Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 7.7s

Alternatives: 14

Speedup: 2.0×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 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 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.7%

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

   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: 71.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* 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) <= 0.7) {
		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) <= 0.7d0) 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) <= 0.7) {
		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) <= 0.7:
		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) <= 0.7)
		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) <= 0.7)
		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], 0.7], 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) < 0.69999999999999996

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. cos-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-/l* 99.5%

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

      5. cos-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a2 around inf 59.2%

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

   7. Applied egg-rr 41.1%

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

   if 0.69999999999999996 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.7%

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

      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 91.7%

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

4. Final simplification 71.9%

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

Alternative 3: 57.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-*l/ 99.6%

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

   4. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 58.8%

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

7. Applied egg-rr 58.8%

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

8. Final simplification 58.8%

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

Alternative 4: 44.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 850:\\ \;\;\;\;a1 \cdot a1 + a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;-{a2}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 850.0) (+ (* a1 a1) (* a2 a2)) (- (pow a2 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 850.0) {
		tmp = (a1 * a1) + (a2 * a2);
	} else {
		tmp = -pow(a2, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 850.0d0) then
        tmp = (a1 * a1) + (a2 * a2)
    else
        tmp = -(a2 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 850.0) {
		tmp = (a1 * a1) + (a2 * a2);
	} else {
		tmp = -Math.pow(a2, 2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 850.0:
		tmp = (a1 * a1) + (a2 * a2)
	else:
		tmp = -math.pow(a2, 2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 850.0)
		tmp = Float64(Float64(a1 * a1) + Float64(a2 * a2));
	else
		tmp = Float64(-(a2 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 850.0)
		tmp = (a1 * a1) + (a2 * a2);
	else
		tmp = -(a2 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 850.0], N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision], (-N[Power[a2, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if th < 850

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.7%

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

      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 53.8%

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

      1. *-inverses 53.8%

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

   10. Simplified 53.8%

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

   if 850 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 22.9%

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

   7. Applied egg-rr 24.7%

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

   8. Taylor expanded in a1 around 0 30.4%

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

      1. neg-mul-1 30.4%

         \[\leadsto \color{blue}{-{a2}^{2}} \]

   10. Simplified 30.4%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 850:\\ \;\;\;\;a1 \cdot a1 + a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;-{a2}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.8% 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 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-*l/ 99.6%

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

   4. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 58.8%

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

7. Applied egg-rr 39.3%

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

8. Final simplification 39.3%

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

Alternative 6: 46.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 8:\\ \;\;\;\;a1 \cdot a1 + a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(-a2\right) - a1 \cdot a1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 8.0) (+ (* a1 a1) (* a2 a2)) (- (* a2 (- a2)) (* a1 a1))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 8.0) {
		tmp = (a1 * a1) + (a2 * a2);
	} else {
		tmp = (a2 * -a2) - (a1 * a1);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 8.0d0) then
        tmp = (a1 * a1) + (a2 * a2)
    else
        tmp = (a2 * -a2) - (a1 * a1)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 8.0) {
		tmp = (a1 * a1) + (a2 * a2);
	} else {
		tmp = (a2 * -a2) - (a1 * a1);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 8.0:
		tmp = (a1 * a1) + (a2 * a2)
	else:
		tmp = (a2 * -a2) - (a1 * a1)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 8.0)
		tmp = Float64(Float64(a1 * a1) + Float64(a2 * a2));
	else
		tmp = Float64(Float64(a2 * Float64(-a2)) - Float64(a1 * a1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 8.0)
		tmp = (a1 * a1) + (a2 * a2);
	else
		tmp = (a2 * -a2) - (a1 * a1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 8.0], N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * (-a2)), $MachinePrecision] - N[(a1 * a1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 8

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.7%

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

      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 53.6%

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

      1. *-inverses 53.6%

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

   10. Simplified 53.6%

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

   if 8 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      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%

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

   8. Applied egg-rr 44.7%

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

      1. +-commutative 44.7%

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

      2. +-inverses 44.7%

         \[\leadsto \frac{\cos \color{blue}{0} + \cos \left(th + th\right)}{-2} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      3. cos-0 44.7%

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

      4. count-2 44.7%

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

      5. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   11. Taylor expanded in th around 0 44.1%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 8:\\ \;\;\;\;a1 \cdot a1 + a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(-a2\right) - a1 \cdot a1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.7% accurate, 31.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 8:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(-a2\right) - a1 \cdot a1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 8.0) (* (+ a1 a2) (+ a1 a2)) (- (* a2 (- a2)) (* a1 a1))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 8.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = (a2 * -a2) - (a1 * a1);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 8.0d0) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = (a2 * -a2) - (a1 * a1)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 8.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = (a2 * -a2) - (a1 * a1);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 8.0:
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = (a2 * -a2) - (a1 * a1)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 8.0)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(a2 * Float64(-a2)) - Float64(a1 * a1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 8.0)
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = (a2 * -a2) - (a1 * a1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 8.0], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * (-a2)), $MachinePrecision] - N[(a1 * a1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 8

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 78.9%

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

   7. Applied egg-rr 46.2%

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

      1. distribute-lft-out 53.5%

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

   9. Simplified 53.5%

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

   if 8 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      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%

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

   8. Applied egg-rr 44.7%

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

      1. +-commutative 44.7%

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

      2. +-inverses 44.7%

         \[\leadsto \frac{\cos \color{blue}{0} + \cos \left(th + th\right)}{-2} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      3. cos-0 44.7%

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

      4. count-2 44.7%

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

      5. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   11. Taylor expanded in th around 0 44.1%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 8:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(-a2\right) - a1 \cdot a1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.3% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 850:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a2 \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 850.0) (* (+ a1 a2) (+ a1 a2)) (* (+ a1 a2) (* a2 -2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 850.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = (a1 + a2) * (a2 * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 850.0d0) then
        tmp = (a1 + a2) * (a1 + a2)
    else
        tmp = (a1 + a2) * (a2 * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 850.0) {
		tmp = (a1 + a2) * (a1 + a2);
	} else {
		tmp = (a1 + a2) * (a2 * -2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 850.0:
		tmp = (a1 + a2) * (a1 + a2)
	else:
		tmp = (a1 + a2) * (a2 * -2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 850.0)
		tmp = Float64(Float64(a1 + a2) * Float64(a1 + a2));
	else
		tmp = Float64(Float64(a1 + a2) * Float64(a2 * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 850.0)
		tmp = (a1 + a2) * (a1 + a2);
	else
		tmp = (a1 + a2) * (a2 * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 850.0], N[(N[(a1 + a2), $MachinePrecision] * N[(a1 + a2), $MachinePrecision]), $MachinePrecision], N[(N[(a1 + a2), $MachinePrecision] * N[(a2 * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 850

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 79.1%

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

   7. Applied egg-rr 46.5%

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

      1. distribute-lft-out 53.8%

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

   9. Simplified 53.8%

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

   if 850 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 22.9%

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

   7. Applied egg-rr 39.7%

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

      1. distribute-lft-out 44.4%

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

      2. +-commutative 44.4%

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

      3. distribute-rgt-out 44.4%

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

   9. Simplified 44.4%

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

   10. Taylor expanded in a2 around inf 36.6%

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

       1. *-commutative 36.6%

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

   12. Simplified 36.6%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 850:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a2 \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 7.7% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 8:\\ \;\;\;\;-2 \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 8.0) (* -2.0 (- a1 a2)) (- a1 (* a2 a2))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 8.0) {
		tmp = -2.0 * (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 <= 8.0d0) then
        tmp = (-2.0d0) * (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 <= 8.0) {
		tmp = -2.0 * (a1 - a2);
	} else {
		tmp = a1 - (a2 * a2);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 8.0:
		tmp = -2.0 * (a1 - a2)
	else:
		tmp = a1 - (a2 * a2)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 8.0)
		tmp = Float64(-2.0 * 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 <= 8.0)
		tmp = -2.0 * (a1 - a2);
	else
		tmp = a1 - (a2 * a2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 8.0], N[(-2.0 * N[(a1 - a2), $MachinePrecision]), $MachinePrecision], N[(a1 - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 8

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 78.9%

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

   7. Applied egg-rr 3.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, a1, -a2 \cdot -2\right) + \mathsf{fma}\left(-a2, -2, a2 \cdot -2\right)} \]
   8. Step-by-step derivation

      1. fma-neg 3.5%

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

      2. *-commutative 3.5%

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

      3. associate-+l- 3.5%

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

      4. fma-undefine 3.5%

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

      5. distribute-lft-neg-in 3.5%

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

      6. distribute-rgt-neg-in 3.5%

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

      7. metadata-eval 3.5%

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

      8. distribute-lft-out 3.5%

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

      9. metadata-eval 3.5%

         \[\leadsto a1 \cdot -2 - \left(a2 \cdot -2 - a2 \cdot \color{blue}{0}\right) \]

      10. distribute-lft-out-- 3.5%

          \[\leadsto a1 \cdot -2 - \color{blue}{a2 \cdot \left(-2 - 0\right)} \]

      11. metadata-eval 3.5%

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

      12. distribute-rgt-out-- 3.5%

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

   9. Simplified 3.5%

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

   if 8 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in th around 0 24.1%

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

   7. Applied egg-rr 24.3%

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

4. Final simplification 8.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 8:\\ \;\;\;\;-2 \cdot \left(a1 - a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 17.7% accurate, 59.3× speedup

Math

\[\begin{array}{l} \\ \left(a1 + a2\right) \cdot \left(a2 \cdot -2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (+ a1 a2) (* a2 -2.0)))

C

double code(double a1, double a2, double th) {
	return (a1 + a2) * (a2 * -2.0);
}

Fortran

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

Java

public static double code(double a1, double a2, double th) {
	return (a1 + a2) * (a2 * -2.0);
}

Python

def code(a1, a2, th):
	return (a1 + a2) * (a2 * -2.0)

Julia

function code(a1, a2, th)
	return Float64(Float64(a1 + a2) * Float64(a2 * -2.0))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a1 + a2) * (a2 * -2.0);
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in th around 0 65.0%

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

7. Applied egg-rr 21.9%

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

   1. distribute-lft-out 23.9%

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

   2. +-commutative 23.9%

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

   3. distribute-rgt-out 23.9%

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

9. Simplified 23.9%

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

10. Taylor expanded in a2 around inf 24.8%

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

    1. *-commutative 24.8%

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

12. Simplified 24.8%

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

13. Final simplification 24.8%

    \[\leadsto \left(a1 + a2\right) \cdot \left(a2 \cdot -2\right) \]
14. Add Preprocessing

Alternative 11: 4.2% accurate, 83.0× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(a1 - a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* -2.0 (- a1 a2)))

C

double code(double a1, double a2, double th) {
	return -2.0 * (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 = (-2.0d0) * (a1 - a2)
end function

Java

public static double code(double a1, double a2, double th) {
	return -2.0 * (a1 - a2);
}

Python

def code(a1, a2, th):
	return -2.0 * (a1 - a2)

Julia

function code(a1, a2, th)
	return Float64(-2.0 * Float64(a1 - a2))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = -2.0 * (a1 - a2);
end

Wolfram

code[a1_, a2_, th_] := N[(-2.0 * N[(a1 - a2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in th around 0 65.0%

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

7. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, a1, -a2 \cdot -2\right) + \mathsf{fma}\left(-a2, -2, a2 \cdot -2\right)} \]
8. Step-by-step derivation

   1. fma-neg 4.1%

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

   2. *-commutative 4.1%

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

   3. associate-+l- 4.1%

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

   4. fma-undefine 4.1%

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

   5. distribute-lft-neg-in 4.1%

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

   6. distribute-rgt-neg-in 4.1%

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

   7. metadata-eval 4.1%

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

   8. distribute-lft-out 4.1%

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

   9. metadata-eval 4.1%

      \[\leadsto a1 \cdot -2 - \left(a2 \cdot -2 - a2 \cdot \color{blue}{0}\right) \]

   10. distribute-lft-out-- 4.1%

       \[\leadsto a1 \cdot -2 - \color{blue}{a2 \cdot \left(-2 - 0\right)} \]

   11. metadata-eval 4.1%

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

   12. distribute-rgt-out-- 4.1%

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

9. Simplified 4.1%

   \[\leadsto \color{blue}{-2 \cdot \left(a1 - a2\right)} \]
10. Add Preprocessing

Alternative 12: 3.7% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a2 \cdot 2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 2.0))

C

double code(double a1, double a2, double th) {
	return a2 * 2.0;
}

Fortran

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

Java

public static double code(double a1, double a2, double th) {
	return a2 * 2.0;
}

Python

def code(a1, a2, th):
	return a2 * 2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in th around 0 65.0%

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

7. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, a1, -a2 \cdot -2\right) + \mathsf{fma}\left(-a2, -2, a2 \cdot -2\right)} \]
8. Step-by-step derivation

   1. fma-neg 4.1%

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

   2. *-commutative 4.1%

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

   3. associate-+l- 4.1%

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

   4. fma-undefine 4.1%

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

   5. distribute-lft-neg-in 4.1%

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

   6. distribute-rgt-neg-in 4.1%

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

   7. metadata-eval 4.1%

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

   8. distribute-lft-out 4.1%

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

   9. metadata-eval 4.1%

      \[\leadsto a1 \cdot -2 - \left(a2 \cdot -2 - a2 \cdot \color{blue}{0}\right) \]

   10. distribute-lft-out-- 4.1%

       \[\leadsto a1 \cdot -2 - \color{blue}{a2 \cdot \left(-2 - 0\right)} \]

   11. metadata-eval 4.1%

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

   12. distribute-rgt-out-- 4.1%

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

9. Simplified 4.1%

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

10. Taylor expanded in a1 around 0 3.4%

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

11. Final simplification 3.4%

    \[\leadsto a2 \cdot 2 \]
12. Add Preprocessing

Alternative 13: 3.7% 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 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in th around 0 65.0%

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

7. Applied egg-rr 4.1%

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

8. Taylor expanded in a2 around inf 3.4%

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

Alternative 14: 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 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in th around 0 65.0%

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

7. Applied egg-rr 4.1%

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

8. Taylor expanded in a2 around 0 3.9%

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

Reproduce

herbie shell --seed 2024139 
(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))))