Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 7.0s

Alternatives: 16

Speedup: 2.0×

• 43.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

9. Simplified 99.7%

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

Alternative 2: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;0.6666666666666666 \cdot \left(\left(a1 + a2\right) \cdot \left(a1 - a2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -1e-309)
   (* 0.6666666666666666 (* (+ a1 a2) (- a1 a2)))
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -1e-309) {
		tmp = 0.6666666666666666 * ((a1 + a2) * (a1 - a2));
	} else {
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= (-1d-309)) then
        tmp = 0.6666666666666666d0 * ((a1 + a2) * (a1 - a2))
    else
        tmp = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -1e-309) {
		tmp = 0.6666666666666666 * ((a1 + a2) * (a1 - a2));
	} else {
		tmp = Math.sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -1e-309:
		tmp = 0.6666666666666666 * ((a1 + a2) * (a1 - a2))
	else:
		tmp = math.sqrt(0.5) * ((a1 * a1) + (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -1e-309)
		tmp = Float64(0.6666666666666666 * Float64(Float64(a1 + a2) * Float64(a1 - a2)));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -1e-309)
		tmp = 0.6666666666666666 * ((a1 + a2) * (a1 - a2));
	else
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -1e-309], N[(0.6666666666666666 * N[(N[(a1 + a2), $MachinePrecision] * N[(a1 - a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -1.000000000000002e-309

   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. Add Preprocessing

   5. Taylor expanded in th around 0 5.8%

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

   7. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{0.6666666666666666} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 3.6%

         \[\leadsto 0.6666666666666666 \cdot \left(a1 \cdot a1 + \color{blue}{\left(\sqrt{a2} \cdot \sqrt{a2}\right)} \cdot a2\right) \]

      2. sqrt-prod 20.4%

         \[\leadsto 0.6666666666666666 \cdot \left(a1 \cdot a1 + \color{blue}{\sqrt{a2 \cdot a2}} \cdot a2\right) \]

      3. sqr-neg 20.4%

         \[\leadsto 0.6666666666666666 \cdot \left(a1 \cdot a1 + \sqrt{\color{blue}{\left(-a2\right) \cdot \left(-a2\right)}} \cdot a2\right) \]

      4. sqrt-unprod 16.8%

         \[\leadsto 0.6666666666666666 \cdot \left(a1 \cdot a1 + \color{blue}{\left(\sqrt{-a2} \cdot \sqrt{-a2}\right)} \cdot a2\right) \]

      5. add-sqr-sqrt 29.4%

         \[\leadsto 0.6666666666666666 \cdot \left(a1 \cdot a1 + \color{blue}{\left(-a2\right)} \cdot a2\right) \]

      6. cancel-sign-sub-inv 29.4%

         \[\leadsto 0.6666666666666666 \cdot \color{blue}{\left(a1 \cdot a1 - a2 \cdot a2\right)} \]

      7. difference-of-squares 31.0%

         \[\leadsto 0.6666666666666666 \cdot \color{blue}{\left(\left(a1 + a2\right) \cdot \left(a1 - a2\right)\right)} \]

   9. Applied egg-rr 31.0%

      \[\leadsto 0.6666666666666666 \cdot \color{blue}{\left(\left(a1 + a2\right) \cdot \left(a1 - a2\right)\right)} \]

   if -1.000000000000002e-309 < (cos.f64 th)

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 86.4%

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

Alternative 3: 59.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

8. Applied egg-rr 61.5%

   \[\leadsto \color{blue}{\left(0 + \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
9. Step-by-step derivation

   1. +-lft-identity 61.5%

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

10. Simplified 61.5%

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

Alternative 4: 47.9% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;t\_1 \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 9.2e+91) (not (<= th 3.55e+134)))
     (* t_1 0.6666666666666666)
     (* t_1 -0.25))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = t_1 * 0.6666666666666666;
	} else {
		tmp = t_1 * -0.25;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if ((th <= 9.2d+91) .or. (.not. (th <= 3.55d+134))) then
        tmp = t_1 * 0.6666666666666666d0
    else
        tmp = t_1 * (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = t_1 * 0.6666666666666666;
	} else {
		tmp = t_1 * -0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if (th <= 9.2e+91) or not (th <= 3.55e+134):
		tmp = t_1 * 0.6666666666666666
	else:
		tmp = t_1 * -0.25
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if ((th <= 9.2e+91) || !(th <= 3.55e+134))
		tmp = Float64(t_1 * 0.6666666666666666);
	else
		tmp = Float64(t_1 * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if ((th <= 9.2e+91) || ~((th <= 3.55e+134)))
		tmp = t_1 * 0.6666666666666666;
	else
		tmp = t_1 * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[th, 9.2e+91], N[Not[LessEqual[th, 3.55e+134]], $MachinePrecision]], N[(t$95$1 * 0.6666666666666666), $MachinePrecision], N[(t$95$1 * -0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 9.19999999999999965e91 or 3.54999999999999995e134 < 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. Add Preprocessing

   5. Taylor expanded in th around 0 69.5%

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

   7. Applied egg-rr 50.1%

      \[\leadsto \color{blue}{0.6666666666666666} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.7% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;0.5 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (or (<= th 9.2e+91) (not (<= th 3.55e+134)))
     (* 0.5 t_1)
     (* t_1 -0.25))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.25;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if ((th <= 9.2d+91) .or. (.not. (th <= 3.55d+134))) then
        tmp = 0.5d0 * t_1
    else
        tmp = t_1 * (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if (th <= 9.2e+91) or not (th <= 3.55e+134):
		tmp = 0.5 * t_1
	else:
		tmp = t_1 * -0.25
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if ((th <= 9.2e+91) || !(th <= 3.55e+134))
		tmp = Float64(0.5 * t_1);
	else
		tmp = Float64(t_1 * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if ((th <= 9.2e+91) || ~((th <= 3.55e+134)))
		tmp = 0.5 * t_1;
	else
		tmp = t_1 * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[th, 9.2e+91], N[Not[LessEqual[th, 3.55e+134]], $MachinePrecision]], N[(0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * -0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 9.19999999999999965e91 or 3.54999999999999995e134 < 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. Add Preprocessing

   5. Taylor expanded in th around 0 69.5%

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

   7. Applied egg-rr 48.9%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 9.2e+91) (not (<= th 3.55e+134)))
   (* (+ a1 a2) (+ a1 a2))
   (* (+ (* a1 a1) (* a2 a2)) -0.25)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.25;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((th <= 9.2d+91) .or. (.not. (th <= 3.55d+134))) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 9.2e+91) or not (th <= 3.55e+134):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.25
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 9.2e+91) || !(th <= 3.55e+134))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 9.2e+91) || ~((th <= 3.55e+134)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 9.2e+91], N[Not[LessEqual[th, 3.55e+134]], $MachinePrecision]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.19999999999999965e91 or 3.54999999999999995e134 < 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. Add Preprocessing

   5. Taylor expanded in th around 0 69.5%

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

   7. Applied egg-rr 44.9%

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

      1. distribute-lft-in 48.6%

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

   9. Simplified 48.6%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 9.2e+91) (not (<= th 3.55e+134)))
   (* (+ a1 a2) (+ a1 a2))
   (* (+ (* a1 a1) (* a2 a2)) -0.3333333333333333)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.3333333333333333;
	}
	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 <= 9.2d+91) .or. (.not. (th <= 3.55d+134))) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 9.2e+91) or not (th <= 3.55e+134):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.3333333333333333
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 9.2e+91) || !(th <= 3.55e+134))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 9.2e+91) || ~((th <= 3.55e+134)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 9.2e+91], N[Not[LessEqual[th, 3.55e+134]], $MachinePrecision]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.19999999999999965e91 or 3.54999999999999995e134 < 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. Add Preprocessing

   5. Taylor expanded in th around 0 69.5%

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

   7. Applied egg-rr 44.9%

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

      1. distribute-lft-in 48.6%

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

   9. Simplified 48.6%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{-0.3333333333333333} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 9.2e+91) (not (<= th 3.55e+134)))
   (* (+ a1 a2) (+ a1 a2))
   (* (+ (* a1 a1) (* a2 a2)) -0.5)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (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 <= 9.2d+91) .or. (.not. (th <= 3.55d+134))) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = ((a1 * a1) + (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 <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 9.2e+91) or not (th <= 3.55e+134):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 9.2e+91) || !(th <= 3.55e+134))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 9.2e+91) || ~((th <= 3.55e+134)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 9.2e+91], N[Not[LessEqual[th, 3.55e+134]], $MachinePrecision]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.19999999999999965e91 or 3.54999999999999995e134 < 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. Add Preprocessing

   5. Taylor expanded in th around 0 69.5%

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

   7. Applied egg-rr 44.9%

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

      1. distribute-lft-in 48.6%

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

   9. Simplified 48.6%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 61.8%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.2 \cdot 10^{+91}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 3.55 \cdot 10^{+134}:\\ \;\;\;\;t\_1 \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 0.375\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 9.2e+91)
     (* (+ a1 a2) (+ a1 a2))
     (if (<= th 3.55e+134) (* t_1 -0.25) (* t_1 0.375)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.2e+91) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 3.55e+134) {
		tmp = t_1 * -0.25;
	} else {
		tmp = t_1 * 0.375;
	}
	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 = (a1 * a1) + (a2 * a2)
    if (th <= 9.2d+91) then
        tmp = (a1 + a2) * (a1 + a2)
    else if (th <= 3.55d+134) then
        tmp = t_1 * (-0.25d0)
    else
        tmp = t_1 * 0.375d0
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.2e+91) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 3.55e+134) {
		tmp = t_1 * -0.25;
	} else {
		tmp = t_1 * 0.375;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 9.2e+91:
		tmp = (a1 + a2) * (a1 + a2)
	elif th <= 3.55e+134:
		tmp = t_1 * -0.25
	else:
		tmp = t_1 * 0.375
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 9.2e+91)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	elseif (th <= 3.55e+134)
		tmp = Float64(t_1 * -0.25);
	else
		tmp = Float64(t_1 * 0.375);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 9.2e+91)
		tmp = (a1 + a2) * (a1 + a2);
	elseif (th <= 3.55e+134)
		tmp = t_1 * -0.25;
	else
		tmp = t_1 * 0.375;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 9.2e+91], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.55e+134], N[(t$95$1 * -0.25), $MachinePrecision], N[(t$95$1 * 0.375), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 9.19999999999999965e91

   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. Add Preprocessing

   5. Taylor expanded in th around 0 75.5%

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

   7. Applied egg-rr 46.8%

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

      1. distribute-lft-in 51.2%

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

   9. Simplified 51.2%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 62.0%

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

   if 3.54999999999999995e134 < th

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in th around 0 34.8%

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

   7. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{0.375} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 3.55 \cdot 10^{+134}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.375\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.2 \cdot 10^{+91}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 3.55 \cdot 10^{+134}:\\ \;\;\;\;t\_1 \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 9.2e+91)
     (* (+ a1 a2) (+ a1 a2))
     (if (<= th 3.55e+134) (* t_1 -0.25) (* t_1 0.3333333333333333)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.2e+91) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 3.55e+134) {
		tmp = t_1 * -0.25;
	} else {
		tmp = t_1 * 0.3333333333333333;
	}
	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 = (a1 * a1) + (a2 * a2)
    if (th <= 9.2d+91) then
        tmp = (a1 + a2) * (a1 + a2)
    else if (th <= 3.55d+134) then
        tmp = t_1 * (-0.25d0)
    else
        tmp = t_1 * 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.2e+91) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 3.55e+134) {
		tmp = t_1 * -0.25;
	} else {
		tmp = t_1 * 0.3333333333333333;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 9.2e+91:
		tmp = (a1 + a2) * (a1 + a2)
	elif th <= 3.55e+134:
		tmp = t_1 * -0.25
	else:
		tmp = t_1 * 0.3333333333333333
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 9.2e+91)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	elseif (th <= 3.55e+134)
		tmp = Float64(t_1 * -0.25);
	else
		tmp = Float64(t_1 * 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 9.2e+91)
		tmp = (a1 + a2) * (a1 + a2);
	elseif (th <= 3.55e+134)
		tmp = t_1 * -0.25;
	else
		tmp = t_1 * 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 9.2e+91], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.55e+134], N[(t$95$1 * -0.25), $MachinePrecision], N[(t$95$1 * 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 9.19999999999999965e91

   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. Add Preprocessing

   5. Taylor expanded in th around 0 75.5%

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

   7. Applied egg-rr 46.8%

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

      1. distribute-lft-in 51.2%

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

   9. Simplified 51.2%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 62.0%

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

   if 3.54999999999999995e134 < th

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in th around 0 34.8%

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

   7. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 3.55 \cdot 10^{+134}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.5% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.2 \cdot 10^{+91}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 3.55 \cdot 10^{+134}:\\ \;\;\;\;t\_1 \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 9.2e+91)
     (* (+ a1 a2) (+ a1 a2))
     (if (<= th 3.55e+134) (* t_1 -0.25) (* t_1 0.25)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.2e+91) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 3.55e+134) {
		tmp = t_1 * -0.25;
	} else {
		tmp = t_1 * 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (th <= 9.2d+91) then
        tmp = (a1 + a2) * (a1 + a2)
    else if (th <= 3.55d+134) then
        tmp = t_1 * (-0.25d0)
    else
        tmp = t_1 * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.2e+91) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 3.55e+134) {
		tmp = t_1 * -0.25;
	} else {
		tmp = t_1 * 0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 9.2e+91:
		tmp = (a1 + a2) * (a1 + a2)
	elif th <= 3.55e+134:
		tmp = t_1 * -0.25
	else:
		tmp = t_1 * 0.25
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 9.2e+91)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	elseif (th <= 3.55e+134)
		tmp = Float64(t_1 * -0.25);
	else
		tmp = Float64(t_1 * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 9.2e+91)
		tmp = (a1 + a2) * (a1 + a2);
	elseif (th <= 3.55e+134)
		tmp = t_1 * -0.25;
	else
		tmp = t_1 * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 9.2e+91], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 3.55e+134], N[(t$95$1 * -0.25), $MachinePrecision], N[(t$95$1 * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 9.19999999999999965e91

   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. Add Preprocessing

   5. Taylor expanded in th around 0 75.5%

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

   7. Applied egg-rr 46.8%

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

      1. distribute-lft-in 51.2%

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

   9. Simplified 51.2%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 62.0%

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

   if 3.54999999999999995e134 < th

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in th around 0 34.8%

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

   7. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 3.55 \cdot 10^{+134}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.1% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 9.2e+91) (not (<= th 3.55e+134)))
   (* (+ a1 a2) (+ a1 a2))
   (- a1 (* a2 a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = a1 - (a2 * a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((th <= 9.2d+91) .or. (.not. (th <= 3.55d+134))) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = a1 - (a2 * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 9.2e+91) || !(th <= 3.55e+134)) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = a1 - (a2 * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 9.2e+91) or not (th <= 3.55e+134):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = a1 - (a2 * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 9.2e+91) || !(th <= 3.55e+134))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(a1 - Float64(a2 * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 9.2e+91) || ~((th <= 3.55e+134)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = a1 - (a2 * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 9.2e+91], N[Not[LessEqual[th, 3.55e+134]], $MachinePrecision]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(a1 - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.19999999999999965e91 or 3.54999999999999995e134 < 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. Add Preprocessing

   5. Taylor expanded in th around 0 69.5%

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

   7. Applied egg-rr 44.9%

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

      1. distribute-lft-in 48.6%

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

   9. Simplified 48.6%

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

   if 9.19999999999999965e91 < th < 3.54999999999999995e134

   1. Initial program 99.9%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in th around 0 12.6%

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

   7. Applied egg-rr 36.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.2 \cdot 10^{+91} \lor \neg \left(th \leq 3.55 \cdot 10^{+134}\right):\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 7.4% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a1 - a2\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.55) (* (- a1 a2) -4.0) (- a1 (* a2 a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a1 - a2) * -4.0;
	} else {
		tmp = a1 - (a2 * a2);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.55d0) then
        tmp = (a1 - a2) * (-4.0d0)
    else
        tmp = a1 - (a2 * a2)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.55) {
		tmp = (a1 - a2) * -4.0;
	} else {
		tmp = a1 - (a2 * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.55:
		tmp = (a1 - a2) * -4.0
	else:
		tmp = a1 - (a2 * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.55)
		tmp = Float64(Float64(a1 - a2) * -4.0);
	else
		tmp = Float64(a1 - Float64(a2 * a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.55)
		tmp = (a1 - a2) * -4.0;
	else
		tmp = a1 - (a2 * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.55], N[(N[(a1 - a2), $MachinePrecision] * -4.0), $MachinePrecision], N[(a1 - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.55000000000000004

   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. Add Preprocessing

   5. Taylor expanded in th around 0 79.0%

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

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, a1, -a2 \cdot -4\right) + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right)} \]
   8. Step-by-step derivation

      1. fma-undefine 4.1%

         \[\leadsto \color{blue}{\left(-4 \cdot a1 + \left(-a2 \cdot -4\right)\right)} + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right) \]

      2. *-commutative 4.1%

         \[\leadsto \left(\color{blue}{a1 \cdot -4} + \left(-a2 \cdot -4\right)\right) + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right) \]

      3. associate-+l+ 4.1%

         \[\leadsto \color{blue}{a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right)\right)} \]

      4. fma-undefine 4.1%

         \[\leadsto a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \color{blue}{\left(\left(-a2\right) \cdot -4 + a2 \cdot -4\right)}\right) \]

      5. distribute-lft-neg-in 4.1%

         \[\leadsto a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \left(\color{blue}{\left(-a2 \cdot -4\right)} + a2 \cdot -4\right)\right) \]

      6. +-commutative 4.1%

         \[\leadsto a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \color{blue}{\left(a2 \cdot -4 + \left(-a2 \cdot -4\right)\right)}\right) \]

      7. associate-+r+ 4.1%

         \[\leadsto a1 \cdot -4 + \color{blue}{\left(\left(\left(-a2 \cdot -4\right) + a2 \cdot -4\right) + \left(-a2 \cdot -4\right)\right)} \]

      8. +-commutative 4.1%

         \[\leadsto a1 \cdot -4 + \left(\color{blue}{\left(a2 \cdot -4 + \left(-a2 \cdot -4\right)\right)} + \left(-a2 \cdot -4\right)\right) \]

      9. sub-neg 4.1%

         \[\leadsto a1 \cdot -4 + \left(\color{blue}{\left(a2 \cdot -4 - a2 \cdot -4\right)} + \left(-a2 \cdot -4\right)\right) \]

      10. +-inverses 4.1%

          \[\leadsto a1 \cdot -4 + \left(\color{blue}{0} + \left(-a2 \cdot -4\right)\right) \]

      11. +-lft-identity 4.1%

          \[\leadsto a1 \cdot -4 + \color{blue}{\left(-a2 \cdot -4\right)} \]

      12. sub-neg 4.1%

          \[\leadsto \color{blue}{a1 \cdot -4 - a2 \cdot -4} \]

      13. distribute-rgt-out-- 4.1%

          \[\leadsto \color{blue}{-4 \cdot \left(a1 - a2\right)} \]

   9. Simplified 4.1%

      \[\leadsto \color{blue}{-4 \cdot \left(a1 - a2\right)} \]

   if 1.55000000000000004 < th

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in th around 0 31.3%

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

   7. Applied egg-rr 18.8%

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

4. Final simplification 7.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.55:\\ \;\;\;\;\left(a1 - a2\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 4.1% accurate, 83.0× speedup

Math

\[\begin{array}{l} \\ \left(a1 - a2\right) \cdot -4 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (- a1 a2) -4.0))

C

double code(double a1, double a2, double th) {
	return (a1 - a2) * -4.0;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a1 - a2) * (-4.0d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a1 - a2) * -4.0;
}

Python

def code(a1, a2, th):
	return (a1 - a2) * -4.0

Julia

function code(a1, a2, th)
	return Float64(Float64(a1 - a2) * -4.0)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a1 - a2) * -4.0;
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 - a2), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 66.9%

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

7. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-4, a1, -a2 \cdot -4\right) + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right)} \]
8. Step-by-step derivation

   1. fma-undefine 4.1%

      \[\leadsto \color{blue}{\left(-4 \cdot a1 + \left(-a2 \cdot -4\right)\right)} + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right) \]

   2. *-commutative 4.1%

      \[\leadsto \left(\color{blue}{a1 \cdot -4} + \left(-a2 \cdot -4\right)\right) + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right) \]

   3. associate-+l+ 4.1%

      \[\leadsto \color{blue}{a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \mathsf{fma}\left(-a2, -4, a2 \cdot -4\right)\right)} \]

   4. fma-undefine 4.1%

      \[\leadsto a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \color{blue}{\left(\left(-a2\right) \cdot -4 + a2 \cdot -4\right)}\right) \]

   5. distribute-lft-neg-in 4.1%

      \[\leadsto a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \left(\color{blue}{\left(-a2 \cdot -4\right)} + a2 \cdot -4\right)\right) \]

   6. +-commutative 4.1%

      \[\leadsto a1 \cdot -4 + \left(\left(-a2 \cdot -4\right) + \color{blue}{\left(a2 \cdot -4 + \left(-a2 \cdot -4\right)\right)}\right) \]

   7. associate-+r+ 4.1%

      \[\leadsto a1 \cdot -4 + \color{blue}{\left(\left(\left(-a2 \cdot -4\right) + a2 \cdot -4\right) + \left(-a2 \cdot -4\right)\right)} \]

   8. +-commutative 4.1%

      \[\leadsto a1 \cdot -4 + \left(\color{blue}{\left(a2 \cdot -4 + \left(-a2 \cdot -4\right)\right)} + \left(-a2 \cdot -4\right)\right) \]

   9. sub-neg 4.1%

      \[\leadsto a1 \cdot -4 + \left(\color{blue}{\left(a2 \cdot -4 - a2 \cdot -4\right)} + \left(-a2 \cdot -4\right)\right) \]

   10. +-inverses 4.1%

       \[\leadsto a1 \cdot -4 + \left(\color{blue}{0} + \left(-a2 \cdot -4\right)\right) \]

   11. +-lft-identity 4.1%

       \[\leadsto a1 \cdot -4 + \color{blue}{\left(-a2 \cdot -4\right)} \]

   12. sub-neg 4.1%

       \[\leadsto \color{blue}{a1 \cdot -4 - a2 \cdot -4} \]

   13. distribute-rgt-out-- 4.1%

       \[\leadsto \color{blue}{-4 \cdot \left(a1 - a2\right)} \]

9. Simplified 4.1%

   \[\leadsto \color{blue}{-4 \cdot \left(a1 - a2\right)} \]

10. Final simplification 4.1%

    \[\leadsto \left(a1 - a2\right) \cdot -4 \]
11. Add Preprocessing

Alternative 15: 3.6% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a2

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 66.9%

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

7. Applied egg-rr 4.3%

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

8. Taylor expanded in a2 around inf 3.9%

   \[\leadsto \color{blue}{a2} \]
9. Add Preprocessing

Alternative 16: 3.6% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a1

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 66.9%

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

7. Applied egg-rr 4.3%

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

8. Taylor expanded in a2 around 0 3.8%

   \[\leadsto \color{blue}{a1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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))))