Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 9.7s

Alternatives: 9

Speedup: 2.0×

• 43.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. associate-*l/ 99.6%

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

   4. div-inv 99.6%

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

   5. associate-*l* 99.6%

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

   6. add-sqr-sqrt 99.5%

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

   7. pow2 99.5%

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

   8. fma-udef 99.5%

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

   9. +-commutative 99.5%

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

   10. hypot-def 99.5%

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

   11. pow1/2 99.5%

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

   12. pow-flip 99.7%

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

   13. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. distribute-lft-in 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-*l/ 99.6%

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

   4. associate-*l/ 99.6%

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

   5. frac-add 99.3%

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

   6. fma-def 99.3%

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

   7. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. fma-udef 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-lft-out 99.6%

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

   4. distribute-lft-in 99.6%

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

   5. unpow2 99.6%

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

   6. unpow2 99.7%

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

   7. *-commutative 99.7%

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

   8. unpow2 99.7%

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

   9. unpow2 99.6%

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

   10. fma-def 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return (cos(th) * pow(2.0, -0.5)) * ((a2 * a2) + (a1 * 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 = (cos(th) * (2.0d0 ** (-0.5d0))) * ((a2 * a2) + (a1 * a1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 6: 56.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. fma-udef 99.6%

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

   3. associate-*l/ 99.6%

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

   4. div-inv 99.6%

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

   5. associate-*l* 99.6%

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

   6. add-sqr-sqrt 99.5%

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

   7. pow2 99.5%

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

   8. fma-udef 99.5%

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

   9. +-commutative 99.5%

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

   10. hypot-def 99.5%

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

   11. pow1/2 99.5%

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

   12. pow-flip 99.7%

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

   13. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in a2 around inf 52.8%

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

   1. unpow2 52.8%

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

   2. *-commutative 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

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

Alternative 7: 63.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{a2 \cdot \left(a2 + \left(a2 \cdot -0.5\right) \cdot \left(th \cdot th\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 1.05e+71)
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))
   (if (<= th 2.6e+133)
     (/ (* a2 (+ a2 (* (* a2 -0.5) (* th th)))) (sqrt 2.0))
     (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.05e+71) {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	} else if (th <= 2.6e+133) {
		tmp = (a2 * (a2 + ((a2 * -0.5) * (th * th)))) / sqrt(2.0);
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 1.05d+71) then
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    else if (th <= 2.6d+133) then
        tmp = (a2 * (a2 + ((a2 * (-0.5d0)) * (th * th)))) / sqrt(2.0d0)
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 1.05e+71) {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	} else if (th <= 2.6e+133) {
		tmp = (a2 * (a2 + ((a2 * -0.5) * (th * th)))) / Math.sqrt(2.0);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 1.05e+71:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	elif th <= 2.6e+133:
		tmp = (a2 * (a2 + ((a2 * -0.5) * (th * th)))) / math.sqrt(2.0)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 1.05e+71)
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	elseif (th <= 2.6e+133)
		tmp = Float64(Float64(a2 * Float64(a2 + Float64(Float64(a2 * -0.5) * Float64(th * th)))) / sqrt(2.0));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 1.05e+71)
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	elseif (th <= 2.6e+133)
		tmp = (a2 * (a2 + ((a2 * -0.5) * (th * th)))) / sqrt(2.0);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 1.05e+71], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[th, 2.6e+133], N[(N[(a2 * N[(a2 + N[(N[(a2 * -0.5), $MachinePrecision] * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 1.04999999999999995e71

   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. +-commutative 99.6%

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

      2. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. pow1/2 99.6%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 77.5%

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

   if 1.04999999999999995e71 < th < 2.5999999999999998e133

   1. Initial program 99.3%

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

      1. distribute-lft-out 99.3%

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

      2. cos-neg 99.3%

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

      3. associate-*l/ 99.5%

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

      4. cos-neg 99.5%

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

      5. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 67.3%

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

      1. unpow2 67.3%

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

      2. *-commutative 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in th around inf 67.3%

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

      1. unpow2 67.3%

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

      2. associate-*r* 67.2%

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

   9. Simplified 67.2%

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

   10. Taylor expanded in th around 0 43.6%

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

       1. associate-*r* 43.6%

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

       2. unpow2 43.6%

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

   12. Simplified 43.6%

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

   if 2.5999999999999998e133 < 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. +-commutative 99.5%

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

      2. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a2 around inf 59.1%

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

      1. unpow2 21.7%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in th around 0 21.7%

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

      1. unpow2 21.7%

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

   9. Simplified 21.7%

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

       1. associate-/l* 21.7%

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

       2. associate-/r/ 21.7%

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

   11. Applied egg-rr 21.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 2.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{a2 \cdot \left(a2 + \left(a2 \cdot -0.5\right) \cdot \left(th \cdot th\right)\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 8: 65.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 68.4%

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

7. Final simplification 68.4%

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

Alternative 9: 39.7% accurate, 4.0× speedup

Math

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

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

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 68.4%

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

7. Taylor expanded in a2 around inf 36.4%

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

   1. unpow2 36.4%

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

9. Simplified 36.4%

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

10. Final simplification 36.4%

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

Reproduce

herbie shell --seed 2023272 
(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))))