Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 7.8s

Alternatives: 10

Speedup: 1.3×

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((2.0 ^ -0.5) * (hypot(a1, a2) ^ 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[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 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. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. associate-*l* 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.6%

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

   7. pow2 99.6%

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

   8. fma-undefine 99.6%

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

   9. hypot-define 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $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. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around inf 99.6%

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

   1. associate-/l* 99.6%

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

   2. rem-square-sqrt 99.6%

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

   3. unpow2 99.6%

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

   4. unpow2 99.6%

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

   5. hypot-undefine 99.6%

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

   6. unpow2 99.6%

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

   7. unpow2 99.6%

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

   8. hypot-undefine 99.6%

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

   9. unpow2 99.6%

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

7. Simplified 99.6%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $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. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 58.1% accurate, 2.0× speedup

Math

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

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(a2 * Float64(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[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $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. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. associate-*l* 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.6%

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

   7. pow2 99.6%

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

   8. fma-undefine 99.6%

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

   9. hypot-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a1 around 0 56.9%

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

   1. *-commutative 56.9%

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

   2. unpow2 56.9%

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

   3. associate-*r* 56.9%

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

9. Applied egg-rr 56.9%

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

10. Final simplification 56.9%

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

Alternative 6: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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) / cos(th)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $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. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. associate-*l* 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.6%

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

   7. pow2 99.6%

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

   8. fma-undefine 99.6%

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

   9. hypot-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a1 around 0 56.9%

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

   1. *-commutative 56.9%

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

   2. *-commutative 56.9%

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

   3. associate-*l* 56.9%

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

   4. metadata-eval 56.9%

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

   5. metadata-eval 56.9%

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

   6. rem-square-sqrt 56.8%

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

   7. frac-times 56.8%

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

   8. sqrt-unprod 56.8%

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

   9. add-sqr-sqrt 56.8%

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

   10. associate-*l* 56.8%

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

   11. *-commutative 56.8%

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

   12. un-div-inv 56.9%

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

   13. associate-/r/ 56.9%

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

   14. unpow2 56.9%

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

   15. associate-/l* 56.8%

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

9. Applied egg-rr 56.8%

   \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}} \]
10. Add Preprocessing

Alternative 7: 40.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\sqrt{2}}{{a2}^{2}}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ 1.0 (/ (sqrt 2.0) (pow a2 2.0))))

C

double code(double a1, double a2, double th) {
	return 1.0 / (sqrt(2.0) / pow(a2, 2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 1.0d0 / (sqrt(2.0d0) / (a2 ** 2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return 1.0 / (Math.sqrt(2.0) / Math.pow(a2, 2.0));
}

Python

def code(a1, a2, th):
	return 1.0 / (math.sqrt(2.0) / math.pow(a2, 2.0))

Julia

function code(a1, a2, th)
	return Float64(1.0 / Float64(sqrt(2.0) / (a2 ^ 2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 1.0 / (sqrt(2.0) / (a2 ^ 2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[a2, 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. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. associate-*l* 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.6%

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

   7. pow2 99.6%

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

   8. fma-undefine 99.6%

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

   9. hypot-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a1 around 0 56.9%

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

8. Taylor expanded in th around 0 38.7%

   \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{0.5}} \]
9. Step-by-step derivation

   1. metadata-eval 38.7%

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

   2. metadata-eval 38.7%

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

   3. rem-square-sqrt 38.7%

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

   4. frac-times 38.7%

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

   5. sqrt-unprod 38.7%

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

   6. add-sqr-sqrt 38.7%

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

   7. un-div-inv 38.7%

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

   8. clear-num 38.7%

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

10. Applied egg-rr 38.7%

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{{a2}^{2}}}} \]
11. Add Preprocessing

Alternative 8: 40.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 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 = a2 * (a2 * sqrt(0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * 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. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a1 around 0 56.9%

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

   1. associate-*r/ 56.8%

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

7. Simplified 56.8%

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

   1. pow2 56.8%

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

   2. associate-*r/ 56.9%

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

   3. div-inv 56.8%

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

   4. pow1/2 56.8%

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

   5. pow-flip 56.9%

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

   6. metadata-eval 56.9%

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

   7. associate-*r* 56.9%

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

   8. *-commutative 56.9%

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

   9. *-commutative 56.9%

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

   10. associate-*r* 56.9%

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

   11. add-sqr-sqrt 56.6%

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

   12. sqrt-unprod 56.9%

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

   13. pow-prod-up 56.9%

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

   14. metadata-eval 56.9%

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

   15. metadata-eval 56.9%

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

9. Applied egg-rr 56.9%

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

10. Taylor expanded in th around 0 38.7%

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

11. Final simplification 38.7%

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

Alternative 9: 40.3% 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 99.6%

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

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. associate-*l* 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.6%

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

   7. pow2 99.6%

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

   8. fma-undefine 99.6%

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

   9. hypot-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a1 around 0 56.9%

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

   1. *-commutative 56.9%

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

   2. *-commutative 56.9%

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

   3. associate-*l* 56.9%

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

   4. metadata-eval 56.9%

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

   5. metadata-eval 56.9%

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

   6. rem-square-sqrt 56.8%

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

   7. frac-times 56.8%

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

   8. sqrt-unprod 56.8%

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

   9. add-sqr-sqrt 56.8%

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

   10. associate-*l* 56.8%

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

   11. *-commutative 56.8%

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

   12. un-div-inv 56.9%

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

   13. associate-/r/ 56.9%

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

   14. unpow2 56.9%

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

   15. associate-/l* 56.8%

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

9. Applied egg-rr 56.8%

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

10. Taylor expanded in th around 0 38.7%

    \[\leadsto a2 \cdot \color{blue}{\frac{a2}{\sqrt{2}}} \]
11. Add Preprocessing

Alternative 10: 39.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a1 \cdot \frac{a1}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 * N[(a1 / 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. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 62.9%

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

6. Taylor expanded in a1 around inf 35.6%

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} \]
7. Step-by-step derivation

   1. unpow2 35.6%

      \[\leadsto \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}} \]

   2. associate-/l* 35.6%

      \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} \]

8. Applied egg-rr 35.6%

   \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))