Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.6s

Alternatives: 13

Speedup: 2.0×

• 44.6% 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, 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 99.6%

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

   5. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 72.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.625)
   (* a2 (* (cos th) a2))
   (* (+ (* a1 a1) (* a2 a2)) (/ 1.0 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.625) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.625d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (1.0d0 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.625) {
		tmp = a2 * (Math.cos(th) * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.625:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.625)
		tmp = Float64(a2 * Float64(cos(th) * a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64(1.0 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.625)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * (1.0 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.625], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.625

   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 a1 around 0 51.0%

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

      1. associate-/l* 51.0%

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

   7. Simplified 51.0%

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

   9. Applied egg-rr 32.6%

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

   if 0.625 < (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. Taylor expanded in th around 0 93.4%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.625)
   (* a2 (* (cos th) a2))
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.625) {
		tmp = a2 * (cos(th) * 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) <= 0.625d0) then
        tmp = a2 * (cos(th) * 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) <= 0.625) {
		tmp = a2 * (Math.cos(th) * 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) <= 0.625:
		tmp = a2 * (math.cos(th) * 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) <= 0.625)
		tmp = Float64(a2 * Float64(cos(th) * 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) <= 0.625)
		tmp = a2 * (cos(th) * 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], 0.625], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $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) < 0.625

   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 a1 around 0 51.0%

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

      1. associate-/l* 51.0%

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

   7. Simplified 51.0%

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

   9. Applied egg-rr 32.6%

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

   if 0.625 < (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 99.6%

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

      5. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0 93.4%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.625) (* a2 (* (cos th) a2)) (* a2 (* a2 (pow 2.0 -0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.625) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = a2 * (a2 * pow(2.0, -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) <= 0.625d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = a2 * (a2 * (2.0d0 ** (-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) <= 0.625) {
		tmp = a2 * (Math.cos(th) * a2);
	} else {
		tmp = a2 * (a2 * Math.pow(2.0, -0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.625:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = a2 * (a2 * math.pow(2.0, -0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.625)
		tmp = Float64(a2 * Float64(cos(th) * a2));
	else
		tmp = Float64(a2 * Float64(a2 * (2.0 ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.625)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = a2 * (a2 * (2.0 ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.625], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.625

   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 a1 around 0 51.0%

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

      1. associate-/l* 51.0%

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

   7. Simplified 51.0%

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

   9. Applied egg-rr 32.6%

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

   if 0.625 < (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. Taylor expanded in th around 0 93.4%

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

   6. Taylor expanded in a1 around 0 53.5%

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

      1. pow2 53.5%

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

      2. div-inv 53.5%

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

      3. associate-*l* 53.5%

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

      4. pow1/2 53.5%

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

      5. pow-flip 53.5%

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

      6. metadata-eval 53.5%

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

   8. Applied egg-rr 53.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.625) (* a2 (* (cos th) a2)) (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.625) {
		tmp = a2 * (cos(th) * a2);
	} 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) <= 0.625d0) then
        tmp = a2 * (cos(th) * a2)
    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) <= 0.625) {
		tmp = a2 * (Math.cos(th) * a2);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.625:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.625)
		tmp = Float64(a2 * Float64(cos(th) * a2));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.625)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.625], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.625

   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 a1 around 0 51.0%

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

      1. associate-/l* 51.0%

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

   7. Simplified 51.0%

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

   9. Applied egg-rr 32.6%

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

   if 0.625 < (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. Taylor expanded in th around 0 93.4%

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

   6. Taylor expanded in a1 around 0 53.5%

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

      1. pow2 53.5%

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

      2. *-un-lft-identity 53.5%

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

      3. times-frac 53.5%

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

   8. Applied egg-rr 53.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.625:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $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. Taylor expanded in a1 around 0 53.5%

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

   1. associate-/l* 53.6%

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

7. Simplified 53.6%

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

9. Applied egg-rr 35.5%

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

10. Final simplification 35.5%

    \[\leadsto a2 \cdot \left(\cos th \cdot a2\right) \]
11. Add Preprocessing

Alternative 7: 46.7% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2050000000000 \lor \neg \left(th \leq 5.5 \cdot 10^{+76}\right) \land th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 2050000000000.0) (and (not (<= th 5.5e+76)) (<= th 2.4e+150)))
   (* (+ a1 a2) (+ a1 a2))
   (* (+ (* a1 a1) (* a2 a2)) -0.75)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 2050000000000.0) || (!(th <= 5.5e+76) && (th <= 2.4e+150))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.75;
	}
	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 <= 2050000000000.0d0) .or. (.not. (th <= 5.5d+76)) .and. (th <= 2.4d+150)) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (-0.75d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 2050000000000.0) || (!(th <= 5.5e+76) && (th <= 2.4e+150))) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.75;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 2050000000000.0) or (not (th <= 5.5e+76) and (th <= 2.4e+150)):
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.75
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 2050000000000.0) || (!(th <= 5.5e+76) && (th <= 2.4e+150)))
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 2050000000000.0) || (~((th <= 5.5e+76)) && (th <= 2.4e+150)))
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 2050000000000.0], And[N[Not[LessEqual[th, 5.5e+76]], $MachinePrecision], LessEqual[th, 2.4e+150]]], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.75), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 2.05e12 or 5.5000000000000001e76 < th < 2.40000000000000003e150

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 72.0%

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

   7. Applied egg-rr 44.2%

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

      1. distribute-lft-out 48.5%

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

   9. Simplified 48.5%

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

   if 2.05e12 < th < 5.5000000000000001e76 or 2.40000000000000003e150 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 32.9%

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

   7. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{-0.75} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2050000000000 \lor \neg \left(th \leq 5.5 \cdot 10^{+76}\right) \land th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.75\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ t_2 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;t\_2 \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (* (+ a1 a2) (+ a1 a2))) (t_2 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 2050000000000.0)
     t_1
     (if (<= th 5.5e+76)
       (* t_2 -0.75)
       (if (<= th 2.4e+150) t_1 (* t_2 -0.5))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 + a2) * (a1 + a2);
	double t_2 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 2050000000000.0) {
		tmp = t_1;
	} else if (th <= 5.5e+76) {
		tmp = t_2 * -0.75;
	} else if (th <= 2.4e+150) {
		tmp = t_1;
	} else {
		tmp = t_2 * -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) :: t_2
    real(8) :: tmp
    t_1 = (a1 + a2) * (a1 + a2)
    t_2 = (a1 * a1) + (a2 * a2)
    if (th <= 2050000000000.0d0) then
        tmp = t_1
    else if (th <= 5.5d+76) then
        tmp = t_2 * (-0.75d0)
    else if (th <= 2.4d+150) then
        tmp = t_1
    else
        tmp = t_2 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 + a2) * (a1 + a2);
	double t_2 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 2050000000000.0) {
		tmp = t_1;
	} else if (th <= 5.5e+76) {
		tmp = t_2 * -0.75;
	} else if (th <= 2.4e+150) {
		tmp = t_1;
	} else {
		tmp = t_2 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 + a2) * (a1 + a2)
	t_2 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 2050000000000.0:
		tmp = t_1
	elif th <= 5.5e+76:
		tmp = t_2 * -0.75
	elif th <= 2.4e+150:
		tmp = t_1
	else:
		tmp = t_2 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 + a2) * Float64(a1 + a2))
	t_2 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 2050000000000.0)
		tmp = t_1;
	elseif (th <= 5.5e+76)
		tmp = Float64(t_2 * -0.75);
	elseif (th <= 2.4e+150)
		tmp = t_1;
	else
		tmp = Float64(t_2 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 + a2) * (a1 + a2);
	t_2 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 2050000000000.0)
		tmp = t_1;
	elseif (th <= 5.5e+76)
		tmp = t_2 * -0.75;
	elseif (th <= 2.4e+150)
		tmp = t_1;
	else
		tmp = t_2 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 2050000000000.0], t$95$1, If[LessEqual[th, 5.5e+76], N[(t$95$2 * -0.75), $MachinePrecision], If[LessEqual[th, 2.4e+150], t$95$1, N[(t$95$2 * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.05e12 or 5.5000000000000001e76 < th < 2.40000000000000003e150

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 72.0%

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

   7. Applied egg-rr 44.2%

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

      1. distribute-lft-out 48.5%

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

   9. Simplified 48.5%

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

   if 2.05e12 < th < 5.5000000000000001e76

   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 23.7%

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

   7. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{-0.75} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 2.40000000000000003e150 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 35.8%

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

   7. Applied egg-rr 45.0%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.7% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ t_2 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;t\_2 \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (* (+ a1 a2) (+ a1 a2))) (t_2 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 2050000000000.0)
     t_1
     (if (<= th 5.5e+76)
       (* t_2 -0.75)
       (if (<= th 2.4e+150) t_1 (* t_2 -0.25))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 + a2) * (a1 + a2);
	double t_2 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 2050000000000.0) {
		tmp = t_1;
	} else if (th <= 5.5e+76) {
		tmp = t_2 * -0.75;
	} else if (th <= 2.4e+150) {
		tmp = t_1;
	} else {
		tmp = t_2 * -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) :: t_2
    real(8) :: tmp
    t_1 = (a1 + a2) * (a1 + a2)
    t_2 = (a1 * a1) + (a2 * a2)
    if (th <= 2050000000000.0d0) then
        tmp = t_1
    else if (th <= 5.5d+76) then
        tmp = t_2 * (-0.75d0)
    else if (th <= 2.4d+150) then
        tmp = t_1
    else
        tmp = t_2 * (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 + a2) * (a1 + a2);
	double t_2 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 2050000000000.0) {
		tmp = t_1;
	} else if (th <= 5.5e+76) {
		tmp = t_2 * -0.75;
	} else if (th <= 2.4e+150) {
		tmp = t_1;
	} else {
		tmp = t_2 * -0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 + a2) * (a1 + a2)
	t_2 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 2050000000000.0:
		tmp = t_1
	elif th <= 5.5e+76:
		tmp = t_2 * -0.75
	elif th <= 2.4e+150:
		tmp = t_1
	else:
		tmp = t_2 * -0.25
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 + a2) * Float64(a1 + a2))
	t_2 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 2050000000000.0)
		tmp = t_1;
	elseif (th <= 5.5e+76)
		tmp = Float64(t_2 * -0.75);
	elseif (th <= 2.4e+150)
		tmp = t_1;
	else
		tmp = Float64(t_2 * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 + a2) * (a1 + a2);
	t_2 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 2050000000000.0)
		tmp = t_1;
	elseif (th <= 5.5e+76)
		tmp = t_2 * -0.75;
	elseif (th <= 2.4e+150)
		tmp = t_1;
	else
		tmp = t_2 * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 2050000000000.0], t$95$1, If[LessEqual[th, 5.5e+76], N[(t$95$2 * -0.75), $MachinePrecision], If[LessEqual[th, 2.4e+150], t$95$1, N[(t$95$2 * -0.25), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if th < 2.05e12 or 5.5000000000000001e76 < th < 2.40000000000000003e150

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 72.0%

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

   7. Applied egg-rr 44.2%

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

      1. distribute-lft-out 48.5%

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

   9. Simplified 48.5%

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

   if 2.05e12 < th < 5.5000000000000001e76

   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 23.7%

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

   7. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{-0.75} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 2.40000000000000003e150 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 35.8%

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

   7. Applied egg-rr 45.0%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.7% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;t\_1 \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;t\_1 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 2050000000000.0)
     (* (+ a1 a2) (+ a1 a2))
     (if (<= th 5.5e+76)
       (* t_1 -0.75)
       (if (<= th 2.4e+150) (* 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 <= 2050000000000.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 5.5e+76) {
		tmp = t_1 * -0.75;
	} else if (th <= 2.4e+150) {
		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 <= 2050000000000.0d0) then
        tmp = (a1 + a2) * (a1 + a2)
    else if (th <= 5.5d+76) then
        tmp = t_1 * (-0.75d0)
    else if (th <= 2.4d+150) 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 <= 2050000000000.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 5.5e+76) {
		tmp = t_1 * -0.75;
	} else if (th <= 2.4e+150) {
		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 <= 2050000000000.0:
		tmp = (a1 + a2) * (a1 + a2)
	elif th <= 5.5e+76:
		tmp = t_1 * -0.75
	elif th <= 2.4e+150:
		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 <= 2050000000000.0)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	elseif (th <= 5.5e+76)
		tmp = Float64(t_1 * -0.75);
	elseif (th <= 2.4e+150)
		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 <= 2050000000000.0)
		tmp = (a1 + a2) * (a1 + a2);
	elseif (th <= 5.5e+76)
		tmp = t_1 * -0.75;
	elseif (th <= 2.4e+150)
		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, 2050000000000.0], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 5.5e+76], N[(t$95$1 * -0.75), $MachinePrecision], If[LessEqual[th, 2.4e+150], N[(t$95$1 * 0.25), $MachinePrecision], N[(t$95$1 * -0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if th < 2.05e12

   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 73.9%

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

   7. Applied egg-rr 44.1%

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

      1. distribute-lft-out 48.7%

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

   9. Simplified 48.7%

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

   if 2.05e12 < th < 5.5000000000000001e76

   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 23.7%

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

   7. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{-0.75} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 5.5000000000000001e76 < th < 2.40000000000000003e150

   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 46.7%

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

   7. Applied egg-rr 45.8%

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

   if 2.40000000000000003e150 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 35.8%

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

   7. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.8% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;t\_1 \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 2050000000000.0)
     (* (+ a1 a2) (+ a1 a2))
     (if (<= th 5.5e+76)
       (* t_1 -0.75)
       (if (<= th 2.4e+150) (* 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 <= 2050000000000.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 5.5e+76) {
		tmp = t_1 * -0.75;
	} else if (th <= 2.4e+150) {
		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 <= 2050000000000.0d0) then
        tmp = (a1 + a2) * (a1 + a2)
    else if (th <= 5.5d+76) then
        tmp = t_1 * (-0.75d0)
    else if (th <= 2.4d+150) 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 <= 2050000000000.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else if (th <= 5.5e+76) {
		tmp = t_1 * -0.75;
	} else if (th <= 2.4e+150) {
		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 <= 2050000000000.0:
		tmp = (a1 + a2) * (a1 + a2)
	elif th <= 5.5e+76:
		tmp = t_1 * -0.75
	elif th <= 2.4e+150:
		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 <= 2050000000000.0)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	elseif (th <= 5.5e+76)
		tmp = Float64(t_1 * -0.75);
	elseif (th <= 2.4e+150)
		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 <= 2050000000000.0)
		tmp = (a1 + a2) * (a1 + a2);
	elseif (th <= 5.5e+76)
		tmp = t_1 * -0.75;
	elseif (th <= 2.4e+150)
		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[LessEqual[th, 2050000000000.0], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 5.5e+76], N[(t$95$1 * -0.75), $MachinePrecision], If[LessEqual[th, 2.4e+150], N[(0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * -0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if th < 2.05e12

   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 73.9%

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

   7. Applied egg-rr 44.1%

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

      1. distribute-lft-out 48.7%

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

   9. Simplified 48.7%

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

   if 2.05e12 < th < 5.5000000000000001e76

   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 23.7%

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

   7. Applied egg-rr 47.3%

      \[\leadsto \color{blue}{-0.75} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 5.5000000000000001e76 < th < 2.40000000000000003e150

   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 46.7%

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

   7. Applied egg-rr 46.1%

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

   if 2.40000000000000003e150 < th

   1. Initial program 99.4%

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

      1. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in th around 0 35.8%

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

   7. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{-0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 2050000000000:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{elif}\;th \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.75\\ \mathbf{elif}\;th \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.0% accurate, 59.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $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. Taylor expanded in th around 0 64.9%

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

7. Applied egg-rr 41.5%

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

   1. distribute-lft-out 45.4%

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

9. Simplified 45.4%

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

10. Final simplification 45.4%

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

Alternative 13: 4.1% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 + a2 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 64.9%

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

7. Applied egg-rr 3.8%

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

8. Final simplification 3.8%

   \[\leadsto a1 + a2 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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))))