2isqrt (example 3.6)

Average Error: 20.0 → 0.3

Time: 7.9s

Precision: binary64

Cost: 26756

Math

\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

↓

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - t_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - t_0\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (+ 1.0 x)))))
   (if (<= (- (/ 1.0 (sqrt x)) t_0) 5e-15)
     (* (pow x -1.5) 0.5)
     (- (pow x -0.5) t_0))))

C

double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}

↓

double code(double x) {
	double t_0 = 1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - t_0) <= 5e-15) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = pow(x, -0.5) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - t_0) <= 5d-15) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = (x ** (-0.5d0)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

↓

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - t_0) <= 5e-15) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = Math.pow(x, -0.5) - t_0;
	}
	return tmp;
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

↓

def code(x):
	t_0 = 1.0 / math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - t_0) <= 5e-15:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = math.pow(x, -0.5) - t_0
	return tmp

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

↓

function code(x)
	t_0 = Float64(1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - t_0) <= 5e-15)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64((x ^ -0.5) - t_0);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

↓

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - t_0) <= 5e-15)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = (x ^ -0.5) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], 5e-15], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - t$95$0), $MachinePrecision]]]

Target

Original	20.0
Target	0.7
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.99999999999999999e-15

   1. Initial program 40.2

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Applied egg-rr 40.2

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - \frac{1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]

   3. Taylor expanded in x around -inf 64.0

      \[\leadsto \color{blue}{\frac{0.5}{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)} \cdot {x}^{2}}} \]

   4. Simplified 11.5

      \[\leadsto \color{blue}{\frac{{\left(\frac{-1}{\frac{-1}{x}}\right)}^{0.5}}{\frac{x \cdot x}{0.5}}} \]
      Proof
      (/.f64 (pow.f64 (/.f64 -1 (/.f64 -1 x)) 1/2) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 (/.f64 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 -1))) (/.f64 -1 x)) 1/2) (/.f64 (*.f64 x x) 1/2)): 195 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 (/.f64 (exp.f64 (log.f64 -1)) (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 (/.f64 -1 x))))) 1/2) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 (Rewrite<= exp-diff_binary64 (exp.f64 (-.f64 (log.f64 -1) (log.f64 (/.f64 -1 x))))) 1/2) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 (exp.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 -1) (neg.f64 (log.f64 (/.f64 -1 x)))))) 1/2) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 (exp.f64 (+.f64 (log.f64 -1) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 -1 x)))))) 1/2) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 (exp.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) 1/2) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) 1/2))) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (*.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) (Rewrite<= metadata-eval (neg.f64 -1/2)))) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)) -1/2)))) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))))) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= rec-exp_binary64 (/.f64 1 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))))) (/.f64 (*.f64 x x) 1/2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) 1/2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 1 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) 1/2) (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 1/2) (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))))) (pow.f64 x 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite=> metadata-eval 1/2) (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1))))) (pow.f64 x 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 1/2 (*.f64 (exp.f64 (*.f64 -1/2 (+.f64 (*.f64 -1 (log.f64 (/.f64 -1 x))) (log.f64 -1)))) (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.2

      \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]

   if 4.99999999999999999e-15 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 0.7

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Applied egg-rr 0.5

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{\sqrt{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13760

\[\frac{\frac{\frac{1}{x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{1 + x} \]

Alternative 2
Error	0.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 3541444.854294857:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]

Alternative 3
Error	1.1
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 2.698043830480501 \cdot 10^{-5}:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]

Alternative 4
Error	2.4
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 2.698043830480501 \cdot 10^{-5}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{{x}^{1.5}}\\ \end{array} \]

Alternative 5
Error	2.0
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 2.698043830480501 \cdot 10^{-5}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]

Alternative 6
Error	1.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 2.698043830480501 \cdot 10^{-5}:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]

Alternative 7
Error	31.6
Cost	6528

\[{x}^{-0.5} \]

Alternative 8
Error	59.3
Cost	192

\[\frac{1}{x} \]

Reproduce

herbie shell --seed 2022295 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))