Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 13.8s

Alternatives: 7

Speedup: N/A×

• 43.9% 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: 99.6% 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, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $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. Final simplification 99.6%

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

Alternative 2: 57.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * a2), $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 60.2%

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

   1. unpow2 60.2%

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

   2. associate-*r/ 60.3%

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

   3. associate-*r* 60.3%

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

6. Simplified 60.3%

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

7. Final simplification 60.3%

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

Alternative 3: 63.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 3800000000:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.5 \cdot 10^{+58} \lor \neg \left(th \leq 1.5 \cdot 10^{+253}\right):\\ \;\;\;\;-0.5 \cdot \frac{a2 \cdot a2}{\frac{\sqrt{2}}{th \cdot th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 3800000000.0)
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))
   (if (or (<= th 1.5e+58) (not (<= th 1.5e+253)))
     (* -0.5 (/ (* a2 a2) (/ (sqrt 2.0) (* th th))))
     (/ 1.0 (/ (sqrt 2.0) (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 3800000000.0) {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	} else if ((th <= 1.5e+58) || !(th <= 1.5e+253)) {
		tmp = -0.5 * ((a2 * a2) / (sqrt(2.0) / (th * th)));
	} else {
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (th <= 3800000000.0d0) then
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    else if ((th <= 1.5d+58) .or. (.not. (th <= 1.5d+253))) then
        tmp = (-0.5d0) * ((a2 * a2) / (sqrt(2.0d0) / (th * th)))
    else
        tmp = 1.0d0 / (sqrt(2.0d0) / (a2 * a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 3800000000.0) {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	} else if ((th <= 1.5e+58) || !(th <= 1.5e+253)) {
		tmp = -0.5 * ((a2 * a2) / (Math.sqrt(2.0) / (th * th)));
	} else {
		tmp = 1.0 / (Math.sqrt(2.0) / (a2 * a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 3800000000.0:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	elif (th <= 1.5e+58) or not (th <= 1.5e+253):
		tmp = -0.5 * ((a2 * a2) / (math.sqrt(2.0) / (th * th)))
	else:
		tmp = 1.0 / (math.sqrt(2.0) / (a2 * a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 3800000000.0)
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	elseif ((th <= 1.5e+58) || !(th <= 1.5e+253))
		tmp = Float64(-0.5 * Float64(Float64(a2 * a2) / Float64(sqrt(2.0) / Float64(th * th))));
	else
		tmp = Float64(1.0 / Float64(sqrt(2.0) / Float64(a2 * a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 3800000000.0)
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	elseif ((th <= 1.5e+58) || ~((th <= 1.5e+253)))
		tmp = -0.5 * ((a2 * a2) / (sqrt(2.0) / (th * th)));
	else
		tmp = 1.0 / (sqrt(2.0) / (a2 * a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 3800000000.0], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[th, 1.5e+58], N[Not[LessEqual[th, 1.5e+253]], $MachinePrecision]], N[(-0.5 * N[(N[(a2 * a2), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] / N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 3.8e9

   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 72.3%

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

   if 3.8e9 < th < 1.5000000000000001e58 or 1.4999999999999999e253 < th

   1. Initial program 99.8%

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

      1. distribute-lft-out 99.8%

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

      2. cos-neg 99.8%

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

      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. Taylor expanded in a1 around 0 51.8%

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

      1. unpow2 51.8%

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

      2. associate-*l* 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in th around 0 32.3%

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

      1. *-commutative 32.3%

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

      2. unpow2 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in th around inf 31.9%

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

       1. unpow2 31.9%

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

       2. associate-/l* 31.9%

          \[\leadsto -0.5 \cdot \color{blue}{\frac{a2 \cdot a2}{\frac{\sqrt{2}}{{th}^{2}}}} \]

       3. unpow2 31.9%

          \[\leadsto -0.5 \cdot \frac{a2 \cdot a2}{\frac{\sqrt{2}}{\color{blue}{th \cdot th}}} \]

   12. Simplified 31.9%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{a2 \cdot a2}{\frac{\sqrt{2}}{th \cdot th}}} \]

   if 1.5000000000000001e58 < th < 1.4999999999999999e253

   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 27.0%

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

   5. Taylor expanded in a2 around inf 18.8%

      \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{{a2}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 18.8%

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

   7. Simplified 18.8%

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

      1. associate-/r/ 18.8%

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

   9. Applied egg-rr 18.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 3800000000:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;th \leq 1.5 \cdot 10^{+58} \lor \neg \left(th \leq 1.5 \cdot 10^{+253}\right):\\ \;\;\;\;-0.5 \cdot \frac{a2 \cdot a2}{\frac{\sqrt{2}}{th \cdot th}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{2}}{a2 \cdot a2}}\\ \end{array} \]

Alternative 4: 66.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (* a2 a2) 5e+288)
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))
   (/ (* a2 (+ a2 (* -0.5 (* a2 (* th th))))) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((a2 * a2) <= 5e+288) {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	} else {
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((a2 * a2) <= 5d+288) then
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    else
        tmp = (a2 * (a2 + ((-0.5d0) * (a2 * (th * th))))) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((a2 * a2) <= 5e+288) {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	} else {
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (a2 * a2) <= 5e+288:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	else:
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (Float64(a2 * a2) <= 5e+288)
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	else
		tmp = Float64(Float64(a2 * Float64(a2 + Float64(-0.5 * Float64(a2 * Float64(th * th))))) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((a2 * a2) <= 5e+288)
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	else
		tmp = (a2 * (a2 + (-0.5 * (a2 * (th * th))))) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[(a2 * a2), $MachinePrecision], 5e+288], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(a2 * N[(a2 + N[(-0.5 * N[(a2 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a2 a2) < 5.0000000000000003e288

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around 0 56.1%

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

   if 5.0000000000000003e288 < (*.f64 a2 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. cos-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around 0 100.0%

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

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

   7. Taylor expanded in th around 0 71.6%

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

      1. *-commutative 71.6%

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

      2. unpow2 71.6%

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

   9. Simplified 71.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)}{\sqrt{2}}\\ \end{array} \]

Alternative 5: 66.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 60.2%

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

7. Final simplification 60.2%

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

Alternative 6: 39.8% 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 60.2%

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

5. Taylor expanded in a2 around inf 39.5%

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

   1. unpow2 39.5%

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

7. Simplified 39.5%

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

   1. div-inv 39.5%

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

   2. add-sqr-sqrt 39.5%

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

   3. sqrt-unprod 39.5%

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

   4. frac-times 39.5%

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

   5. metadata-eval 39.5%

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

   6. add-sqr-sqrt 39.6%

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

   7. metadata-eval 39.6%

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

9. Applied egg-rr 39.6%

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

10. Final simplification 39.6%

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

Alternative 7: 39.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-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 60.2%

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

5. Taylor expanded in a2 around inf 39.5%

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

   1. unpow2 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in a2 around 0 39.5%

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

   1. unpow2 39.5%

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

   2. associate-*l/ 39.6%

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

10. Simplified 39.6%

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

11. Final simplification 39.6%

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

Reproduce

herbie shell --seed 2023287 
(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))))