Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 9.3s

Alternatives: 10

Speedup: 2.0×

• 41.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision] * N[(N[Sqrt[a1 ^ 2 + a2 ^ 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. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-def 99.6%

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

   2. div-inv 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. associate-*l* 99.5%

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

   5. hypot-def 99.5%

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

   6. hypot-def 99.5%

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

   7. pow1/2 99.5%

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

   8. pow-flip 99.6%

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

   9. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-def 99.6%

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

   2. associate-*r/ 99.6%

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

   3. associate-*l/ 99.5%

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

   4. distribute-lft-in 99.5%

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

   5. +-commutative 99.5%

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

   6. associate-*l/ 99.6%

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

   7. associate-*l/ 99.6%

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

   8. frac-add 99.3%

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

   9. fma-def 99.3%

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

   10. *-commutative 99.3%

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

   11. add-sqr-sqrt 99.6%

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

5. Applied egg-rr 99.6%

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

   1. fma-udef 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-rgt-out 99.6%

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

   4. *-commutative 99.6%

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

   5. distribute-lft-in 99.6%

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

   6. unpow2 99.6%

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

   7. unpow2 99.6%

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

   8. *-commutative 99.6%

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

   9. unpow2 99.6%

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

   10. unpow2 99.6%

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

   11. fma-def 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

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

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

Wolfram

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

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $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. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around inf 99.6%

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

   1. associate-/l* 99.6%

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

   2. unpow2 99.6%

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

   3. unpow2 99.6%

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

   4. +-commutative 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 58.2% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 51.6%

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

   1. unpow2 51.6%

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

   2. associate-/l* 51.6%

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

   3. associate-/r/ 51.6%

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

6. Simplified 51.6%

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

   1. div-inv 51.6%

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

   2. pow1/2 51.6%

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

   3. pow-flip 51.7%

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

   4. metadata-eval 51.7%

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

   5. add-sqr-sqrt 51.5%

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

   6. sqrt-unprod 51.7%

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

   7. pow-prod-up 51.7%

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

   8. metadata-eval 51.7%

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

   9. metadata-eval 51.7%

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

8. Applied egg-rr 51.7%

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

9. Final simplification 51.7%

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

Alternative 7: 66.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 4.6 \cdot 10^{+185}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 4.6e+185)
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))
   (* (sqrt 0.5) (* (* a2 a2) (+ (* -0.5 (* th th)) 1.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 4.6e+185) {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	} else {
		tmp = sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0));
	}
	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 (a2 <= 4.6d+185) then
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    else
        tmp = sqrt(0.5d0) * ((a2 * a2) * (((-0.5d0) * (th * th)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 4.6e+185) {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	} else {
		tmp = Math.sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 4.6e+185:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	else:
		tmp = math.sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 4.6e+185)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	else
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) * Float64(Float64(-0.5 * Float64(th * th)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 4.6e+185)
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	else
		tmp = sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 4.6e+185], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[(N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 4.6000000000000003e185

   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. Taylor expanded in th around 0 65.0%

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

      1. add-sqr-sqrt 65.0%

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

      2. sqrt-unprod 65.0%

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

      3. frac-times 65.0%

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

      4. metadata-eval 65.0%

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

      5. add-sqr-sqrt 65.1%

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

      6. metadata-eval 65.1%

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

      7. expm1-log1p-u 64.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      8. expm1-udef 64.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. Applied egg-rr 64.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   7. Step-by-step derivation

      1. expm1-def 64.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      2. expm1-log1p 65.1%

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

   8. Simplified 65.1%

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

   if 4.6000000000000003e185 < a2

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around 0 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. pow1/2 100.0%

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

      3. pow-flip 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. pow-prod-up 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in th around 0 0.0%

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

       1. associate-*r* 0.0%

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

       2. distribute-rgt1-in 87.5%

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

       3. unpow2 87.5%

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

       4. unpow2 87.5%

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

   11. Simplified 87.5%

       \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left(th \cdot th\right) + 1\right) \cdot \left(a2 \cdot a2\right)\right)} \cdot \sqrt{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 4.6 \cdot 10^{+185}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \end{array} \]

Alternative 8: 66.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $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. Taylor expanded in th around 0 64.7%

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

   1. add-sqr-sqrt 64.7%

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

   2. sqrt-unprod 64.7%

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

   3. frac-times 64.7%

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

   4. metadata-eval 64.7%

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

   5. add-sqr-sqrt 64.8%

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

   6. metadata-eval 64.8%

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

   7. expm1-log1p-u 64.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. expm1-udef 64.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
7. Step-by-step derivation

   1. expm1-def 64.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. expm1-log1p 64.8%

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

8. Simplified 64.8%

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

9. Final simplification 64.8%

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

Alternative 9: 40.4% 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. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 64.8%

   \[\leadsto \color{blue}{\frac{{a2}^{2} + {a1}^{2}}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. unpow2 64.8%

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

   2. unpow2 64.8%

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

   3. +-commutative 64.8%

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

6. Simplified 64.8%

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

7. Taylor expanded in a1 around 0 33.9%

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

   1. unpow2 33.9%

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

   2. associate-*r/ 33.9%

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

9. Simplified 33.9%

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

10. Final simplification 33.9%

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

Alternative 10: 40.4% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 51.7%

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

   1. unpow2 51.7%

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

   2. associate-*l* 51.7%

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

6. Simplified 51.7%

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

   1. div-inv 51.6%

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

   2. pow1/2 51.6%

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

   3. pow-flip 51.7%

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

   4. metadata-eval 51.7%

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

   5. add-sqr-sqrt 51.4%

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

   6. sqrt-unprod 51.7%

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

   7. pow-prod-up 51.7%

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

   8. metadata-eval 51.7%

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

   9. metadata-eval 51.7%

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

8. Applied egg-rr 51.7%

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

9. Taylor expanded in th around 0 33.9%

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

10. Final simplification 33.9%

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

Reproduce

herbie shell --seed 2023252 
(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))))