Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 17.1s

Precision: binary64

Cost: 20160

• Unsound rule application detected in e-graph. Results may not be sound.

• 42.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

FPCore

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

↓

(FPCore (a1 a2 th)
 :precision binary64
 (* (/ (/ (cos th) (pow 2.0 0.25)) (pow 2.0 0.25)) (+ (* a1 a1) (* a2 a2))))

C

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

↓

double code(double a1, double a2, double th) {
	return ((cos(th) / pow(2.0, 0.25)) / pow(2.0, 0.25)) * ((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)) + ((cos(th) / sqrt(2.0d0)) * (a2 * a2))
end function

↓

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) / (2.0d0 ** 0.25d0)) / (2.0d0 ** 0.25d0)) * ((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)) + ((Math.cos(th) / Math.sqrt(2.0)) * (a2 * a2));
}

↓

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) / Math.pow(2.0, 0.25)) / Math.pow(2.0, 0.25)) * ((a1 * a1) + (a2 * a2));
}

Python

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

↓

def code(a1, a2, th):
	return ((math.cos(th) / math.pow(2.0, 0.25)) / math.pow(2.0, 0.25)) * ((a1 * a1) + (a2 * a2))

Julia

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

↓

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * Float64(Float64(a1 * a1) + Float64(a2 * a2)))
end

MATLAB

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

↓

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * ((a1 * a1) + (a2 * a2));
end

Wolfram

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

↓

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $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. Simplified 99.5%

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

   [Start] 99.6%	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   distribute-lft-out [=>] 99.5%	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   Step-by-step derivation

   [Start] 99.5%	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   *-un-lft-identity [=>] 99.5%	\[ \frac{\color{blue}{1 \cdot \cos th}}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   add-sqr-sqrt [=>] 99.6%	\[ \frac{1 \cdot \cos th}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   times-frac [=>] 99.2%	\[ \color{blue}{\left(\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   pow1/2 [=>] 99.2%	\[ \left(\frac{1}{\sqrt{\color{blue}{{2}^{0.5}}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   sqrt-pow1 [=>] 99.2%	\[ \left(\frac{1}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   metadata-eval [=>] 99.2%	\[ \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   pow1/2 [=>] 99.2%	\[ \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   sqrt-pow1 [=>] 99.2%	\[ \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   metadata-eval [=>] 99.2%	\[ \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

4. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   Step-by-step derivation

   [Start] 99.2%	\[ \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   associate-*l/ [=>] 99.6%	\[ \color{blue}{\frac{1 \cdot \frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   *-lft-identity [=>] 99.6%	\[ \frac{\color{blue}{\frac{\cos th}{{2}^{0.25}}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

5. Final simplification 99.6%

   \[\leadsto \frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	20160

\[\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

Alternative 2
Accuracy	79.7%
Cost	13508

\[\begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq -0.005:\\ \;\;\;\;t_1 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sqrt{2}}\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	13504

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

Alternative 4
Accuracy	70.1%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 6.6 \cdot 10^{-50}:\\ \;\;\;\;\cos th \cdot \frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 5
Accuracy	70.1%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.4 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\frac{\sqrt{2}}{\cos th}}\\ \end{array} \]

Alternative 6
Accuracy	66.2%
Cost	7372

\[\begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := t_1 \cdot \sqrt{0.5}\\ \mathbf{if}\;th \leq 1.6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;t_1 \cdot -0.5\\ \mathbf{elif}\;th \leq 1.1 \cdot 10^{+281}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \sqrt{\frac{a1 \cdot a1}{2}}\\ \end{array} \]

Alternative 7
Accuracy	47.5%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.6 \cdot 10^{-50}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5\\ \end{array} \]

Alternative 8
Accuracy	47.5%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 7.1 \cdot 10^{-50}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5\\ \end{array} \]

Alternative 9
Accuracy	47.3%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.6 \cdot 10^{-102}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 10
Accuracy	47.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.6 \cdot 10^{-102}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 11
Accuracy	45.9%
Cost	972

\[\begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ t_2 := t_1 \cdot 0.5\\ \mathbf{if}\;th \leq 1.6:\\ \;\;\;\;t_2\\ \mathbf{elif}\;th \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;t_1 \cdot -0.5\\ \mathbf{elif}\;th \leq 1.1 \cdot 10^{+281}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.25\\ \end{array} \]

Alternative 12
Accuracy	45.6%
Cost	909

\[\begin{array}{l} \mathbf{if}\;th \leq 1.6 \lor \neg \left(th \leq 1.5 \cdot 10^{+157}\right) \land th \leq 1.1 \cdot 10^{+281}:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(-a1\right) - a2 \cdot a2\\ \end{array} \]

Alternative 13
Accuracy	45.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;th \leq 1.6:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(a1 + a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 14
Accuracy	30.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 5 \cdot 10^{+272}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;a1 - a2 \cdot a2\\ \end{array} \]

Alternative 15
Accuracy	46.0%
Cost	448

\[\left(a1 + a2\right) \cdot \left(a1 + a2\right) \]

Alternative 16
Accuracy	3.7%
Cost	192

\[a1 \cdot -2 \]

Alternative 17
Accuracy	29.9%
Cost	192

\[a1 \cdot a1 \]

Alternative 18
Accuracy	3.7%
Cost	128

\[-a1 \]

Alternative 19
Accuracy	3.6%
Cost	64

\[a1 \]

Reproduce

herbie shell --seed 2023263 
(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))))