Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 9.6s

Alternatives: 8

Speedup: 2.0×

• 42.7% 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: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 2: 71.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* (cos th) (* a2 a2))
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	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 (cos(th) <= 0.7d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   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. Add Preprocessing

   5. Taylor expanded in a1 around 0 61.1%

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

      1. associate-/l* 61.1%

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

      2. associate-/r/ 61.2%

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

   7. Simplified 61.2%

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

   9. Applied egg-rr 41.0%

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

   if 0.69999999999999996 < (cos.f64 th)

   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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 99.6%

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

      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%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in th around 0 89.8%

      \[\leadsto \color{blue}{\sqrt{0.5}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.5%

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

   5. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in th around inf 99.5%

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

   1. *-commutative 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

    \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot \sqrt{0.5}\right) \]
11. Add Preprocessing

Alternative 4: 42.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;a1 \cdot \left(-a1\right) - a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -1e-309) (- (* a1 (- a1)) (* a2 a2)) (* a2 a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -1e-309) {
		tmp = (a1 * -a1) - (a2 * a2);
	} else {
		tmp = 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 (cos(th) <= (-1d-309)) then
        tmp = (a1 * -a1) - (a2 * a2)
    else
        tmp = a2 * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -1e-309) {
		tmp = (a1 * -a1) - (a2 * a2);
	} else {
		tmp = a2 * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -1e-309:
		tmp = (a1 * -a1) - (a2 * a2)
	else:
		tmp = a2 * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -1e-309)
		tmp = Float64(Float64(a1 * Float64(-a1)) - Float64(a2 * a2));
	else
		tmp = Float64(a2 * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -1e-309)
		tmp = (a1 * -a1) - (a2 * a2);
	else
		tmp = a2 * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -1e-309], N[(N[(a1 * (-a1)), $MachinePrecision] - N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -1.000000000000002e-309

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   8. Applied egg-rr 58.5%

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

      1. +-commutative 58.5%

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

      2. +-inverses 58.5%

         \[\leadsto \frac{\cos \color{blue}{0} + \cos \left(th + th\right)}{-2} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

      3. cos-0 58.5%

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

      4. count-2 58.5%

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

   10. Simplified 58.5%

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

   11. Taylor expanded in th around 0 57.5%

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

   if -1.000000000000002e-309 < (cos.f64 th)

   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. Add Preprocessing

   5. Taylor expanded in th around 0 83.9%

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

   6. Taylor expanded in a1 around 0 54.7%

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

   8. Applied egg-rr 42.3%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -1 \cdot 10^{-309}:\\ \;\;\;\;a1 \cdot \left(-a1\right) - a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot a2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return cos(th) * (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 = cos(th) * (a2 * a2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(a2 * 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. 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. Add Preprocessing

5. Taylor expanded in a1 around 0 62.7%

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

   1. associate-/l* 62.7%

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

   2. associate-/r/ 62.8%

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

7. Simplified 62.8%

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

9. Applied egg-rr 41.8%

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

10. Final simplification 41.8%

    \[\leadsto \cos th \cdot \left(a2 \cdot a2\right) \]
11. Add Preprocessing

Alternative 6: 29.5% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a2 \cdot a2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 a2))

C

double code(double a1, double a2, double th) {
	return 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 = a2 * a2
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * a2;
}

Python

def code(a1, a2, th):
	return a2 * a2

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * a2), $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. Add Preprocessing

5. Taylor expanded in th around 0 63.2%

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

6. Taylor expanded in a1 around 0 42.0%

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

8. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{a2 \cdot a2} \]

9. Final simplification 33.0%

   \[\leadsto a2 \cdot a2 \]
10. Add Preprocessing

Alternative 7: 3.6% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ a1 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 a1)

C

double code(double a1, double a2, double th) {
	return 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 = a1
end function

Java

public static double code(double a1, double a2, double th) {
	return a1;
}

Python

def code(a1, a2, th):
	return a1

Julia

function code(a1, a2, th)
	return a1
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a1;
end

Wolfram

code[a1_, a2_, th_] := a1

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. Add Preprocessing

5. Taylor expanded in th around 0 63.2%

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

7. Applied egg-rr 5.2%

   \[\leadsto \color{blue}{\left(a1 + a2\right) + \mathsf{fma}\left(a2, -2, a2\right)} \]
8. Step-by-step derivation

   1. associate-+l+ 3.5%

      \[\leadsto \color{blue}{a1 + \left(a2 + \mathsf{fma}\left(a2, -2, a2\right)\right)} \]

   2. fma-udef 3.5%

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

   3. *-commutative 3.5%

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

   4. distribute-lft1-in 3.5%

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

   5. metadata-eval 3.5%

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

   6. distribute-rgt1-in 3.5%

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

   7. metadata-eval 3.5%

      \[\leadsto a1 + \color{blue}{0} \cdot a2 \]

   8. mul0-lft 3.5%

      \[\leadsto a1 + \color{blue}{0} \]

   9. +-rgt-identity 3.5%

      \[\leadsto \color{blue}{a1} \]

9. Simplified 3.5%

   \[\leadsto \color{blue}{a1} \]

10. Final simplification 3.5%

    \[\leadsto a1 \]
11. Add Preprocessing

Alternative 8: 3.6% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ a2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 a2)

C

double code(double a1, double a2, double th) {
	return 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
end function

Java

public static double code(double a1, double a2, double th) {
	return a2;
}

Python

def code(a1, a2, th):
	return a2

Julia

function code(a1, a2, th)
	return a2
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2;
end

Wolfram

code[a1_, a2_, th_] := a2

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. Add Preprocessing

5. Taylor expanded in th around 0 63.2%

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

6. Taylor expanded in a1 around 0 42.0%

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

8. Applied egg-rr 4.8%

   \[\leadsto \color{blue}{\left|a2\right|} \]
9. Step-by-step derivation

   1. unpow1 4.8%

      \[\leadsto \left|\color{blue}{{a2}^{1}}\right| \]

   2. sqr-pow 2.2%

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

   3. fabs-sqr 2.2%

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

   4. sqr-pow 3.6%

      \[\leadsto \color{blue}{{a2}^{1}} \]

   5. unpow1 3.6%

      \[\leadsto \color{blue}{a2} \]

10. Simplified 3.6%

    \[\leadsto \color{blue}{a2} \]

11. Final simplification 3.6%

    \[\leadsto a2 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(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))))