Migdal et al, Equation (64)

Average Error: 0.8% → 0.69%

Time: 19.3s

Precision: binary64

Cost: 26176

Math

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

↓

\[\frac{\mathsf{hypot}\left(a1, a2\right)}{\frac{\frac{\sqrt{2}}{\cos th}}{\mathsf{hypot}\left(a1, a2\right)}} \]

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
 (/ (hypot a1 a2) (/ (/ (sqrt 2.0) (cos th)) (hypot a1 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 hypot(a1, a2) / ((sqrt(2.0) / cos(th)) / hypot(a1, a2));
}

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.hypot(a1, a2) / ((Math.sqrt(2.0) / Math.cos(th)) / Math.hypot(a1, 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.hypot(a1, a2) / ((math.sqrt(2.0) / math.cos(th)) / math.hypot(a1, 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(hypot(a1, a2) / Float64(Float64(sqrt(2.0) / cos(th)) / hypot(a1, 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 = hypot(a1, a2) / ((sqrt(2.0) / cos(th)) / hypot(a1, 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[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.8

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

2. Simplified 0.8

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

   [Start] 0.8	\[ \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   distribute-lft-out [=>] 0.8	\[ \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

3. Applied egg-rr 0.69

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

4. Final simplification 0.69

   \[\leadsto \frac{\mathsf{hypot}\left(a1, a2\right)}{\frac{\frac{\sqrt{2}}{\cos th}}{\mathsf{hypot}\left(a1, a2\right)}} \]

Alternatives

Alternative 1
Error	22.96%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\cos th \leq 0.98:\\ \;\;\;\;a1 \cdot \left(\left(a1 \cdot \cos th\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 2
Error	32.82%
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147} \lor \neg \left(a2 \leq 4.8 \cdot 10^{-135}\right) \land a2 \leq 4 \cdot 10^{-89}:\\ \;\;\;\;a1 \cdot \left(\left(a1 \cdot \cos th\right) \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 3
Error	32.84%
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \left(\left(a1 \cdot \cos th\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 4.7 \cdot 10^{-135} \lor \neg \left(a2 \leq 9.8 \cdot 10^{-88}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}}\\ \end{array} \]

Alternative 4
Error	32.85%
Cost	13645

\[\begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \left(\left(a1 \cdot \cos th\right) \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;a2 \leq 5.8 \cdot 10^{-135} \lor \neg \left(a2 \leq 2.25 \cdot 10^{-88}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot t_1\\ \end{array} \]

Alternative 5
Error	32.85%
Cost	13645

\[\begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \frac{\cos th}{\frac{\sqrt{2}}{a1}}\\ \mathbf{elif}\;a2 \leq 4.7 \cdot 10^{-135} \lor \neg \left(a2 \leq 1.08 \cdot 10^{-88}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot t_1\\ \end{array} \]

Alternative 6
Error	32.84%
Cost	13645

\[\begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 5 \cdot 10^{-135} \lor \neg \left(a2 \leq 4.5 \cdot 10^{-88}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1\right) \cdot t_1\\ \end{array} \]

Alternative 7
Error	32.85%
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 7.6 \cdot 10^{-135} \lor \neg \left(a2 \leq 8.5 \cdot 10^{-88}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th}{\frac{\sqrt{2}}{a1 \cdot a1}}\\ \end{array} \]

Alternative 8
Error	32.85%
Cost	13645

\[\begin{array}{l} t_1 := a1 \cdot \cos th\\ \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \frac{t_1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 1.2 \cdot 10^{-134} \lor \neg \left(a2 \leq 1.25 \cdot 10^{-88}\right):\\ \;\;\;\;a2 \cdot \left(a2 \cdot \frac{\cos th}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot t_1}{\sqrt{2}}\\ \end{array} \]

Alternative 9
Error	32.82%
Cost	13645

\[\begin{array}{l} t_1 := a1 \cdot \cos th\\ \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \frac{t_1}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 5.1 \cdot 10^{-135} \lor \neg \left(a2 \leq 4.4 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{a2 \cdot \left(a2 \cdot \cos th\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot t_1}{\sqrt{2}}\\ \end{array} \]

Alternative 10
Error	32.83%
Cost	13645

\[\begin{array}{l} \mathbf{if}\;a2 \leq 2.35 \cdot 10^{-147}:\\ \;\;\;\;a1 \cdot \frac{a1 \cdot \cos th}{\sqrt{2}}\\ \mathbf{elif}\;a2 \leq 4.7 \cdot 10^{-135} \lor \neg \left(a2 \leq 1.8 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{a2 \cdot \left(a2 \cdot \cos th\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}\\ \end{array} \]

Alternative 11
Error	0.8%
Cost	13504

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

Alternative 12
Error	0.72%
Cost	13504

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

Alternative 13
Error	40.56%
Cost	6976

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

Alternative 14
Error	40.54%
Cost	6976

\[\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}} \]

Alternative 15
Error	57.46%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.6 \cdot 10^{-156}:\\ \;\;\;\;a1 \cdot \frac{a1}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 16
Error	57.45%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a1 \leq -1.85 \cdot 10^{-156}:\\ \;\;\;\;\frac{a1}{\frac{\sqrt{2}}{a1}}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 17
Error	63.87%
Cost	6720

\[a1 \cdot \frac{a1}{\sqrt{2}} \]

Reproduce

herbie shell --seed 2023121 
(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))))