Kahan p9 Example

Percentage Accurate: 68.4% → 91.6%

Time: 28.3s

Precision: binary64

Cost: 1357

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-67}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-156} \lor \neg \left(y \leq 1.55 \cdot 10^{-162}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e-67)
   -1.0
   (if (or (<= y -2.35e-156) (not (<= y 1.55e-162)))
     (* (- x y) (/ (+ x y) (+ (* x x) (* y y))))
     (+ 1.0 (* -2.0 (* (/ y x) (/ y x)))))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e-67) {
		tmp = -1.0;
	} else if ((y <= -2.35e-156) || !(y <= 1.55e-162)) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	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) :: tmp
    if (y <= (-4.7d-67)) then
        tmp = -1.0d0
    else if ((y <= (-2.35d-156)) .or. (.not. (y <= 1.55d-162))) then
        tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
    else
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    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 tmp;
	if (y <= -4.7e-67) {
		tmp = -1.0;
	} else if ((y <= -2.35e-156) || !(y <= 1.55e-162)) {
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	} else {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if y <= -4.7e-67:
		tmp = -1.0
	elif (y <= -2.35e-156) or not (y <= 1.55e-162):
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)))
	else:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	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)
	tmp = 0.0
	if (y <= -4.7e-67)
		tmp = -1.0;
	elseif ((y <= -2.35e-156) || !(y <= 1.55e-162))
		tmp = Float64(Float64(x - y) * Float64(Float64(x + y) / Float64(Float64(x * x) + Float64(y * y))));
	else
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	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)
	tmp = 0.0;
	if (y <= -4.7e-67)
		tmp = -1.0;
	elseif ((y <= -2.35e-156) || ~((y <= 1.55e-162)))
		tmp = (x - y) * ((x + y) / ((x * x) + (y * y)));
	else
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	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_] := If[LessEqual[y, -4.7e-67], -1.0, If[Or[LessEqual[y, -2.35e-156], N[Not[LessEqual[y, 1.55e-162]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	68.4%
Target	99.9%
Herbie	91.6%

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

Derivation

1. Split input into 3 regimes

2. if y < -4.70000000000000004e-67

   1. Initial program 57.1%

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

   2. Simplified 58.3%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
      Step-by-step derivation

      [Start] 57.1%	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      associate-*r/ [<=] 58.3%	\[ \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
      +-commutative [=>] 58.3%	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{y \cdot y + x \cdot x}} \]
      fma-def [=>] 58.3%	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]

   3. Taylor expanded in x around 0 100.0%

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

   if -4.70000000000000004e-67 < y < -2.35000000000000023e-156 or 1.5499999999999999e-162 < y

   1. Initial program 100.0%

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

   2. Simplified 99.5%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      associate-*r/ [<=] 99.5%	\[ \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
      +-commutative [=>] 99.5%	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{y \cdot y + x \cdot x}} \]
      fma-def [=>] 99.5%	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]

   3. Applied egg-rr 99.5%

      \[\leadsto \left(x - y\right) \cdot \frac{x + y}{\color{blue}{y \cdot y + x \cdot x}} \]
      Step-by-step derivation

      [Start] 99.5%	\[ \left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)} \]
      fma-udef [=>] 99.5%	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{y \cdot y + x \cdot x}} \]

   if -2.35000000000000023e-156 < y < 1.5499999999999999e-162

   1. Initial program 54.3%

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

   2. Simplified 54.9%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
      Step-by-step derivation

      [Start] 54.3%	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
      associate-*r/ [<=] 54.9%	\[ \color{blue}{\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}} \]
      +-commutative [=>] 54.9%	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{y \cdot y + x \cdot x}} \]
      fma-def [=>] 54.9%	\[ \left(x - y\right) \cdot \frac{x + y}{\color{blue}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]

   3. Taylor expanded in y around 0 54.3%

      \[\leadsto \color{blue}{1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}}} \]

   4. Simplified 54.3%

      \[\leadsto \color{blue}{1 + -2 \cdot \frac{y \cdot y}{x \cdot x}} \]
      Step-by-step derivation

      [Start] 54.3%	\[ 1 + -2 \cdot \frac{{y}^{2}}{{x}^{2}} \]
      unpow2 [=>] 54.3%	\[ 1 + -2 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]
      unpow2 [=>] 54.3%	\[ 1 + -2 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

   5. Applied egg-rr 81.1%

      \[\leadsto 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
      Step-by-step derivation

      [Start] 54.3%	\[ 1 + -2 \cdot \frac{y \cdot y}{x \cdot x} \]
      times-frac [=>] 81.1%	\[ 1 + -2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-67}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-156} \lor \neg \left(y \leq 1.55 \cdot 10^{-162}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	91.6%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-67}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-156} \lor \neg \left(y \leq 1.55 \cdot 10^{-162}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{x + y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	92.8%
Cost	8004

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 3
Accuracy	92.6%
Cost	1988

\[\begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \]

Alternative 4
Accuracy	80.9%
Cost	1371

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-136} \lor \neg \left(y \leq -8.5 \cdot 10^{-169}\right) \land \left(y \leq -5.4 \cdot 10^{-233} \lor \neg \left(y \leq 6 \cdot 10^{-171}\right) \land \left(y \leq 2.9 \cdot 10^{-142} \lor \neg \left(y \leq 1.82 \cdot 10^{-121}\right)\right)\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	80.9%
Cost	1370

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;1 - \frac{y \cdot y}{x \cdot x}\\ \mathbf{elif}\;\neg \left(y \leq -5.4 \cdot 10^{-233}\right) \land \left(y \leq 7.2 \cdot 10^{-171} \lor \neg \left(y \leq 2.1 \cdot 10^{-143}\right) \land y \leq 1.85 \cdot 10^{-120}\right):\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	83.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-136} \lor \neg \left(y \leq 1.2 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 7
Accuracy	83.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-121}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{y \cdot y}\\ \end{array} \]

Alternative 8
Accuracy	83.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-136}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + -1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-120}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \frac{x + y}{y}}{y}\\ \end{array} \]

Alternative 9
Accuracy	82.7%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-136}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10
Accuracy	66.7%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023277 
(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))))