Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.4s

Alternatives: 10

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: 99.5% 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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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. +-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 56.8% 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. 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 52.9%

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

   1. unpow2 52.9%

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

   2. associate-*l* 52.9%

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

   3. associate-*r/ 52.9%

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

   4. associate-/l* 52.9%

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

6. Simplified 52.9%

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

7. Final simplification 52.9%

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

Alternative 4: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / 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. 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 52.9%

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

   1. unpow2 52.9%

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

   2. associate-/l* 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

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

Alternative 5: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $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. 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 52.9%

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

   1. unpow2 52.9%

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

6. Simplified 52.9%

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

7. Final simplification 52.9%

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

Alternative 6: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   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 52.9%

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

   1. unpow2 52.9%

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

   2. associate-*l* 52.9%

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

6. Simplified 52.9%

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

   1. div-inv 52.8%

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

   2. associate-*r* 52.9%

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

   3. associate-*r* 52.9%

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

   4. pow1/2 52.9%

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

   5. pow-flip 52.8%

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

   6. metadata-eval 52.8%

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

   7. *-commutative 52.8%

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

   8. *-commutative 52.8%

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

   9. metadata-eval 52.8%

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

   10. pow-flip 52.9%

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

   11. pow1/2 52.9%

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

   12. add-sqr-sqrt 52.9%

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

   13. sqrt-unprod 52.9%

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

   14. frac-times 52.9%

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

   15. metadata-eval 52.9%

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

   16. add-sqr-sqrt 52.8%

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

   17. metadata-eval 52.8%

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

8. Applied egg-rr 52.8%

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

9. Final simplification 52.8%

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

Alternative 7: 65.9% 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 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 67.5%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
7. Step-by-step derivation

   1. unpow2 67.5%

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

   2. unpow2 67.5%

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

8. Simplified 67.5%

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

9. Final simplification 67.5%

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

Alternative 8: 39.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 67.4%

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

5. Taylor expanded in a1 around 0 37.7%

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

   1. unpow2 37.7%

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

   2. associate-*r/ 37.7%

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

7. Simplified 37.7%

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

   1. associate-*r/ 37.7%

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

   2. clear-num 37.7%

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

9. Applied egg-rr 37.7%

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

10. Final simplification 37.7%

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

Alternative 9: 39.1% 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%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 67.5%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
7. Step-by-step derivation

   1. unpow2 67.5%

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

   2. unpow2 67.5%

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

8. Simplified 67.5%

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

9. Taylor expanded in a2 around inf 37.7%

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

    1. unpow2 37.7%

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

    2. *-commutative 37.7%

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

    3. associate-*l* 37.7%

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

11. Simplified 37.7%

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

12. Final simplification 37.7%

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

Alternative 10: 39.1% 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. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 67.5%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \left({a2}^{2} + {a1}^{2}\right)} \]
7. Step-by-step derivation

   1. unpow2 67.5%

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

   2. unpow2 67.5%

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

8. Simplified 67.5%

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

9. Taylor expanded in a2 around inf 37.7%

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

    1. unpow2 37.7%

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

11. Simplified 37.7%

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

12. Final simplification 37.7%

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

Reproduce

herbie shell --seed 2023214 
(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))))