Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 9.1s

Alternatives: 12

Speedup: 2.0×

• 43.6% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.7%

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

   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. Add Preprocessing

Alternative 2: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;t\_1 \cdot \left(\cos th \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.68)
     (* t_1 (* (cos th) 0.5))
     (* t_1 (/ 1.0 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.68) {
		tmp = t_1 * (cos(th) * 0.5);
	} else {
		tmp = t_1 * (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) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.68d0) then
        tmp = t_1 * (cos(th) * 0.5d0)
    else
        tmp = t_1 * (1.0d0 / sqrt(2.0d0))
    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 (Math.cos(th) <= 0.68) {
		tmp = t_1 * (Math.cos(th) * 0.5);
	} else {
		tmp = t_1 * (1.0 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.68:
		tmp = t_1 * (math.cos(th) * 0.5)
	else:
		tmp = t_1 * (1.0 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.68)
		tmp = Float64(t_1 * Float64(cos(th) * 0.5));
	else
		tmp = Float64(t_1 * Float64(1.0 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.68)
		tmp = t_1 * (cos(th) * 0.5);
	else
		tmp = t_1 * (1.0 / sqrt(2.0));
	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[N[Cos[th], $MachinePrecision], 0.68], N[(t$95$1 * N[(N[Cos[th], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.680000000000000049

   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. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   8. Applied egg-rr 60.2%

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

   9. Taylor expanded in th around inf 60.2%

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

   if 0.680000000000000049 < (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 93.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{1}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;t\_1 \cdot \left(\cos th \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.68) (* t_1 (* (cos th) 0.5)) (* t_1 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.68) {
		tmp = t_1 * (cos(th) * 0.5);
	} else {
		tmp = t_1 * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.68d0) then
        tmp = t_1 * (cos(th) * 0.5d0)
    else
        tmp = t_1 * sqrt(0.5d0)
    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 (Math.cos(th) <= 0.68) {
		tmp = t_1 * (Math.cos(th) * 0.5);
	} else {
		tmp = t_1 * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.68:
		tmp = t_1 * (math.cos(th) * 0.5)
	else:
		tmp = t_1 * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.68)
		tmp = Float64(t_1 * Float64(cos(th) * 0.5));
	else
		tmp = Float64(t_1 * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.68)
		tmp = t_1 * (cos(th) * 0.5);
	else
		tmp = t_1 * sqrt(0.5);
	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[N[Cos[th], $MachinePrecision], 0.68], N[(t$95$1 * N[(N[Cos[th], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.680000000000000049

   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. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   8. Applied egg-rr 60.2%

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

   9. Taylor expanded in th around inf 60.2%

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

   if 0.680000000000000049 < (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 99.5%

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

      5. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in th around 0 92.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.68) (* (cos th) t_1) (* t_1 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.68) {
		tmp = cos(th) * t_1;
	} else {
		tmp = t_1 * sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.68d0) then
        tmp = cos(th) * t_1
    else
        tmp = t_1 * sqrt(0.5d0)
    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 (Math.cos(th) <= 0.68) {
		tmp = Math.cos(th) * t_1;
	} else {
		tmp = t_1 * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.68:
		tmp = math.cos(th) * t_1
	else:
		tmp = t_1 * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.68)
		tmp = Float64(cos(th) * t_1);
	else
		tmp = Float64(t_1 * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.68)
		tmp = cos(th) * t_1;
	else
		tmp = t_1 * sqrt(0.5);
	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[N[Cos[th], $MachinePrecision], 0.68], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.680000000000000049

   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. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   8. Applied egg-rr 60.2%

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

      1. +-lft-identity 60.2%

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

   10. Simplified 60.2%

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

   if 0.680000000000000049 < (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 99.5%

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

      5. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in th around 0 92.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $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. Add Preprocessing

Alternative 6: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $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. Final simplification 99.6%

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

Alternative 7: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.7%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 78.2%

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

   3. frac-times 78.1%

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

   4. pow2 78.1%

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

   5. add-sqr-sqrt 78.1%

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

   6. pow2 78.1%

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

   7. fma-undefine 78.1%

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

   8. hypot-define 78.1%

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

   9. rem-square-sqrt 78.3%

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

6. Applied egg-rr 78.3%

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

   1. unpow2 78.3%

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

   2. pow-sqr 78.3%

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

   3. hypot-undefine 78.3%

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

   4. unpow2 78.3%

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

   5. unpow2 78.3%

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

   6. +-commutative 78.3%

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

   7. unpow2 78.3%

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

   8. unpow2 78.3%

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

   9. hypot-define 78.3%

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

   10. metadata-eval 78.3%

       \[\leadsto \cos th \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{\color{blue}{4}}}{2}} \]

8. Simplified 78.3%

   \[\leadsto \cos th \cdot \color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{4}}{2}}} \]

9. Taylor expanded in a1 around 0 58.4%

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

    1. *-commutative 58.4%

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

    2. associate-*l* 58.4%

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

    3. *-commutative 58.4%

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

11. Simplified 58.4%

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

    1. pow2 58.4%

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

13. Applied egg-rr 58.4%

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

Alternative 8: 59.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 59.8%

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

   1. +-lft-identity 59.8%

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

10. Simplified 59.8%

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

Alternative 9: 46.1% accurate, 46.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * 0.5), $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. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 59.9%

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

9. Taylor expanded in th around 0 48.2%

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

10. Final simplification 48.2%

    \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5 \]
11. Add Preprocessing

Alternative 10: 30.0% accurate, 138.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * 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. Step-by-step derivation

   1. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 48.0%

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

   1. *-inverses 48.0%

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

10. Simplified 48.0%

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

11. Taylor expanded in a1 around 0 28.4%

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

    1. pow2 58.4%

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

13. Applied egg-rr 28.4%

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

Alternative 11: 3.6% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a2 \cdot 4 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.7%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 78.2%

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

   3. frac-times 78.1%

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

   4. pow2 78.1%

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

   5. add-sqr-sqrt 78.1%

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

   6. pow2 78.1%

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

   7. fma-undefine 78.1%

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

   8. hypot-define 78.1%

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

   9. rem-square-sqrt 78.3%

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

6. Applied egg-rr 78.3%

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

   1. unpow2 78.3%

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

   2. pow-sqr 78.3%

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

   3. hypot-undefine 78.3%

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

   4. unpow2 78.3%

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

   5. unpow2 78.3%

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

   6. +-commutative 78.3%

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

   7. unpow2 78.3%

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

   8. unpow2 78.3%

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

   9. hypot-define 78.3%

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

   10. metadata-eval 78.3%

       \[\leadsto \cos th \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{\color{blue}{4}}}{2}} \]

8. Simplified 78.3%

   \[\leadsto \cos th \cdot \color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{4}}{2}}} \]

10. Applied egg-rr 3.8%

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

    1. fmm-undef 3.8%

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

    2. *-commutative 3.8%

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

    3. associate-+l- 3.8%

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

    4. fma-undefine 3.8%

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

    5. distribute-lft-neg-in 3.8%

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

    6. distribute-rgt-neg-in 3.8%

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

    7. metadata-eval 3.8%

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

    8. distribute-lft-out 3.8%

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

    9. metadata-eval 3.8%

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

    10. distribute-lft-out-- 3.8%

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

    11. metadata-eval 3.8%

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

    12. distribute-rgt-out-- 3.8%

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

12. Simplified 3.8%

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

13. Taylor expanded in th around 0 4.3%

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

14. Taylor expanded in a1 around 0 3.9%

    \[\leadsto \color{blue}{4 \cdot a2} \]
15. Step-by-step derivation

    1. *-commutative 3.9%

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

16. Simplified 3.9%

    \[\leadsto \color{blue}{a2 \cdot 4} \]
17. Add Preprocessing

Alternative 12: 3.7% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 \cdot -4 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.7%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 78.2%

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

   3. frac-times 78.1%

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

   4. pow2 78.1%

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

   5. add-sqr-sqrt 78.1%

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

   6. pow2 78.1%

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

   7. fma-undefine 78.1%

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

   8. hypot-define 78.1%

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

   9. rem-square-sqrt 78.3%

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

6. Applied egg-rr 78.3%

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

   1. unpow2 78.3%

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

   2. pow-sqr 78.3%

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

   3. hypot-undefine 78.3%

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

   4. unpow2 78.3%

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

   5. unpow2 78.3%

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

   6. +-commutative 78.3%

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

   7. unpow2 78.3%

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

   8. unpow2 78.3%

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

   9. hypot-define 78.3%

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

   10. metadata-eval 78.3%

       \[\leadsto \cos th \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{\color{blue}{4}}}{2}} \]

8. Simplified 78.3%

   \[\leadsto \cos th \cdot \color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{4}}{2}}} \]

10. Applied egg-rr 3.8%

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

    1. fmm-undef 3.8%

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

    2. *-commutative 3.8%

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

    3. associate-+l- 3.8%

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

    4. fma-undefine 3.8%

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

    5. distribute-lft-neg-in 3.8%

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

    6. distribute-rgt-neg-in 3.8%

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

    7. metadata-eval 3.8%

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

    8. distribute-lft-out 3.8%

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

    9. metadata-eval 3.8%

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

    10. distribute-lft-out-- 3.8%

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

    11. metadata-eval 3.8%

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

    12. distribute-rgt-out-- 3.8%

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

12. Simplified 3.8%

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

13. Taylor expanded in th around 0 4.3%

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

14. Taylor expanded in a1 around inf 3.4%

    \[\leadsto \color{blue}{-4 \cdot a1} \]
15. Step-by-step derivation

    1. *-commutative 3.4%

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

16. Simplified 3.4%

    \[\leadsto \color{blue}{a1 \cdot -4} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))))