Kahan p9 Example

Average Error: 20.1 → 5.4

Time: 3.9s

Precision: binary64

Cost: 1488

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-172}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ y x)) (+ (* x x) (* y y)))))
   (if (<= y -1.5e+154)
     -1.0
     (if (<= y -4.1e-168)
       t_0
       (if (<= y -1.62e-189)
         -1.0
         (if (<= y 4.5e-172) (- (+ (/ 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) * (y + x)) / ((x * x) + (y * y));
	double tmp;
	if (y <= -1.5e+154) {
		tmp = -1.0;
	} else if (y <= -4.1e-168) {
		tmp = t_0;
	} else if (y <= -1.62e-189) {
		tmp = -1.0;
	} else if (y <= 4.5e-172) {
		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) * (y + x)) / ((x * x) + (y * y))
    if (y <= (-1.5d+154)) then
        tmp = -1.0d0
    else if (y <= (-4.1d-168)) then
        tmp = t_0
    else if (y <= (-1.62d-189)) then
        tmp = -1.0d0
    else if (y <= 4.5d-172) 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) * (y + x)) / ((x * x) + (y * y));
	double tmp;
	if (y <= -1.5e+154) {
		tmp = -1.0;
	} else if (y <= -4.1e-168) {
		tmp = t_0;
	} else if (y <= -1.62e-189) {
		tmp = -1.0;
	} else if (y <= 4.5e-172) {
		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) * (y + x)) / ((x * x) + (y * y))
	tmp = 0
	if y <= -1.5e+154:
		tmp = -1.0
	elif y <= -4.1e-168:
		tmp = t_0
	elif y <= -1.62e-189:
		tmp = -1.0
	elif y <= 4.5e-172:
		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(y + x)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (y <= -1.5e+154)
		tmp = -1.0;
	elseif (y <= -4.1e-168)
		tmp = t_0;
	elseif (y <= -1.62e-189)
		tmp = -1.0;
	elseif (y <= 4.5e-172)
		tmp = Float64(Float64(Float64(y / x) + 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) * (y + x)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (y <= -1.5e+154)
		tmp = -1.0;
	elseif (y <= -4.1e-168)
		tmp = t_0;
	elseif (y <= -1.62e-189)
		tmp = -1.0;
	elseif (y <= 4.5e-172)
		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[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e+154], -1.0, If[LessEqual[y, -4.1e-168], t$95$0, If[LessEqual[y, -1.62e-189], -1.0, If[LessEqual[y, 4.5e-172], N[(N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Target

Original	20.1
Target	0.1
Herbie	5.4

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

Derivation

1. Split input into 3 regimes

2. if y < -1.50000000000000013e154 or -4.0999999999999998e-168 < y < -1.62e-189

   1. Initial program 59.3

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

   2. Taylor expanded in x around 0 5.1

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

   if -1.50000000000000013e154 < y < -4.0999999999999998e-168 or 4.50000000000000004e-172 < y

   1. Initial program 1.2

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

   if -1.62e-189 < y < 4.50000000000000004e-172

   1. Initial program 30.2

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

   2. Taylor expanded in x around inf 13.9

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

   3. Simplified 13.9

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

      [Start] 13.9	\[ \frac{y}{x} + \left(1 + -1 \cdot \frac{y}{x}\right) \]
      associate-+r+ [=>] 13.9	\[ \color{blue}{\left(\frac{y}{x} + 1\right) + -1 \cdot \frac{y}{x}} \]
      mul-1-neg [=>] 13.9	\[ \left(\frac{y}{x} + 1\right) + \color{blue}{\left(-\frac{y}{x}\right)} \]

   4. Applied egg-rr 13.9

      \[\leadsto \color{blue}{\left(\frac{y}{x} + 1\right) - \frac{y}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.62 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-172}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.8
Cost	1105

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-133}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-168} \lor \neg \left(y \leq -8 \cdot 10^{-190}\right) \land y \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	11.7
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-147}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-189}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	21.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023018 
(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))))