Given's Rotation SVD example, simplified

Average Error: 15.4 → 0.1

Time: 11.3s

Precision: binary64

Cost: 26756

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+
    (* 0.125 (pow x 2.0))
    (+ (* 0.0673828125 (pow x 6.0)) (* -0.0859375 (pow x 4.0))))
   (/
    (+ 0.5 (/ -0.5 (hypot 1.0 x)))
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 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 tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (0.125 * pow(x, 2.0)) + ((0.0673828125 * pow(x, 6.0)) + (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = (0.5 + (-0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, 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 tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (0.125 * Math.pow(x, 2.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (-0.0859375 * Math.pow(x, 4.0)));
	} else {
		tmp = (0.5 + (-0.5 / Math.hypot(1.0, x))) / (1.0 + Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, 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):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (0.125 * math.pow(x, 2.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (-0.0859375 * math.pow(x, 4.0)))
	else:
		tmp = (0.5 + (-0.5 / math.hypot(1.0, x))) / (1.0 + math.sqrt((0.5 + (0.5 / math.hypot(1.0, 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)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(0.125 * (x ^ 2.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(-0.0859375 * (x ^ 4.0))));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.5 / hypot(1.0, x))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, 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)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (0.125 * (x ^ 2.0)) + ((0.0673828125 * (x ^ 6.0)) + (-0.0859375 * (x ^ 4.0)));
	else
		tmp = (0.5 + (-0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, 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_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 2

   1. Initial program 29.7

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

   2. Simplified 29.7

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

      [Start] 29.7	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 29.7	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 29.7	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 29.7	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 29.7	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around 0 0.3

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)} \]

   if 2 < (hypot.f64 1 x)

   1. Initial program 1.0

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

   2. Simplified 1.0

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

      [Start] 1.0	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 1.0	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 1.0	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 1.0	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 1.0	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Applied egg-rr 1.0

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

   4. Simplified 0.0

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

      [Start] 1.0	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      sqr-neg [=>] 1.0	\[ \frac{1 - \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      rem-square-sqrt [=>] 0.0	\[ \frac{1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate--r+ [=>] 0.0	\[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 0.0	\[ \frac{\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	20360

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{t_0}{\frac{1}{t_1} \cdot \left(t_1 \cdot \left(1 + \sqrt{t_1}\right)\right)}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 + \sqrt{t_0}}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	13576

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{t_0}{\frac{1}{t_1} \cdot \left(t_1 \cdot \left(1 + \sqrt{t_1}\right)\right)}\\ \mathbf{elif}\;x \leq 1.12:\\ \;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 + \sqrt{t_0}}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	8260

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{t_0}{\frac{1}{t_1} \cdot \left(t_1 \cdot \left(1 + \sqrt{t_1}\right)\right)}\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 + \sqrt{t_0}}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	7496

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{t_0}{1 + \sqrt{t_1}}\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 + \sqrt{t_0}}\\ \end{array} \]

Alternative 5
Error	0.8
Cost	7364

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 6
Error	1.0
Cost	6985

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

Alternative 7
Error	1.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{-0.5}{x}}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 8
Error	1.5
Cost	6857

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

Alternative 9
Error	25.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 10
Error	26.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;0.18181818181818182\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \]

Alternative 11
Error	26.1
Cost	576

\[\frac{1}{5.5 + \frac{8}{x \cdot x}} \]

Alternative 12
Error	40.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-77}:\\ \;\;\;\;0.18181818181818182\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \]

Alternative 13
Error	56.4
Cost	64

\[0.18181818181818182 \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))