Given's Rotation SVD example, simplified

Average Error: 15.6 → 0.2

Time: 6.3s

Precision: binary64

Cost: 26756

Math

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

↓

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

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)
   (+ (* (pow x 4.0) -0.0859375) (* (* x x) 0.125))
   (/
    (+ 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 = (pow(x, 4.0) * -0.0859375) + ((x * x) * 0.125);
	} 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 = (Math.pow(x, 4.0) * -0.0859375) + ((x * x) * 0.125);
	} 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 = (math.pow(x, 4.0) * -0.0859375) + ((x * x) * 0.125)
	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((x ^ 4.0) * -0.0859375) + Float64(Float64(x * x) * 0.125));
	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 = ((x ^ 4.0) * -0.0859375) + ((x * x) * 0.125);
	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[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.125), $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 29.9

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

   2. Simplified 29.9

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof
      (-.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 1 1/2)) (/.f64 1/2 (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (+.f64 (*.f64 1 1/2) (/.f64 (Rewrite<= metadata-eval (*.f64 1 1/2)) (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (+.f64 (*.f64 1 1/2) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (hypot.f64 1 x)) 1/2))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/2 (+.f64 1 (/.f64 1 (hypot.f64 1 x))))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 29.9

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

   4. Simplified 29.9

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      Proof
      (/.f64 (-.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))) (+.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))) 1)) (+.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))) (/.f64 1 (+.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around 0 0.4

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

   6. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
      Proof
      (fma.f64 (pow.f64 x 4) -11/128 (*.f64 (*.f64 x x) 1/8)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 x 4) -11/128 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) 1/8)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 x 4) -11/128 (Rewrite<= *-commutative_binary64 (*.f64 1/8 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 x 4) -11/128) (*.f64 1/8 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -11/128 (pow.f64 x 4))) (*.f64 1/8 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/8 (pow.f64 x 2)) (*.f64 -11/128 (pow.f64 x 4)))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 0.4

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

   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
      (-.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 1 1/2)) (/.f64 1/2 (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (+.f64 (*.f64 1 1/2) (/.f64 (Rewrite<= metadata-eval (*.f64 1 1/2)) (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (+.f64 (*.f64 1 1/2) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (hypot.f64 1 x)) 1/2))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (sqrt.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/2 (+.f64 1 (/.f64 1 (hypot.f64 1 x))))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.0

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

   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
      (/.f64 (-.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))) (+.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (-.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))) 1)) (+.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 1/2 (/.f64 1/2 (hypot.f64 1 x))) (/.f64 1 (+.f64 1 (sqrt.f64 (+.f64 1/2 (/.f64 1/2 (hypot.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.9
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 2
Error	0.9
Cost	7240

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

Alternative 3
Error	1.1
Cost	6984

\[\begin{array}{l} t_0 := \frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.5
Cost	6856

\[\begin{array}{l} t_0 := 1 - \sqrt{0.5}\\ \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	24.8
Cost	584

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

Alternative 6
Error	31.1
Cost	320

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

Alternative 7
Error	46.2
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))