Given's Rotation SVD example, simplified

Percentage Accurate: 75.9% → 99.9%

Time: 10.5s

Precision: binary64

Cost: 40132

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;{x}^{4} \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t_0} - \frac{0.5}{\mathsf{hypot}\left(1, x\right) \cdot t_0}\\ \end{array} \]

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 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))
   (if (<= (hypot 1.0 x) 1.0002)
     (+ (* (pow x 4.0) -0.0859375) (* (* x x) 0.125))
     (- (/ 0.5 t_0) (/ 0.5 (* (hypot 1.0 x) t_0))))))

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 = 1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x))));
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (pow(x, 4.0) * -0.0859375) + ((x * x) * 0.125);
	} else {
		tmp = (0.5 / t_0) - (0.5 / (hypot(1.0, x) * t_0));
	}
	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 = 1.0 + Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0002) {
		tmp = (Math.pow(x, 4.0) * -0.0859375) + ((x * x) * 0.125);
	} else {
		tmp = (0.5 / t_0) - (0.5 / (Math.hypot(1.0, x) * t_0));
	}
	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 = 1.0 + math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	tmp = 0
	if math.hypot(1.0, x) <= 1.0002:
		tmp = (math.pow(x, 4.0) * -0.0859375) + ((x * x) * 0.125)
	else:
		tmp = (0.5 / t_0) - (0.5 / (math.hypot(1.0, x) * t_0))
	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(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(Float64((x ^ 4.0) * -0.0859375) + Float64(Float64(x * x) * 0.125));
	else
		tmp = Float64(Float64(0.5 / t_0) - Float64(0.5 / Float64(hypot(1.0, x) * t_0)));
	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 = 1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x))));
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0002)
		tmp = ((x ^ 4.0) * -0.0859375) + ((x * x) * 0.125);
	else
		tmp = (0.5 / t_0) - (0.5 / (hypot(1.0, x) * t_0));
	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[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / t$95$0), $MachinePrecision] - N[(0.5 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 48.0%

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

   2. Simplified 48.0%

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

      [Start] 48.0%	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 48.0%	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 48.0%	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 48.0%	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 48.0%	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Applied egg-rr 48.1%

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

      [Start] 48.0%	\[ 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \]
      flip-- [=>] 48.0%	\[ \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      metadata-eval [=>] 48.0%	\[ \frac{\color{blue}{1} - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      add-sqr-sqrt [<=] 48.1%	\[ \frac{1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate--r+ [=>] 48.1%	\[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 48.1%	\[ \frac{\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   4. Taylor expanded in x around 0 100.0%

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

   5. Simplified 100.0%

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

      [Start] 100.0%	\[ 0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4} \]
      +-commutative [=>] 100.0%	\[ \color{blue}{-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}} \]
      *-commutative [=>] 100.0%	\[ \color{blue}{{x}^{4} \cdot -0.0859375} + 0.125 \cdot {x}^{2} \]
      fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0859375, 0.125 \cdot {x}^{2}\right)} \]
      *-commutative [=>] 100.0%	\[ \mathsf{fma}\left({x}^{4}, -0.0859375, \color{blue}{{x}^{2} \cdot 0.125}\right) \]
      unpow2 [=>] 100.0%	\[ \mathsf{fma}\left({x}^{4}, -0.0859375, \color{blue}{\left(x \cdot x\right)} \cdot 0.125\right) \]
      associate-*l* [=>] 100.0%	\[ \mathsf{fma}\left({x}^{4}, -0.0859375, \color{blue}{x \cdot \left(x \cdot 0.125\right)}\right) \]

   6. Applied egg-rr 100.0%

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

      [Start] 100.0%	\[ \mathsf{fma}\left({x}^{4}, -0.0859375, x \cdot \left(x \cdot 0.125\right)\right) \]
      fma-udef [=>] 100.0%	\[ \color{blue}{{x}^{4} \cdot -0.0859375 + x \cdot \left(x \cdot 0.125\right)} \]
      associate-*r* [=>] 100.0%	\[ {x}^{4} \cdot -0.0859375 + \color{blue}{\left(x \cdot x\right) \cdot 0.125} \]

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.3%

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

   2. Simplified 98.3%

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

      [Start] 98.3%	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 98.3%	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.3%	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 98.3%	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.3%	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Applied egg-rr 99.9%

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

      [Start] 98.3%	\[ 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \]
      flip-- [=>] 98.4%	\[ \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      metadata-eval [=>] 98.4%	\[ \frac{\color{blue}{1} - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      add-sqr-sqrt [<=] 99.8%	\[ \frac{1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate--r+ [=>] 99.9%	\[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 99.9%	\[ \frac{\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   4. Applied egg-rr 99.9%

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

      [Start] 99.9%	\[ \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      div-sub [=>] 99.9%	\[ \color{blue}{\frac{0.5}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} - \frac{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      associate-/l/ [=>] 99.9%	\[ \frac{0.5}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} - \color{blue}{\frac{0.5}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \mathsf{hypot}\left(1, x\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;{x}^{4} \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} - \frac{0.5}{\mathsf{hypot}\left(1, x\right) \cdot \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	40132

\[\begin{array}{l} t_0 := 1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;{x}^{4} \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{t_0} - \frac{0.5}{\mathsf{hypot}\left(1, x\right) \cdot t_0}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	26756

\[\begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;{x}^{4} \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	13956

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

Alternative 4
Accuracy	99.0%
Cost	13828

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

Alternative 5
Accuracy	99.1%
Cost	13828

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

Alternative 6
Accuracy	98.7%
Cost	13636

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

Alternative 7
Accuracy	98.5%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \end{array} \]

Alternative 8
Accuracy	98.5%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{0.5}{x}}\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 9
Accuracy	97.7%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \end{array} \]

Alternative 10
Accuracy	60.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{2}\\ \end{array} \]

Alternative 11
Accuracy	60.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.78:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{2}\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{2}\\ \end{array} \]

Alternative 12
Accuracy	60.8%
Cost	584

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

Alternative 13
Accuracy	51.5%
Cost	320

\[\left(x \cdot x\right) \cdot 0.125 \]

Alternative 14
Accuracy	27.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023165 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))