Migdal et al, Equation (64)

Average Accuracy: 99.6% → 99.6%

Time: 15.8s

Precision: binary64

Cost: 13504

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

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

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

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

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

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

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

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

Math

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

↓

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

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
 (* (* (sqrt 0.5) (cos th)) (+ (* 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 (sqrt(0.5) * cos(th)) * ((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 = (sqrt(0.5d0) * cos(th)) * ((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.sqrt(0.5) * Math.cos(th)) * ((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.sqrt(0.5) * math.cos(th)) * ((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(sqrt(0.5) * cos(th)) * 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 = (sqrt(0.5) * cos(th)) * ((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[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $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 99.6%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   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 [=>] 99.6	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Applied egg-rr 99.7%

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

   [Start] 99.6	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   clear-num [=>] 99.6	\[ \color{blue}{\frac{1}{\frac{\sqrt{2}}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   associate-/r/ [=>] 99.6	\[ \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   pow1/2 [=>] 99.6	\[ \left(\frac{1}{\color{blue}{{2}^{0.5}}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   pow-flip [=>] 99.7	\[ \left(\color{blue}{{2}^{\left(-0.5\right)}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   metadata-eval [=>] 99.7	\[ \left({2}^{\color{blue}{-0.5}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

4. Taylor expanded in th around inf 99.7%

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

5. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	80.1%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq -0.001:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 2
Accuracy	70.3%
Cost	13444

\[\begin{array}{l} \mathbf{if}\;a1 \leq -5.4 \cdot 10^{-75}:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot \left(a1 \cdot {2}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	70.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -3.55 \cdot 10^{-75}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	70.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -3.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(\cos th \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	70.3%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;a1 \leq -8 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\cos th \cdot \left(a1 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \left(\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	67.3%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot \left(a1 + -0.5 \cdot \left(a1 \cdot \left(th \cdot th\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]

Alternative 7
Accuracy	47.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{1}{\frac{\sqrt{2}}{a2}}\\ \end{array} \]

Alternative 8
Accuracy	67.4%
Cost	6976

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

Alternative 9
Accuracy	39.7%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;th \leq 500000000000:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{elif}\;th \leq 4.2 \cdot 10^{+230}:\\ \;\;\;\;a1 \cdot \left(a1 \cdot \left(0.5 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a1}{\frac{-2}{a1}}\\ \end{array} \]

Alternative 10
Accuracy	47.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 3.3 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 11
Accuracy	47.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 5.2 \cdot 10^{-112}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2\right)\\ \end{array} \]

Alternative 12
Accuracy	47.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 4.8 \cdot 10^{-113}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 13
Accuracy	47.2%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a2 \leq 5.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 14
Accuracy	29.7%
Cost	712

\[\begin{array}{l} t_1 := a1 \cdot \left(0.5 \cdot a1\right)\\ \mathbf{if}\;th \leq 410000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 4.2 \cdot 10^{+230}:\\ \;\;\;\;a1 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 + a1\right)\\ \end{array} \]

Alternative 15
Accuracy	29.7%
Cost	712

\[\begin{array}{l} t_1 := a1 \cdot \left(0.5 \cdot a1\right)\\ \mathbf{if}\;th \leq 500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;th \leq 4.2 \cdot 10^{+230}:\\ \;\;\;\;a1 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{a1}{\frac{-2}{a1}}\\ \end{array} \]

Alternative 16
Accuracy	30.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;th \leq 500000000000:\\ \;\;\;\;a1 \cdot \left(0.5 \cdot a1\right)\\ \mathbf{elif}\;th \leq 4.2 \cdot 10^{+230}:\\ \;\;\;\;a1 \cdot \frac{a1}{-2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(a1 + a1\right)\\ \end{array} \]

Alternative 17
Accuracy	29.8%
Cost	320

\[a1 \cdot \left(0.5 \cdot a1\right) \]

Alternative 18
Accuracy	29.7%
Cost	192

\[a1 \cdot a1 \]

Reproduce

herbie shell --seed 2023158 
(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))))