Given's Rotation SVD example, simplified

Average Error: 15.6 → 0.6

Time: 4.3s

Precision: binary64

Cost: 20036

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\\ \end{array} \]

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) 1.2)
   (+ (* 0.125 (pow x 2.0)) (* -0.0859375 (pow x 4.0)))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (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) <= 1.2) {
		tmp = (0.125 * pow(x, 2.0)) + (-0.0859375 * pow(x, 4.0));
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / 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) <= 1.2) {
		tmp = (0.125 * Math.pow(x, 2.0)) + (-0.0859375 * Math.pow(x, 4.0));
	} else {
		tmp = 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / 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) <= 1.2:
		tmp = (0.125 * math.pow(x, 2.0)) + (-0.0859375 * math.pow(x, 4.0))
	else:
		tmp = 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / 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) <= 1.2)
		tmp = Float64(Float64(0.125 * (x ^ 2.0)) + Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / 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) <= 1.2)
		tmp = (0.125 * (x ^ 2.0)) + (-0.0859375 * (x ^ 4.0));
	else
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / 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], 1.2], N[(N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.19999999999999996

   1. Initial program 30.6

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

   2. Simplified 30.6

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

      [Start] 30.6	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 30.6	\[ 1 - \sqrt{0.5 \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)}} \]
      metadata-eval [<=] 30.6	\[ 1 - \sqrt{\color{blue}{\left(--0.5\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)} \]
      metadata-eval [<=] 30.6	\[ 1 - \sqrt{\left(-\color{blue}{0.5 \cdot -1}\right) \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} + 1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-11 [=>] 30.6	\[ 1 - \sqrt{\left(-0.5 \cdot -1\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - -1\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-87 [<=] 30.6	\[ 1 - \sqrt{\color{blue}{\left(0.5 \cdot -1\right) \cdot \left(-1 - \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 30.6	\[ 1 - \sqrt{\color{blue}{-1 \cdot \left(0.5 \cdot -1\right) - \left(0.5 \cdot -1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 30.6	\[ 1 - \sqrt{-1 \cdot \color{blue}{-0.5} - \left(0.5 \cdot -1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      metadata-eval [=>] 30.6	\[ 1 - \sqrt{\color{blue}{0.5} - \left(0.5 \cdot -1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 30.6	\[ 1 - \sqrt{0.5 - \color{blue}{\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(0.5 \cdot -1\right)}} \]
      metadata-eval [=>] 30.6	\[ 1 - \sqrt{0.5 - \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \color{blue}{-0.5}} \]

   3. Taylor expanded in x around 0 0.3

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

   if 1.19999999999999996 < (hypot.f64 1 x)

   1. Initial program 1.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{-1}{x} \cdot -0.5}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{1}{x} \cdot -0.5}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{-1}{x} \cdot -0.5}\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{1}{x} \cdot -0.5}\\ \end{array} \]

Alternative 3
Error	1.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{-1}{x} \cdot -0.5}\\ \mathbf{elif}\;x \leq 1.52:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	1.4
Cost	6920

\[\begin{array}{l} t_0 := 1 - \sqrt{0.5}\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.52:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	31.3
Cost	6592

\[1 - \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))