Kahan p9 Example

Average Error: 19.8 → 4.8

Time: 11.0s

Precision: binary64

Cost: 1356

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

↓

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(-\frac{y}{x}\right) + \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= y -1e+154)
     -1.0
     (if (<= y -1.55e-162)
       t_0
       (if (<= y 1.6e-162) (+ (- (/ y x)) (+ 1.0 (/ y x))) t_0)))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

↓

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (y <= -1e+154) {
		tmp = -1.0;
	} else if (y <= -1.55e-162) {
		tmp = t_0;
	} else if (y <= 1.6e-162) {
		tmp = -(y / x) + (1.0 + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (y <= (-1d+154)) then
        tmp = -1.0d0
    else if (y <= (-1.55d-162)) then
        tmp = t_0
    else if (y <= 1.6d-162) then
        tmp = -(y / x) + (1.0d0 + (y / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

↓

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (y <= -1e+154) {
		tmp = -1.0;
	} else if (y <= -1.55e-162) {
		tmp = t_0;
	} else if (y <= 1.6e-162) {
		tmp = -(y / x) + (1.0 + (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

↓

def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if y <= -1e+154:
		tmp = -1.0
	elif y <= -1.55e-162:
		tmp = t_0
	elif y <= 1.6e-162:
		tmp = -(y / x) + (1.0 + (y / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

↓

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (y <= -1e+154)
		tmp = -1.0;
	elseif (y <= -1.55e-162)
		tmp = t_0;
	elseif (y <= 1.6e-162)
		tmp = Float64(Float64(-Float64(y / x)) + Float64(1.0 + Float64(y / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

↓

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (y <= -1e+154)
		tmp = -1.0;
	elseif (y <= -1.55e-162)
		tmp = t_0;
	elseif (y <= 1.6e-162)
		tmp = -(y / x) + (1.0 + (y / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e+154], -1.0, If[LessEqual[y, -1.55e-162], t$95$0, If[LessEqual[y, 1.6e-162], N[((-N[(y / x), $MachinePrecision]) + N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Target

Original	19.8
Target	0.0
Herbie	4.8

\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000004e154

   1. Initial program 64.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

   2. Simplified 62.0

      \[\leadsto \color{blue}{\left(x + y\right) \cdot \frac{x - y}{x \cdot x + y \cdot y}} \]
      Proof

      [Start] 64.0	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-49 [=>] 62.0	\[ \color{blue}{\left(x + y\right) \cdot \frac{x - y}{x \cdot x + y \cdot y}} \]

   3. Taylor expanded in x around 0 0

      \[\leadsto \color{blue}{-1} \]

   if -1.00000000000000004e154 < y < -1.5499999999999999e-162 or 1.59999999999999988e-162 < y

   1. Initial program 0.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

   if -1.5499999999999999e-162 < y < 1.59999999999999988e-162

   1. Initial program 29.8

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]

   2. Simplified 30.5

      \[\leadsto \color{blue}{\left(x + y\right) \cdot \frac{x - y}{x \cdot x + y \cdot y}} \]
      Proof

      [Start] 29.8	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      rational.json-simplify-49 [=>] 30.5	\[ \color{blue}{\left(x + y\right) \cdot \frac{x - y}{x \cdot x + y \cdot y}} \]

   3. Taylor expanded in x around inf 15.0

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

   4. Simplified 15.0

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

      [Start] 15.0	\[ \frac{y}{x} + \left(1 + -1 \cdot \frac{y}{x}\right) \]
      rational.json-simplify-41 [<=] 15.0	\[ \color{blue}{-1 \cdot \frac{y}{x} + \left(\frac{y}{x} + 1\right)} \]
      rational.json-simplify-2 [=>] 15.0	\[ \color{blue}{\frac{y}{x} \cdot -1} + \left(\frac{y}{x} + 1\right) \]
      rational.json-simplify-9 [=>] 15.0	\[ \color{blue}{\left(-\frac{y}{x}\right)} + \left(\frac{y}{x} + 1\right) \]
      rational.json-simplify-1 [<=] 15.0	\[ \left(-\frac{y}{x}\right) + \color{blue}{\left(1 + \frac{y}{x}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(-\frac{y}{x}\right) + \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.1
Cost	1356

\[\begin{array}{l} t_0 := \left(x + y\right) \cdot \frac{x - y}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(-\frac{y}{x}\right) + \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.8
Cost	904

\[\begin{array}{l} t_0 := \left(-\frac{x}{y}\right) + \left(-1 + \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-153}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.8
Cost	904

\[\begin{array}{l} t_0 := \left(-\frac{x}{y}\right) + \left(-1 + \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-144}:\\ \;\;\;\;\left(-\frac{y}{x}\right) + \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	10.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-132}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-148}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Error	22.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))