Given's Rotation SVD example

Average Accuracy: 79.0% → 91.4%

Time: 12.7s

Precision: binary64

Cost: 20932

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]

↓

\[\begin{array}{l} t_0 := \frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}}\\ \mathbf{if}\;t_0 \leq -0.999999995:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(t_0 + 1\right)}\\ \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

↓

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p (* 4.0 p)) (* x x))))))
   (if (<= t_0 -0.999999995)
     (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
     (sqrt (* 0.5 (+ t_0 1.0))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

↓

double code(double p, double x) {
	double t_0 = x / sqrt(((p * (4.0 * p)) + (x * x)));
	double tmp;
	if (t_0 <= -0.999999995) {
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = sqrt((0.5 * (t_0 + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

↓

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / sqrt(((p * (4.0d0 * p)) + (x * x)))
    if (t_0 <= (-0.999999995d0)) then
        tmp = sqrt((0.5d0 * (2.0d0 * ((p / x) * (p / x)))))
    else
        tmp = sqrt((0.5d0 * (t_0 + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

↓

public static double code(double p, double x) {
	double t_0 = x / Math.sqrt(((p * (4.0 * p)) + (x * x)));
	double tmp;
	if (t_0 <= -0.999999995) {
		tmp = Math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = Math.sqrt((0.5 * (t_0 + 1.0)));
	}
	return tmp;
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

↓

def code(p, x):
	t_0 = x / math.sqrt(((p * (4.0 * p)) + (x * x)))
	tmp = 0
	if t_0 <= -0.999999995:
		tmp = math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))))
	else:
		tmp = math.sqrt((0.5 * (t_0 + 1.0)))
	return tmp

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

↓

function code(p, x)
	t_0 = Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.999999995)
		tmp = sqrt(Float64(0.5 * Float64(2.0 * Float64(Float64(p / x) * Float64(p / x)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(t_0 + 1.0)));
	end
	return tmp
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

↓

function tmp_2 = code(p, x)
	t_0 = x / sqrt(((p * (4.0 * p)) + (x * x)));
	tmp = 0.0;
	if (t_0 <= -0.999999995)
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	else
		tmp = sqrt((0.5 * (t_0 + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[p_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999999995], N[Sqrt[N[(0.5 * N[(2.0 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	79.0%
Target	79.0%
Herbie	91.4%

\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99999999500000003

   1. Initial program 16.1%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]

   2. Taylor expanded in x around -inf 52.7%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

   3. Simplified 66.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}} \]
      Proof

      [Start] 52.7	\[ \sqrt{0.5 \cdot \left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)} \]
      unpow2 [=>] 52.7	\[ \sqrt{0.5 \cdot \left(2 \cdot \frac{\color{blue}{p \cdot p}}{{x}^{2}}\right)} \]
      unpow2 [=>] 52.7	\[ \sqrt{0.5 \cdot \left(2 \cdot \frac{p \cdot p}{\color{blue}{x \cdot x}}\right)} \]
      times-frac [=>] 66.0	\[ \sqrt{0.5 \cdot \left(2 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)}\right)} \]

   if -0.99999999500000003 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999999995:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	78.2%
Cost	13969

\[\begin{array}{l} t_0 := \sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \mathbf{if}\;p \leq -4.3 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq -7.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-198} \lor \neg \left(p \leq 2.7 \cdot 10^{-130}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 2
Accuracy	78.2%
Cost	13969

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, p \cdot 2\right)\\ \mathbf{if}\;p \leq -4.9 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{t_0}}\\ \mathbf{elif}\;p \leq -9 \cdot 10^{-145}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 6 \cdot 10^{-198} \lor \neg \left(p \leq 5.4 \cdot 10^{-130}\right):\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 3
Accuracy	67.0%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;p \leq -1.55 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{p \cdot p}{\frac{x}{-2}} - x}\right)}\\ \mathbf{elif}\;p \leq -1.15 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -9.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-234}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Accuracy	67.5%
Cost	7124

\[\begin{array}{l} \mathbf{if}\;p \leq -1.4 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.4 \cdot 10^{-112}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq -1.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-234}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Accuracy	44.2%
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-155}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	44.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-165}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	35.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023126 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))