Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.0s

Alternatives: 10

Speedup: 1.9×

• 43.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

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

Initial Program: 99.6% 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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -5 \cdot 10^{-96}:\\ \;\;\;\;\frac{a2 \cdot \mathsf{fma}\left(a2, th \cdot \left(th \cdot -0.5\right), a2\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -5e-96)
     (/ (* a2 (fma a2 (* th (* th -0.5)) a2)) (sqrt 2.0))
     (/ (fma (* a2 a2) (sqrt 2.0) (* (sqrt 2.0) (* a1 a1))) 2.0))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -5e-96) {
		tmp = (a2 * fma(a2, (th * (th * -0.5)), a2)) / sqrt(2.0);
	} else {
		tmp = fma((a2 * a2), sqrt(2.0), (sqrt(2.0) * (a1 * a1))) / 2.0;
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -5e-96)
		tmp = Float64(Float64(a2 * fma(a2, Float64(th * Float64(th * -0.5)), a2)) / sqrt(2.0));
	else
		tmp = Float64(fma(Float64(a2 * a2), sqrt(2.0), Float64(sqrt(2.0) * Float64(a1 * a1))) / 2.0);
	end
	return 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[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -5e-96], N[(N[(a2 * N[(a2 * N[(th * N[(th * -0.5), $MachinePrecision]), $MachinePrecision] + a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -4.99999999999999995e-96

   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

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

      1. times-frac N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot \left({a1}^{2} \cdot \left({th}^{2} \cdot \left(1 + \frac{{a2}^{2}}{{a1}^{2}}\right)\right)\right) + {a1}^{2} \cdot \left(1 + \frac{{a2}^{2}}{{a1}^{2}}\right)}{\sqrt{\color{blue}{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.5%

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

   9. Taylor expanded in a1 around 0

      \[\leadsto \frac{{a2}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}{\sqrt{2}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.4%

       \[\leadsto \frac{a2 \cdot \mathsf{fma}\left(a2, th \cdot \left(th \cdot -0.5\right), a2\right)}{\sqrt{2}} \]

   if -4.99999999999999995e-96 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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 th around 0

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 83.7

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

   5. Applied rewrites 83.7%

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

   7. Applied rewrites 83.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -5 \cdot 10^{-96}:\\ \;\;\;\;\frac{a2 \cdot \mathsf{fma}\left(a2, th \cdot \left(th \cdot -0.5\right), a2\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(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))))