Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.8s

Alternatives: 9

Speedup: 2.0×

• 44.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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. Step-by-step derivation

   1. clear-num 98.9%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

   6. fma-udef 99.7%

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

   7. +-commutative 99.7%

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

   8. add-sqr-sqrt 99.7%

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

   9. pow2 99.7%

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

   10. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Power[2.0, -0.5], $MachinePrecision] * N[Cos[th], $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/ 99.6%

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

   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 99.6%

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

   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%

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

Alternative 3: 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 4: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $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. +-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. Taylor expanded in a2 around inf 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r/ 56.8%

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

   3. associate-*r* 56.8%

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

6. Simplified 56.8%

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

7. Final simplification 56.8%

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

Alternative 5: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-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. Taylor expanded in a2 around inf 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*l/ 56.9%

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

6. Simplified 56.9%

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

   1. div-inv 44.4%

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

   2. pow1/2 44.4%

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

   3. pow-flip 44.5%

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

   4. metadata-eval 44.5%

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

   5. associate-*l* 44.5%

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

   6. add-sqr-sqrt 44.4%

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

   7. sqrt-unprod 44.5%

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

   8. pow-prod-up 44.5%

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

   9. metadata-eval 44.5%

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

   10. metadata-eval 44.5%

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

8. Applied egg-rr 56.9%

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

9. Final simplification 56.9%

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

Alternative 6: 57.8% 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. +-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. Taylor expanded in a2 around inf 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*l/ 56.9%

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

6. Simplified 56.9%

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

   1. div-inv 44.4%

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

   2. pow1/2 44.4%

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

   3. pow-flip 44.5%

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

   4. metadata-eval 44.5%

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

   5. add-sqr-sqrt 44.4%

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

   6. sqrt-unprod 44.5%

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

   7. pow-prod-up 44.5%

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

   8. metadata-eval 44.5%

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

   9. metadata-eval 44.5%

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

8. Applied egg-rr 56.9%

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

9. Final simplification 56.9%

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

Alternative 7: 67.0% 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/ 99.6%

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

   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 99.6%

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

   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 74.4%

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

7. Final simplification 74.4%

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

Alternative 8: 40.5% 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. +-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. Taylor expanded in th around 0 74.3%

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

   1. unpow2 74.3%

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

   2. unpow2 74.3%

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

   3. +-commutative 74.3%

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

6. Simplified 74.3%

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

7. Taylor expanded in a2 around inf 44.5%

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

   1. unpow2 44.5%

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

9. Simplified 44.5%

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

    1. div-inv 44.4%

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

    2. pow1/2 44.4%

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

    3. pow-flip 44.5%

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

    4. metadata-eval 44.5%

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

    5. associate-*l* 44.5%

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

    6. add-sqr-sqrt 44.4%

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

    7. sqrt-unprod 44.5%

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

    8. pow-prod-up 44.5%

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

    9. metadata-eval 44.5%

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

    10. metadata-eval 44.5%

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

11. Applied egg-rr 44.5%

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

12. Final simplification 44.5%

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

Alternative 9: 40.5% 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. Taylor expanded in th around 0 74.3%

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

   1. unpow2 74.3%

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

   2. unpow2 74.3%

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

   3. +-commutative 74.3%

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

6. Simplified 74.3%

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

7. Taylor expanded in a2 around inf 44.5%

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

   1. unpow2 44.5%

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

9. Simplified 44.5%

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

    1. div-inv 44.4%

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

    2. pow1/2 44.4%

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

    3. pow-flip 44.5%

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

    4. metadata-eval 44.5%

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

    5. add-sqr-sqrt 44.4%

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

    6. sqrt-unprod 44.5%

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

    7. pow-prod-up 44.5%

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

    8. metadata-eval 44.5%

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

    9. metadata-eval 44.5%

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

11. Applied egg-rr 44.5%

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

12. Final simplification 44.5%

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

Reproduce

herbie shell --seed 2023282 
(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))))