Given's Rotation SVD example

Average Error: 12.8 → 7.3

Time: 8.3s

Precision: binary64

Cost: 26692

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\\ \mathbf{if}\;t_0 \leq -0.98:\\ \;\;\;\;\sqrt{\frac{{p}^{2}}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + t_0\right)}\\ \end{array} \]

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 (+ (* (* 4.0 p) p) (* x x))))))
   (if (<= t_0 -0.98)
     (sqrt (/ (pow p 2.0) (pow x 2.0)))
     (sqrt (* 0.5 (+ 1.0 t_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((((4.0 * p) * p) + (x * x)));
	double tmp;
	if (t_0 <= -0.98) {
		tmp = sqrt((pow(p, 2.0) / pow(x, 2.0)));
	} else {
		tmp = sqrt((0.5 * (1.0 + t_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((((4.0d0 * p) * p) + (x * x)))
    if (t_0 <= (-0.98d0)) then
        tmp = sqrt(((p ** 2.0d0) / (x ** 2.0d0)))
    else
        tmp = sqrt((0.5d0 * (1.0d0 + t_0)))
    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((((4.0 * p) * p) + (x * x)));
	double tmp;
	if (t_0 <= -0.98) {
		tmp = Math.sqrt((Math.pow(p, 2.0) / Math.pow(x, 2.0)));
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + t_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((((4.0 * p) * p) + (x * x)))
	tmp = 0
	if t_0 <= -0.98:
		tmp = math.sqrt((math.pow(p, 2.0) / math.pow(x, 2.0)))
	else:
		tmp = math.sqrt((0.5 * (1.0 + t_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(Float64(4.0 * p) * p) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.98)
		tmp = sqrt(Float64((p ^ 2.0) / (x ^ 2.0)));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + t_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((((4.0 * p) * p) + (x * x)));
	tmp = 0.0;
	if (t_0 <= -0.98)
		tmp = sqrt(((p ^ 2.0) / (x ^ 2.0)));
	else
		tmp = sqrt((0.5 * (1.0 + t_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[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.98], N[Sqrt[N[(N[Power[p, 2.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	12.8
Target	12.9
Herbie	7.3

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.97999999999999998

   1. Initial program 53.5

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

   2. Simplified 53.5

      \[\leadsto \color{blue}{\sqrt{0.5 - -0.5 \cdot \frac{x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}} \]
      Proof

      [Start] 53.5	\[ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 53.5	\[ \sqrt{0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-11 [=>] 53.5	\[ \sqrt{0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - -1\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-87 [=>] 53.5	\[ \sqrt{\color{blue}{\left(-0.5\right) \cdot \left(-1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]
      metadata-eval [=>] 53.5	\[ \sqrt{\color{blue}{-0.5} \cdot \left(-1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      metadata-eval [<=] 53.5	\[ \sqrt{\color{blue}{\left(0.5 \cdot -1\right)} \cdot \left(-1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 53.5	\[ \sqrt{\color{blue}{-1 \cdot \left(0.5 \cdot -1\right) - \left(0.5 \cdot -1\right) \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} \]
      metadata-eval [=>] 53.5	\[ \sqrt{-1 \cdot \color{blue}{-0.5} - \left(0.5 \cdot -1\right) \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      metadata-eval [=>] 53.5	\[ \sqrt{\color{blue}{0.5} - \left(0.5 \cdot -1\right) \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      metadata-eval [=>] 53.5	\[ \sqrt{0.5 - \color{blue}{-0.5} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 53.5	\[ \sqrt{0.5 - -0.5 \cdot \frac{x}{\sqrt{\color{blue}{p \cdot \left(4 \cdot p\right)} + x \cdot x}}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 53.5	\[ \sqrt{0.5 - -0.5 \cdot \frac{x}{\sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)} + x \cdot x}}} \]

   3. Taylor expanded in x around -inf 30.6

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

   if -0.97999999999999998 < (/.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 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \leq -0.98:\\ \;\;\;\;\sqrt{\frac{{p}^{2}}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.5
Cost	20932

\[\begin{array}{l} t_0 := \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\\ \mathbf{if}\;t_0 \leq -0.98:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + t_0\right)}\\ \end{array} \]

Alternative 2
Error	16.0
Cost	14092

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 - -0.5 \cdot \frac{x}{2 \cdot \frac{{p}^{2}}{x} + x}}\\ \end{array} \]

Alternative 3
Error	19.8
Cost	6992

\[\begin{array}{l} \mathbf{if}\;p \leq -1.12 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 3.5 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.4 \cdot 10^{-94}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4
Error	21.0
Cost	6860

\[\begin{array}{l} \mathbf{if}\;p \leq -1.06 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-304}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 5.2 \cdot 10^{-94}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5
Error	47.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;p \leq 2.3 \cdot 10^{-304}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;-\frac{p}{x}\\ \end{array} \]

Alternative 6
Error	53.6
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))))))))