Given's Rotation SVD example, simplified

Average Accuracy: 75.3% → 99.7%

Time: 11.0s

Precision: binary64

Cost: 20420

Math

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

↓

\[\begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;x \leq -0.0024:\\ \;\;\;\;\left(0.5 - t_0\right) \cdot \frac{1}{1 + \sqrt{0.5 + t_0}}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= x -0.0024)
     (* (- 0.5 t_0) (/ 1.0 (+ 1.0 (sqrt (+ 0.5 t_0)))))
     (if (<= x 1.15)
       (+ (* 0.125 (* x x)) (* -0.0859375 (pow x 4.0)))
       (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

↓

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (x <= -0.0024) {
		tmp = (0.5 - t_0) * (1.0 / (1.0 + sqrt((0.5 + t_0))));
	} else if (x <= 1.15) {
		tmp = (0.125 * (x * x)) + (-0.0859375 * pow(x, 4.0));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

↓

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (x <= -0.0024) {
		tmp = (0.5 - t_0) * (1.0 / (1.0 + Math.sqrt((0.5 + t_0))));
	} else if (x <= 1.15) {
		tmp = (0.125 * (x * x)) + (-0.0859375 * Math.pow(x, 4.0));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

↓

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if x <= -0.0024:
		tmp = (0.5 - t_0) * (1.0 / (1.0 + math.sqrt((0.5 + t_0))))
	elif x <= 1.15:
		tmp = (0.125 * (x * x)) + (-0.0859375 * math.pow(x, 4.0))
	else:
		tmp = (0.5 - (0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

↓

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (x <= -0.0024)
		tmp = Float64(Float64(0.5 - t_0) * Float64(1.0 / Float64(1.0 + sqrt(Float64(0.5 + t_0)))));
	elseif (x <= 1.15)
		tmp = Float64(Float64(0.125 * Float64(x * x)) + Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

↓

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (x <= -0.0024)
		tmp = (0.5 - t_0) * (1.0 / (1.0 + sqrt((0.5 + t_0))));
	elseif (x <= 1.15)
		tmp = (0.125 * (x * x)) + (-0.0859375 * (x ^ 4.0));
	else
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0024], N[(N[(0.5 - t$95$0), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15], N[(N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00239999999999999979

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

   2. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start] 98.4	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 98.4	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.4	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 98.4	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.4	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      Proof

      [Start] 98.4	\[ 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \]
      flip-- [=>] 98.4	\[ \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      div-inv [=>] 98.4	\[ \color{blue}{\left(1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      metadata-eval [=>] 98.4	\[ \left(\color{blue}{1} - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      add-sqr-sqrt [<=] 99.9	\[ \left(1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate--r+ [=>] 99.9	\[ \color{blue}{\left(\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 99.9	\[ \left(\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   if -0.00239999999999999979 < x < 1.1499999999999999

   1. Initial program 51.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

   2. Simplified 51.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start] 51.5	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 51.5	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 51.5	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 51.5	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 51.5	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x \cdot x, -0.0859375 \cdot {x}^{4}\right)} \]
      Proof

      [Start] 99.7	\[ 0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4} \]
      fma-def [=>] 99.7	\[ \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]
      unpow2 [=>] 99.7	\[ \mathsf{fma}\left(0.125, \color{blue}{x \cdot x}, -0.0859375 \cdot {x}^{4}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}} \]
      Proof

      [Start] 99.7	\[ \mathsf{fma}\left(0.125, x \cdot x, -0.0859375 \cdot {x}^{4}\right) \]
      fma-udef [=>] 99.7	\[ \color{blue}{0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}} \]

   if 1.1499999999999999 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

   2. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start] 98.5	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 98.5	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.5	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 98.5	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.5	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around inf 97.9%

      \[\leadsto 1 - \sqrt{0.5 + \color{blue}{\frac{0.5}{x}}} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(0.5 - \frac{0.5}{x}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}}} \]
      Proof

      [Start] 97.9	\[ 1 - \sqrt{0.5 + \frac{0.5}{x}} \]
      flip-- [=>] 97.9	\[ \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}} \]
      div-inv [=>] 97.9	\[ \color{blue}{\left(1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}}} \]
      metadata-eval [=>] 97.9	\[ \left(\color{blue}{1} - \sqrt{0.5 + \frac{0.5}{x}} \cdot \sqrt{0.5 + \frac{0.5}{x}}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \]
      add-sqr-sqrt [<=] 99.4	\[ \left(1 - \color{blue}{\left(0.5 + \frac{0.5}{x}\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \]
      associate--r+ [=>] 99.4	\[ \color{blue}{\left(\left(1 - 0.5\right) - \frac{0.5}{x}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \]
      metadata-eval [=>] 99.4	\[ \left(\color{blue}{0.5} - \frac{0.5}{x}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}} \]
      Proof

      [Start] 99.4	\[ \left(0.5 - \frac{0.5}{x}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \]
      associate-*r/ [=>] 99.4	\[ \color{blue}{\frac{\left(0.5 - \frac{0.5}{x}\right) \cdot 1}{1 + \sqrt{0.5 + \frac{0.5}{x}}}} \]
      *-rgt-identity [=>] 99.4	\[ \frac{\color{blue}{0.5 - \frac{0.5}{x}}}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0024:\\ \;\;\;\;\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	26756

\[\begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	13444

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]

Alternative 3
Accuracy	98.6%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]

Alternative 5
Accuracy	98.8%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}\\ \end{array} \]

Alternative 6
Accuracy	97.8%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]

Alternative 7
Accuracy	60.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1.75\right):\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]

Alternative 8
Accuracy	60.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 9
Accuracy	60.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 10
Accuracy	13.6%
Cost	64

\[0.25 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))