Given's Rotation SVD example

Average Error: 13.3 → 5.6

Time: 13.7s

Precision: binary64

Cost: 20932

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

Math

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

↓

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

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 -1.0)
     (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 <= -1.0) {
		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 <= (-1.0d0)) 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 <= -1.0) {
		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 <= -1.0:
		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 <= -1.0)
		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 <= -1.0)
		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, -1.0], 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	13.3
Target	13.3
Herbie	5.6

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 53.9

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

   2. Taylor expanded in x around -inf 30.6

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

   3. Simplified 22.6

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 1/2 (*.f64 2 (*.f64 (/.f64 p x) (/.f64 p x))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 1/2 (*.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 p p) (*.f64 x x)))))): 0 points increase in error, 4 points decrease in error
      (sqrt.f64 (*.f64 1/2 (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 p 2)) (*.f64 x x))))): 4 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 1/2 (*.f64 2 (/.f64 (pow.f64 p 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))))): 0 points increase in error, 4 points decrease in error

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 0.2

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

4. Final simplification 5.6

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

Alternatives

Alternative 1
Error	13.8
Cost	13705

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-21} \lor \neg \left(x \leq -2.1 \cdot 10^{-33}\right):\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \end{array} \]

Alternative 2
Error	19.8
Cost	7888

\[\begin{array}{l} \mathbf{if}\;p \leq -6.5 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{0.5 + \frac{x}{p} \cdot -0.25}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-288}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.2 \cdot 10^{-192}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\frac{p \cdot p}{\frac{x}{-2}} - x}\right)}\\ \end{array} \]

Alternative 3
Error	19.4
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -2 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-288}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.65 \cdot 10^{-43}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	19.6
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -3 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{0.5 + \frac{x}{p} \cdot -0.25}\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{-288}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	20.7
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -4.6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-289}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 2.7 \cdot 10^{-102}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	47.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 3.3 \cdot 10^{-289}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 7
Error	53.4
Cost	192

\[\frac{p}{x} \]

Reproduce

herbie shell --seed 2022349 
(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))))))))