Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 1.9s

Alternatives: 6

Speedup: 1.0×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\mathsf{TRUE}\left(\right)\]

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

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]]

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;a1 \cdot a1 \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;t\_1 \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(a1 \cdot a1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (* a1 a1) 4.3e+98) (* t_1 (* a2 a2)) (* t_1 (* a1 a1)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((a1 * a1) <= 4.3e+98) {
		tmp = t_1 * (a2 * a2);
	} else {
		tmp = t_1 * (a1 * a1);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = cos(th) / sqrt(2.0d0)
    if ((a1 * a1) <= 4.3d+98) then
        tmp = t_1 * (a2 * a2)
    else
        tmp = t_1 * (a1 * a1)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((a1 * a1) <= 4.3e+98) {
		tmp = t_1 * (a2 * a2);
	} else {
		tmp = t_1 * (a1 * a1);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (a1 * a1) <= 4.3e+98:
		tmp = t_1 * (a2 * a2)
	else:
		tmp = t_1 * (a1 * a1)
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(a1 * a1) <= 4.3e+98)
		tmp = Float64(t_1 * Float64(a2 * a2));
	else
		tmp = Float64(t_1 * Float64(a1 * a1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((a1 * a1) <= 4.3e+98)
		tmp = t_1 * (a2 * a2);
	else
		tmp = t_1 * (a1 * a1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a1 * a1), $MachinePrecision], 4.3e+98], N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a1 a1) < 4.3000000000000001e98

   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. Add Preprocessing

   3. Taylor expanded in a2 around inf

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

   5. Applied rewrites 81.7%

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

   if 4.3000000000000001e98 < (*.f64 a1 a1)

   1. Initial program 99.7%

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

   3. Taylor expanded in a1 around 0

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

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in a1 around 0

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

5. Applied rewrites 57.8%

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

Alternative 4: 4.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in a1 around inf

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

5. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\cos th} \]

6. Taylor expanded in a2 around inf

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

8. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}}} \]
9. Add Preprocessing

Alternative 5: 4.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \cos th \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in a1 around inf

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

5. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\cos th} \]
6. Add Preprocessing

Alternative 6: 3.5% accurate, 24.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[Sqrt[2.0], $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. Add Preprocessing

3. Taylor expanded in a1 around inf

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

5. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\cos th} \]

6. Taylor expanded in a2 around inf

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

8. Applied rewrites 4.1%

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

9. Taylor expanded in a1 around inf

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

11. Applied rewrites 3.4%

    \[\leadsto \color{blue}{\sqrt{2}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  :pre (TRUE)
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))