Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 10.7s

Alternatives: 12

Speedup: 2.0×

• 44.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-*l/ 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l/ 99.7%

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

   6. fma-def 99.7%

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

   7. cos-neg 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 72.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.55)
   (* a2 (* a2 (cos th)))
   (* (sqrt 0.5) (+ (* a2 a2) (* a1 a1)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.55) {
		tmp = a2 * (a2 * cos(th));
	} else {
		tmp = sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.55d0) then
        tmp = a2 * (a2 * cos(th))
    else
        tmp = sqrt(0.5d0) * ((a2 * a2) + (a1 * a1))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.55) {
		tmp = a2 * (a2 * Math.cos(th));
	} else {
		tmp = Math.sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.55:
		tmp = a2 * (a2 * math.cos(th))
	else:
		tmp = math.sqrt(0.5) * ((a2 * a2) + (a1 * a1))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.55)
		tmp = Float64(a2 * Float64(a2 * cos(th)));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) + Float64(a1 * a1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.55)
		tmp = a2 * (a2 * cos(th));
	else
		tmp = sqrt(0.5) * ((a2 * a2) + (a1 * a1));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.55], N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.55000000000000004

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. cos-neg 99.7%

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

      3. associate-/r/ 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 59.1%

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

   6. Applied egg-rr 40.7%

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

   if 0.55000000000000004 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 91.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 46.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(-a2\right)}{-\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.55)
   (* a2 (* a2 (cos th)))
   (/ (* a2 (- a2)) (- (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.55) {
		tmp = a2 * (a2 * cos(th));
	} else {
		tmp = (a2 * -a2) / -sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.55d0) then
        tmp = a2 * (a2 * cos(th))
    else
        tmp = (a2 * -a2) / -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.55) {
		tmp = a2 * (a2 * Math.cos(th));
	} else {
		tmp = (a2 * -a2) / -Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.55:
		tmp = a2 * (a2 * math.cos(th))
	else:
		tmp = (a2 * -a2) / -math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.55)
		tmp = Float64(a2 * Float64(a2 * cos(th)));
	else
		tmp = Float64(Float64(a2 * Float64(-a2)) / Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.55)
		tmp = a2 * (a2 * cos(th));
	else
		tmp = (a2 * -a2) / -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.55], N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * (-a2)), $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.55000000000000004

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. cos-neg 99.7%

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

      3. associate-/r/ 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 59.1%

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

   6. Applied egg-rr 40.7%

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

   if 0.55000000000000004 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 91.7%

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

   5. Taylor expanded in a1 around 0 46.8%

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

      1. pow2 46.8%

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

      2. *-un-lft-identity 46.8%

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

      3. times-frac 46.7%

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

   7. Applied egg-rr 46.7%

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

      1. /-rgt-identity 46.7%

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

      2. *-commutative 46.7%

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

      3. frac-2neg 46.7%

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

      4. associate-*l/ 46.8%

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

   9. Applied egg-rr 46.8%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(-a2\right)}{-\sqrt{2}}\\ \end{array} \]

Alternative 5: 46.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.55) (* a2 (* a2 (cos th))) (* a2 (* a2 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.55) {
		tmp = a2 * (a2 * cos(th));
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.55d0) then
        tmp = a2 * (a2 * cos(th))
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.55) {
		tmp = a2 * (a2 * Math.cos(th));
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.55:
		tmp = a2 * (a2 * math.cos(th))
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.55)
		tmp = Float64(a2 * Float64(a2 * cos(th)));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.55)
		tmp = a2 * (a2 * cos(th));
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.55], N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.55000000000000004

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. cos-neg 99.7%

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

      3. associate-/r/ 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 59.1%

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

   6. Applied egg-rr 40.7%

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

   if 0.55000000000000004 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 91.7%

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

   5. Taylor expanded in a1 around 0 46.8%

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

      1. pow2 46.8%

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

      2. div-inv 46.7%

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

      3. pow1/2 46.7%

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

      4. pow-flip 46.8%

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

      5. metadata-eval 46.8%

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

      6. associate-*l* 46.8%

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

      7. add-sqr-sqrt 46.6%

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

      8. sqrt-unprod 46.8%

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

      9. pow-prod-up 46.8%

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

      10. metadata-eval 46.8%

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

      11. metadata-eval 46.8%

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

   7. Applied egg-rr 46.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 6: 46.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.55) (* a2 (* a2 (cos th))) (/ a2 (/ (sqrt 2.0) a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.55) {
		tmp = a2 * (a2 * cos(th));
	} else {
		tmp = a2 / (sqrt(2.0) / a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.55d0) then
        tmp = a2 * (a2 * cos(th))
    else
        tmp = a2 / (sqrt(2.0d0) / a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.55) {
		tmp = a2 * (a2 * Math.cos(th));
	} else {
		tmp = a2 / (Math.sqrt(2.0) / a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.55:
		tmp = a2 * (a2 * math.cos(th))
	else:
		tmp = a2 / (math.sqrt(2.0) / a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.55)
		tmp = Float64(a2 * Float64(a2 * cos(th)));
	else
		tmp = Float64(a2 / Float64(sqrt(2.0) / a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.55)
		tmp = a2 * (a2 * cos(th));
	else
		tmp = a2 / (sqrt(2.0) / a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.55], N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.55000000000000004

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. cos-neg 99.7%

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

      3. associate-/r/ 99.6%

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

      4. cos-neg 99.6%

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

      5. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 59.1%

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

   6. Applied egg-rr 40.7%

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

   if 0.55000000000000004 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 91.7%

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

   5. Taylor expanded in a1 around 0 46.8%

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

      1. pow2 46.8%

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

      2. *-un-lft-identity 46.8%

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

      3. times-frac 46.7%

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

   7. Applied egg-rr 46.7%

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

      1. /-rgt-identity 46.7%

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

      2. clear-num 46.7%

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

      3. un-div-inv 46.7%

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

   9. Applied egg-rr 46.7%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.55:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \cos th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 7: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in a1 around 0 53.1%

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

   1. clear-num 53.1%

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

   2. pow2 53.1%

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

   3. associate-/r/ 53.0%

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

   4. pow1/2 53.0%

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

   5. pow-flip 53.1%

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

   6. metadata-eval 53.1%

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

   7. *-commutative 53.1%

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

   8. associate-*l* 53.0%

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

   9. associate-*r* 53.1%

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

   10. add-sqr-sqrt 52.8%

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

   11. sqrt-unprod 53.1%

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

   12. pow-prod-up 53.1%

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

   13. metadata-eval 53.1%

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

   14. metadata-eval 53.1%

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

6. Applied egg-rr 53.1%

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

7. Final simplification 53.1%

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

Alternative 8: 41.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 10^{+226}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 1.6e+16) (and (not (<= th 1e+226)) (<= th 3.7e+287)))
   (* a2 (* a2 (sqrt 0.5)))
   (* (+ (* a2 a2) (* a1 a1)) -0.5)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 1e+226) && (th <= 3.7e+287))) {
		tmp = a2 * (a2 * sqrt(0.5));
	} else {
		tmp = ((a2 * a2) + (a1 * a1)) * -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 <= 1.6d+16) .or. (.not. (th <= 1d+226)) .and. (th <= 3.7d+287)) then
        tmp = a2 * (a2 * sqrt(0.5d0))
    else
        tmp = ((a2 * a2) + (a1 * a1)) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 1e+226) && (th <= 3.7e+287))) {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	} else {
		tmp = ((a2 * a2) + (a1 * a1)) * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 1.6e+16) or (not (th <= 1e+226) and (th <= 3.7e+287)):
		tmp = a2 * (a2 * math.sqrt(0.5))
	else:
		tmp = ((a2 * a2) + (a1 * a1)) * -0.5
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 1.6e+16) || (!(th <= 1e+226) && (th <= 3.7e+287)))
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	else
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 1.6e+16) || (~((th <= 1e+226)) && (th <= 3.7e+287)))
		tmp = a2 * (a2 * sqrt(0.5));
	else
		tmp = ((a2 * a2) + (a1 * a1)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 1.6e+16], And[N[Not[LessEqual[th, 1e+226]], $MachinePrecision], LessEqual[th, 3.7e+287]]], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.6e16 or 9.99999999999999961e225 < th < 3.69999999999999997e287

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 74.9%

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

   5. Taylor expanded in a1 around 0 39.2%

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

      1. pow2 39.2%

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

      2. div-inv 39.1%

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

      3. pow1/2 39.1%

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

      4. pow-flip 39.2%

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

      5. metadata-eval 39.2%

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

      6. associate-*l* 39.2%

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

      7. add-sqr-sqrt 39.0%

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

      8. sqrt-unprod 39.2%

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

      9. pow-prod-up 39.2%

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

      10. metadata-eval 39.2%

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

      11. metadata-eval 39.2%

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

   7. Applied egg-rr 39.2%

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

   if 1.6e16 < th < 9.99999999999999961e225 or 3.69999999999999997e287 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 34.6%

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

   6. Applied egg-rr 43.7%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 10^{+226}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot -0.5\\ \end{array} \]

Alternative 9: 45.5% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 3.2 \cdot 10^{+267}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;t_1 \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a2 a2) (* a1 a1))))
   (if (or (<= th 1.6e+16) (and (not (<= th 3.2e+267)) (<= th 3.7e+287)))
     (* t_1 0.125)
     (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287))) {
		tmp = t_1 * 0.125;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a2 * a2) + (a1 * a1)
    if ((th <= 1.6d+16) .or. (.not. (th <= 3.2d+267)) .and. (th <= 3.7d+287)) then
        tmp = t_1 * 0.125d0
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287))) {
		tmp = t_1 * 0.125;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) + (a1 * a1)
	tmp = 0
	if (th <= 1.6e+16) or (not (th <= 3.2e+267) and (th <= 3.7e+287)):
		tmp = t_1 * 0.125
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) + Float64(a1 * a1))
	tmp = 0.0
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287)))
		tmp = Float64(t_1 * 0.125);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) + (a1 * a1);
	tmp = 0.0;
	if ((th <= 1.6e+16) || (~((th <= 3.2e+267)) && (th <= 3.7e+287)))
		tmp = t_1 * 0.125;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[th, 1.6e+16], And[N[Not[LessEqual[th, 3.2e+267]], $MachinePrecision], LessEqual[th, 3.7e+287]]], N[(t$95$1 * 0.125), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 1.6e16 or 3.2000000000000001e267 < th < 3.69999999999999997e287

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 76.1%

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

   6. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{0.125} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 1.6e16 < th < 3.2000000000000001e267 or 3.69999999999999997e287 < th

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 38.9%

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

   6. Applied egg-rr 44.1%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 3.2 \cdot 10^{+267}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot -0.5\\ \end{array} \]

Alternative 10: 45.9% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 3.2 \cdot 10^{+267}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;t_1 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a2 a2) (* a1 a1))))
   (if (or (<= th 1.6e+16) (and (not (<= th 3.2e+267)) (<= th 3.7e+287)))
     (* t_1 0.25)
     (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287))) {
		tmp = t_1 * 0.25;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a2 * a2) + (a1 * a1)
    if ((th <= 1.6d+16) .or. (.not. (th <= 3.2d+267)) .and. (th <= 3.7d+287)) then
        tmp = t_1 * 0.25d0
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287))) {
		tmp = t_1 * 0.25;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) + (a1 * a1)
	tmp = 0
	if (th <= 1.6e+16) or (not (th <= 3.2e+267) and (th <= 3.7e+287)):
		tmp = t_1 * 0.25
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) + Float64(a1 * a1))
	tmp = 0.0
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287)))
		tmp = Float64(t_1 * 0.25);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) + (a1 * a1);
	tmp = 0.0;
	if ((th <= 1.6e+16) || (~((th <= 3.2e+267)) && (th <= 3.7e+287)))
		tmp = t_1 * 0.25;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[th, 1.6e+16], And[N[Not[LessEqual[th, 3.2e+267]], $MachinePrecision], LessEqual[th, 3.7e+287]]], N[(t$95$1 * 0.25), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 1.6e16 or 3.2000000000000001e267 < th < 3.69999999999999997e287

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 76.1%

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

   6. Applied egg-rr 48.0%

      \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 1.6e16 < th < 3.2000000000000001e267 or 3.69999999999999997e287 < th

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 38.9%

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

   6. Applied egg-rr 44.1%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 3.2 \cdot 10^{+267}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot -0.5\\ \end{array} \]

Alternative 11: 46.6% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 3.2 \cdot 10^{+267}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a2 a2) (* a1 a1))))
   (if (or (<= th 1.6e+16) (and (not (<= th 3.2e+267)) (<= th 3.7e+287)))
     (* 0.5 t_1)
     (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287))) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a2 * a2) + (a1 * a1)
    if ((th <= 1.6d+16) .or. (.not. (th <= 3.2d+267)) .and. (th <= 3.7d+287)) then
        tmp = 0.5d0 * t_1
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287))) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) + (a1 * a1)
	tmp = 0
	if (th <= 1.6e+16) or (not (th <= 3.2e+267) and (th <= 3.7e+287)):
		tmp = 0.5 * t_1
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) + Float64(a1 * a1))
	tmp = 0.0
	if ((th <= 1.6e+16) || (!(th <= 3.2e+267) && (th <= 3.7e+287)))
		tmp = Float64(0.5 * t_1);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) + (a1 * a1);
	tmp = 0.0;
	if ((th <= 1.6e+16) || (~((th <= 3.2e+267)) && (th <= 3.7e+287)))
		tmp = 0.5 * t_1;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[th, 1.6e+16], And[N[Not[LessEqual[th, 3.2e+267]], $MachinePrecision], LessEqual[th, 3.7e+287]]], N[(0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 1.6e16 or 3.2000000000000001e267 < th < 3.69999999999999997e287

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 76.1%

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

   6. Applied egg-rr 49.0%

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

   if 1.6e16 < th < 3.2000000000000001e267 or 3.69999999999999997e287 < th

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 38.9%

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

   6. Applied egg-rr 44.1%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.6 \cdot 10^{+16} \lor \neg \left(th \leq 3.2 \cdot 10^{+267}\right) \land th \leq 3.7 \cdot 10^{+287}:\\ \;\;\;\;0.5 \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot -0.5\\ \end{array} \]

Alternative 12: 19.8% accurate, 46.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around 0 68.1%

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

6. Applied egg-rr 24.1%

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

7. Final simplification 24.1%

   \[\leadsto \left(a2 \cdot a2 + a1 \cdot a1\right) \cdot -0.5 \]

Reproduce

herbie shell --seed 2023307 
(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))))