%

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

   5. Taylor expanded in a2 around inf 46.1%

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

      1. unpow2 46.1%

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

      2. associate-*r/ 46.1%

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

   7. Simplified 46.1%

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

      1. frac-2neg 46.1%

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

      2. div-inv 46.0%

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

   9. Applied egg-rr 46.0%

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

   11. Applied egg-rr 33.9%

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

   if 1.55000000000000004 < th < 7.5e18 or 4.4e73 < th

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 22.2%

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

   5. Taylor expanded in a2 around inf 12.3%

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

      1. unpow2 12.3%

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

   7. Simplified 12.3%

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

      1. div-inv 12.3%

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

      2. add-sqr-sqrt 12.3%

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

      3. sqrt-unprod 12.3%

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

      4. frac-times 12.3%

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

      5. metadata-eval 12.3%

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

      6. add-sqr-sqrt 12.3%

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

      7. metadata-eval 12.3%

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

   9. Applied egg-rr 12.3%

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

   11. Applied egg-rr 20.3%

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

       1. *-lft-identity 20.3%

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

       2. *-inverses 20.3%

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

       3. associate-*l/ 20.3%

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

       4. associate-/r/ 20.3%

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

       5. *-inverses 20.3%

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

       6. metadata-eval 20.3%

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

       7. *-commutative 20.3%

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

       8. *-commutative 20.3%

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

   13. Simplified 20.3%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55 \lor \neg \left(th \leq 7.5 \cdot 10^{+18}\right) \land th \leq 4.4 \cdot 10^{+73}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot -0.5\right)\\ \end{array} \]

Alternative 16: 15.2% accurate, 83.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in th around 0 63.3%

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

5. Taylor expanded in a2 around inf 39.6%

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

   1. unpow2 39.6%

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

7. Simplified 39.6%

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

   1. div-inv 39.6%

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

   2. add-sqr-sqrt 39.6%

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

   3. sqrt-unprod 39.6%

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

   4. frac-times 39.6%

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

   5. metadata-eval 39.6%

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

   6. add-sqr-sqrt 39.6%

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

   7. metadata-eval 39.6%

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

9. Applied egg-rr 39.6%

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

11. Applied egg-rr 14.6%

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

    1. *-lft-identity 14.6%

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

    2. *-inverses 14.6%

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

    3. associate-*l/ 14.6%

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

    4. associate-/r/ 14.6%

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

    5. *-inverses 14.6%

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

    6. metadata-eval 14.6%

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

    7. *-commutative 14.6%

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

    8. *-commutative 14.6%

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

13. Simplified 14.6%

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

14. Final simplification 14.6%

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

Alternative 17: 3.6% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a1

Derivation

1. Initial program 99.3%

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

   1. +-commutative 99.3%

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

   2. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in th around 0 63.3%

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

5. Taylor expanded in a2 around 0 37.6%

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

   1. unpow2 37.6%

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

7. Simplified 37.6%

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

9. Applied egg-rr 4.6%

   \[\leadsto \color{blue}{\left|a1\right|} \]
10. Step-by-step derivation

    1. unpow1 4.6%

       \[\leadsto \left|\color{blue}{{a1}^{1}}\right| \]

    2. *-inverses 4.6%

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

    3. sqr-pow 2.4%

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

    4. fabs-sqr 2.4%

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

    5. sqr-pow 3.7%

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

    6. *-inverses 3.7%

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

    7. unpow1 3.7%

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

11. Simplified 3.7%

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

12. Final simplification 3.7%

    \[\leadsto a1 \]

Alternative 18: 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 99.3%

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

   1. +-commutative 99.3%

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

   2. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in th around 0 63.3%

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

5. Taylor expanded in a2 around inf 39.6%

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

   1. unpow2 39.6%

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

7. Simplified 39.6%

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

   1. div-inv 39.6%

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

   2. add-sqr-sqrt 39.6%

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

   3. sqrt-unprod 39.6%

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

   4. frac-times 39.6%

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

   5. metadata-eval 39.6%

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

   6. add-sqr-sqrt 39.6%

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

   7. metadata-eval 39.6%

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

9. Applied egg-rr 39.6%

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

11. Applied egg-rr 3.7%

    \[\leadsto \color{blue}{0 + a2} \]

12. Final simplification 3.7%

    \[\leadsto a2 \]

Reproduce

herbie shell --seed 2023277 
(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))))