Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.9s

Alternatives: 12

Speedup: 2.0×

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left(\mathsf{hypot}\left(a2, a1\right) \cdot \left(\mathsf{hypot}\left(a2, a1\right) \cdot {2}^{-0.5}\right)\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (hypot a2 a1) (* (hypot a2 a1) (pow 2.0 -0.5)))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (hypot(a2, a1) * (hypot(a2, a1) * pow(2.0, -0.5)));
}

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (Math.hypot(a2, a1) * (Math.hypot(a2, a1) * Math.pow(2.0, -0.5)));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (math.hypot(a2, a1) * (math.hypot(a2, a1) * math.pow(2.0, -0.5)))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(hypot(a2, a1) * Float64(hypot(a2, a1) * (2.0 ^ -0.5))))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * (hypot(a2, a1) * (hypot(a2, a1) * (2.0 ^ -0.5)));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision] * N[(N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   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* 99.5%

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

   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. Step-by-step derivation

   1. div-inv 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. associate-*l* 99.5%

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

   4. fma-undefine 99.5%

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

   5. hypot-define 99.5%

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

   6. fma-undefine 99.5%

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

   7. hypot-define 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-flip 99.6%

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

   10. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \cos th \cdot \left(\mathsf{hypot}\left(a2, a1\right) \cdot \left(\mathsf{hypot}\left(a2, a1\right) \cdot {2}^{-0.5}\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left({2}^{-0.5} \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (pow 2.0 -0.5) (pow (hypot a2 a1) 2.0))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (pow(2.0, -0.5) * pow(hypot(a2, a1), 2.0));
}

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * (Math.pow(2.0, -0.5) * Math.pow(Math.hypot(a2, a1), 2.0));
}

Python

def code(a1, a2, th):
	return math.cos(th) * (math.pow(2.0, -0.5) * math.pow(math.hypot(a2, a1), 2.0))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64((2.0 ^ -0.5) * (hypot(a2, a1) ^ 2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((2.0 ^ -0.5) * (hypot(a2, a1) ^ 2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[2.0, -0.5], $MachinePrecision] * N[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $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%

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

   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* 99.5%

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

   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. Step-by-step derivation

   1. div-inv 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. pow2 99.5%

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

   4. fma-undefine 99.5%

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

   5. hypot-define 99.5%

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

   6. pow1/2 99.5%

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

   7. pow-flip 99.6%

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

   8. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \cos th \cdot \left({2}^{-0.5} \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}\right) \]
8. Add Preprocessing

Alternative 3: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.69:\\ \;\;\;\;t\_1 \cdot \left(\cos th \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.69) (* t_1 (* (cos th) (- -0.5))) (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.69) {
		tmp = t_1 * (cos(th) * -(-0.5));
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	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) <= 0.69d0) then
        tmp = t_1 * (cos(th) * -(-0.5d0))
    else
        tmp = sqrt(0.5d0) * t_1
    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) <= 0.69) {
		tmp = t_1 * (Math.cos(th) * -(-0.5));
	} else {
		tmp = Math.sqrt(0.5) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.69:
		tmp = t_1 * (math.cos(th) * -(-0.5))
	else:
		tmp = math.sqrt(0.5) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.69)
		tmp = Float64(t_1 * Float64(cos(th) * Float64(-(-0.5))));
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.69)
		tmp = t_1 * (cos(th) * -(-0.5));
	else
		tmp = sqrt(0.5) * t_1;
	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], 0.69], N[(t$95$1 * N[(N[Cos[th], $MachinePrecision] * (--0.5)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.68999999999999995

   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 99.5%

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

   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 61.0%

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

   if 0.68999999999999995 < (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. Taylor expanded in th around 0 91.2%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   6. Step-by-step derivation

      1. *-un-lft-identity 91.2%

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

      2. add-sqr-sqrt 91.2%

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

      3. sqrt-unprod 91.2%

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

      4. frac-times 91.2%

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

      5. metadata-eval 91.2%

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

      6. rem-square-sqrt 91.3%

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

      7. metadata-eval 91.3%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{0.5}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   8. Step-by-step derivation

      1. *-lft-identity 91.3%

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

   9. Simplified 91.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.69:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.692:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.692)
   (* (cos th) (+ (* a1 a1) (* a2 a2)))
   (* a2 (* a2 (pow 2.0 -0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.692) {
		tmp = cos(th) * ((a1 * a1) + (a2 * a2));
	} else {
		tmp = a2 * (a2 * pow(2.0, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.692d0) then
        tmp = cos(th) * ((a1 * a1) + (a2 * a2))
    else
        tmp = a2 * (a2 * (2.0d0 ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.692) {
		tmp = Math.cos(th) * ((a1 * a1) + (a2 * a2));
	} else {
		tmp = a2 * (a2 * Math.pow(2.0, -0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.692:
		tmp = math.cos(th) * ((a1 * a1) + (a2 * a2))
	else:
		tmp = a2 * (a2 * math.pow(2.0, -0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.692)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	else
		tmp = Float64(a2 * Float64(a2 * (2.0 ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.692)
		tmp = cos(th) * ((a1 * a1) + (a2 * a2));
	else
		tmp = a2 * (a2 * (2.0 ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.692], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69199999999999995

   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 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   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 60.5%

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

      1. +-lft-identity 60.5%

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

   10. Simplified 60.5%

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

   if 0.69199999999999995 < (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%

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

      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* 99.5%

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

      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 55.8%

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

      1. clear-num 55.7%

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

      2. inv-pow 55.7%

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

      3. pow2 55.7%

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

      4. *-commutative 55.7%

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

      5. associate-/r* 55.7%

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

      6. pow2 55.7%

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

   7. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\cos th}}{{a2}^{2}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 55.7%

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

      2. associate-/l/ 55.7%

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

   9. Simplified 55.7%

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

   10. Taylor expanded in th around 0 53.1%

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

       1. pow2 53.1%

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

       2. associate-/r/ 53.2%

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

       3. associate-*r* 53.3%

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

       4. pow1/2 53.3%

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

       5. pow-flip 53.3%

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

       6. metadata-eval 53.3%

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

   12. Applied egg-rr 53.3%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.692:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot {2}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.692:\\ \;\;\;\;\cos th \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.692) (* (cos th) t_1) (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.692) {
		tmp = cos(th) * t_1;
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	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) <= 0.692d0) then
        tmp = cos(th) * t_1
    else
        tmp = sqrt(0.5d0) * t_1
    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) <= 0.692) {
		tmp = Math.cos(th) * t_1;
	} else {
		tmp = Math.sqrt(0.5) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.692:
		tmp = math.cos(th) * t_1
	else:
		tmp = math.sqrt(0.5) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.692)
		tmp = Float64(cos(th) * t_1);
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.692)
		tmp = cos(th) * t_1;
	else
		tmp = sqrt(0.5) * t_1;
	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], 0.692], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69199999999999995

   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 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   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 60.5%

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

      1. +-lft-identity 60.5%

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

   10. Simplified 60.5%

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

   if 0.69199999999999995 < (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. Taylor expanded in th around 0 91.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   6. Step-by-step derivation

      1. *-un-lft-identity 91.6%

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

      2. add-sqr-sqrt 91.6%

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

      3. sqrt-unprod 91.6%

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

      4. frac-times 91.6%

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

      5. metadata-eval 91.6%

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

      6. rem-square-sqrt 91.8%

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

      7. metadata-eval 91.8%

         \[\leadsto \left(1 \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{0.5}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   8. Step-by-step derivation

      1. *-lft-identity 91.8%

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

   9. Simplified 91.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.692:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 99.6%

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

   5. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 7: 57.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   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* 99.5%

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

   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 59.6%

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

   1. pow2 59.6%

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

   2. associate-/l* 59.5%

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

   3. associate-*l* 59.6%

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

   4. div-inv 59.6%

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

   5. add-sqr-sqrt 59.6%

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

   6. sqrt-unprod 59.6%

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

   7. frac-times 59.6%

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

   8. metadata-eval 59.6%

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

   9. rem-square-sqrt 59.6%

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

   10. metadata-eval 59.6%

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

7. Applied egg-rr 59.6%

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

8. Final simplification 59.6%

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

Alternative 8: 39.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \left(a2 \cdot {2}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (* a2 (pow 2.0 -0.5))))

C

double code(double a1, double a2, double th) {
	return a2 * (a2 * pow(2.0, -0.5));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 * (2.0d0 ** (-0.5d0)))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 * Math.pow(2.0, -0.5));
}

Python

def code(a1, a2, th):
	return a2 * (a2 * math.pow(2.0, -0.5))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 * (2.0 ^ -0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 * (2.0 ^ -0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[Power[2.0, -0.5], $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%

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

   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* 99.5%

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

   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 59.6%

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

   1. clear-num 59.5%

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

   2. inv-pow 59.5%

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

   3. pow2 59.5%

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

   4. *-commutative 59.5%

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

   5. associate-/r* 59.5%

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

   6. pow2 59.5%

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

7. Applied egg-rr 59.5%

   \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\cos th}}{{a2}^{2}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 59.5%

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

   2. associate-/l/ 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in th around 0 41.6%

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

    1. pow2 41.6%

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

    2. associate-/r/ 41.7%

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

    3. associate-*r* 41.7%

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

    4. pow1/2 41.7%

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

    5. pow-flip 41.7%

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

    6. metadata-eval 41.7%

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

12. Applied egg-rr 41.7%

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

13. Final simplification 41.7%

    \[\leadsto a2 \cdot \left(a2 \cdot {2}^{-0.5}\right) \]
14. Add Preprocessing

Alternative 9: 39.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $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%

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

   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* 99.5%

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

   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 59.6%

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

   1. clear-num 59.5%

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

   2. inv-pow 59.5%

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

   3. pow2 59.5%

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

   4. *-commutative 59.5%

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

   5. associate-/r* 59.5%

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

   6. pow2 59.5%

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

7. Applied egg-rr 59.5%

   \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{2}}{\cos th}}{{a2}^{2}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 59.5%

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

   2. associate-/l/ 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in th around 0 41.6%

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

    1. pow2 41.6%

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

    2. clear-num 41.7%

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

    3. associate-/l* 41.7%

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

12. Applied egg-rr 41.7%

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

13. Final simplification 41.7%

    \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]
14. Add Preprocessing

Alternative 10: 30.4% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 67.0%

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

6. Taylor expanded in a1 around 0 41.7%

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

8. Applied egg-rr 31.5%

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

9. Final simplification 31.5%

   \[\leadsto a2 \cdot a2 \]
10. Add Preprocessing

Alternative 11: 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 67.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
6. Step-by-step derivation

   1. *-un-lft-identity 67.0%

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

   2. add-sqr-sqrt 67.0%

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

   3. sqrt-unprod 67.0%

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

   4. frac-times 67.0%

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

   5. metadata-eval 67.0%

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

   6. rem-square-sqrt 67.1%

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

   7. metadata-eval 67.1%

      \[\leadsto \left(1 \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Applied egg-rr 67.1%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{0.5}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
8. Step-by-step derivation

   1. *-lft-identity 67.1%

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

9. Simplified 67.1%

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

11. Applied egg-rr 6.8%

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

    1. associate-+l+ 3.9%

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

    2. fma-undefine 3.9%

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

    3. *-commutative 3.9%

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

    4. distribute-lft1-in 3.9%

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

    5. metadata-eval 3.9%

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

    6. neg-mul-1 3.9%

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

    7. neg-mul-1 3.9%

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

    8. distribute-rgt1-in 3.9%

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

    9. metadata-eval 3.9%

       \[\leadsto a1 + \color{blue}{0} \cdot a2 \]

    10. mul0-lft 3.9%

        \[\leadsto a1 + \color{blue}{0} \]

    11. +-rgt-identity 3.9%

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

13. Simplified 3.9%

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

14. Final simplification 3.9%

    \[\leadsto a1 \]
15. Add Preprocessing

Alternative 12: 3.6% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a2

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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 67.0%

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

6. Taylor expanded in a1 around 0 41.7%

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

8. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{\left|a2\right|} \]
9. Step-by-step derivation

   1. unpow1 4.7%

      \[\leadsto \left|\color{blue}{{a2}^{1}}\right| \]

   2. sqr-pow 2.2%

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

   3. fabs-sqr 2.2%

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

   4. sqr-pow 3.8%

      \[\leadsto \color{blue}{{a2}^{1}} \]

   5. unpow1 3.8%

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

10. Simplified 3.8%

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

11. Final simplification 3.8%

    \[\leadsto a2 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))))