Given's Rotation SVD example

Average Error: 13.9 → 5.9

Time: 10.6s

Precision: binary64

Cost: 20932

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}}\\ \mathbf{if}\;t_0 \leq -0.9999999:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(t_0 + 1\right)}\\ \end{array} \]

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 -0.9999999)
     (sqrt (* (/ 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 <= -0.9999999) {
		tmp = sqrt(((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 <= (-0.9999999d0)) then
        tmp = sqrt(((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 <= -0.9999999) {
		tmp = Math.sqrt(((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 <= -0.9999999:
		tmp = math.sqrt(((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 <= -0.9999999)
		tmp = sqrt(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 <= -0.9999999)
		tmp = sqrt(((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, -0.9999999], N[Sqrt[N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	13.9
Target	13.9
Herbie	5.9

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999900000000053

   1. Initial program 54.1

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

   2. Applied egg-rr 54.1

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

   3. Applied egg-rr 55.2

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(2 \cdot p, x\right)}}\right)} - 1} \]

   4. Simplified 61.9

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

      [Start] 55.2	\[ e^{\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(2 \cdot p, x\right)}}\right)} - 1 \]
      expm1-def [=>] 55.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(2 \cdot p, x\right)}}\right)\right)} \]
      expm1-log1p [=>] 55.2	\[ \color{blue}{\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(2 \cdot p, x\right)}}} \]
      +-commutative [=>] 55.2	\[ \sqrt{\color{blue}{x \cdot \frac{0.5}{\mathsf{hypot}\left(2 \cdot p, x\right)} + 0.5}} \]
      fma-def [=>] 61.9	\[ \sqrt{\color{blue}{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(2 \cdot p, x\right)}, 0.5\right)}} \]
      *-commutative [=>] 61.9	\[ \sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(\color{blue}{p \cdot 2}, x\right)}, 0.5\right)} \]

   5. Taylor expanded in x around -inf 30.2

      \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

   6. Simplified 22.8

      \[\leadsto \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]
      Proof

      [Start] 30.2	\[ \sqrt{\frac{{p}^{2}}{{x}^{2}}} \]
      unpow2 [=>] 30.2	\[ \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]
      unpow2 [=>] 30.2	\[ \sqrt{\frac{p \cdot p}{\color{blue}{x \cdot x}}} \]
      times-frac [=>] 22.8	\[ \sqrt{\color{blue}{\frac{p}{x} \cdot \frac{p}{x}}} \]

   if -0.999999900000000053 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 0.0

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999:\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.4
Cost	14102

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+128} \lor \neg \left(x \leq -1.45 \cdot 10^{+93}\right) \land \left(x \leq -8.2 \cdot 10^{+78} \lor \neg \left(x \leq -9.2 \cdot 10^{-24}\right) \land x \leq -3.1 \cdot 10^{-30}\right):\\ \;\;\;\;\sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 2
Error	14.5
Cost	14101

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(p \cdot 2, x\right)\\ t_1 := \sqrt{\frac{p}{x} \cdot \frac{p}{x}}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{t_0}}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+78} \lor \neg \left(x \leq -9 \cdot 10^{-24}\right) \land x \leq -3.8 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{t_0}\right)}\\ \end{array} \]

Alternative 3
Error	20.8
Cost	7244

\[\begin{array}{l} \mathbf{if}\;p \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 6.8 \cdot 10^{-304}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.25}{p}}\\ \end{array} \]

Alternative 4
Error	20.8
Cost	7244

\[\begin{array}{l} \mathbf{if}\;p \leq -7.4:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot -0.25}{p}}\\ \mathbf{elif}\;p \leq 3.35 \cdot 10^{-302}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.25}{p}}\\ \end{array} \]

Alternative 5
Error	20.6
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.4 \cdot 10^{-300}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6
Error	24.5
Cost	6728

\[\begin{array}{l} \mathbf{if}\;p \leq -2.2 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7
Error	53.3
Cost	256

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

Reproduce

herbie shell --seed 2023018 
(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))))))))