Kahan p9 Example

Average Accuracy: 67.9% → 100.0%

Time: 13.7s

Precision: binary64

Cost: 1216

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

Math

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

↓

\[\frac{x - y}{\frac{y}{\frac{x + y}{y}} + x \cdot \frac{x}{x + y}} \]

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

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)
	return Float64(Float64(x - y) / Float64(Float64(y / Float64(Float64(x + y) / y)) + Float64(x * Float64(x / Float64(x + y)))))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (x - y) / ((y / ((x + y) / y)) + (x * (x / (x + y))));
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_] := N[(N[(x - y), $MachinePrecision] / N[(N[(y / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	67.9%
Target	99.9%
Herbie	100.0%

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

Derivation

1. Initial program 67.9%

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

2. Simplified 67.8%

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

   [Start] 67.9	\[ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   associate-/l* [=>] 67.8	\[ \color{blue}{\frac{x - y}{\frac{x \cdot x + y \cdot y}{x + y}}} \]
   fma-def [=>] 67.8	\[ \frac{x - y}{\frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)}}{x + y}} \]

3. Applied egg-rr 67.5%

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

   [Start] 67.8	\[ \frac{x - y}{\frac{\mathsf{fma}\left(x, x, y \cdot y\right)}{x + y}} \]
   div-inv [=>] 67.5	\[ \frac{x - y}{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right) \cdot \frac{1}{x + y}}} \]
   *-commutative [=>] 67.5	\[ \frac{x - y}{\color{blue}{\frac{1}{x + y} \cdot \mathsf{fma}\left(x, x, y \cdot y\right)}} \]
   fma-udef [=>] 67.5	\[ \frac{x - y}{\frac{1}{x + y} \cdot \color{blue}{\left(x \cdot x + y \cdot y\right)}} \]
   +-commutative [=>] 67.5	\[ \frac{x - y}{\frac{1}{x + y} \cdot \color{blue}{\left(y \cdot y + x \cdot x\right)}} \]
   distribute-lft-in [=>] 67.5	\[ \frac{x - y}{\color{blue}{\frac{1}{x + y} \cdot \left(y \cdot y\right) + \frac{1}{x + y} \cdot \left(x \cdot x\right)}} \]

4. Simplified 67.5%

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

   [Start] 67.5	\[ \frac{x - y}{\frac{1}{x + y} \cdot \left(y \cdot y\right) + \frac{1}{x + y} \cdot \left(x \cdot x\right)} \]
   distribute-lft-out [=>] 67.5	\[ \frac{x - y}{\color{blue}{\frac{1}{x + y} \cdot \left(y \cdot y + x \cdot x\right)}} \]
   +-commutative [=>] 67.5	\[ \frac{x - y}{\frac{1}{\color{blue}{y + x}} \cdot \left(y \cdot y + x \cdot x\right)} \]

5. Applied egg-rr 75.6%

   \[\leadsto \frac{x - y}{\color{blue}{\frac{y \cdot y}{y + x} + \frac{x}{\frac{y + x}{x}}}} \]
   Proof

   [Start] 67.5	\[ \frac{x - y}{\frac{1}{y + x} \cdot \left(y \cdot y + x \cdot x\right)} \]
   distribute-rgt-in [=>] 67.5	\[ \frac{x - y}{\color{blue}{\left(y \cdot y\right) \cdot \frac{1}{y + x} + \left(x \cdot x\right) \cdot \frac{1}{y + x}}} \]
   un-div-inv [=>] 67.7	\[ \frac{x - y}{\color{blue}{\frac{y \cdot y}{y + x}} + \left(x \cdot x\right) \cdot \frac{1}{y + x}} \]
   un-div-inv [=>] 67.8	\[ \frac{x - y}{\frac{y \cdot y}{y + x} + \color{blue}{\frac{x \cdot x}{y + x}}} \]
   associate-/l* [=>] 75.6	\[ \frac{x - y}{\frac{y \cdot y}{y + x} + \color{blue}{\frac{x}{\frac{y + x}{x}}}} \]

6. Simplified 100.0%

   \[\leadsto \frac{x - y}{\color{blue}{\frac{y}{\frac{x + y}{y}} + \frac{x}{x + y} \cdot x}} \]
   Proof

   [Start] 75.6	\[ \frac{x - y}{\frac{y \cdot y}{y + x} + \frac{x}{\frac{y + x}{x}}} \]
   associate-/l* [=>] 100.0	\[ \frac{x - y}{\color{blue}{\frac{y}{\frac{y + x}{y}}} + \frac{x}{\frac{y + x}{x}}} \]
   +-commutative [=>] 100.0	\[ \frac{x - y}{\frac{y}{\frac{\color{blue}{x + y}}{y}} + \frac{x}{\frac{y + x}{x}}} \]
   associate-/r/ [=>] 100.0	\[ \frac{x - y}{\frac{y}{\frac{x + y}{y}} + \color{blue}{\frac{x}{y + x} \cdot x}} \]
   +-commutative [=>] 100.0	\[ \frac{x - y}{\frac{y}{\frac{x + y}{y}} + \frac{x}{\color{blue}{x + y}} \cdot x} \]

7. Final simplification 100.0%

   \[\leadsto \frac{x - y}{\frac{y}{\frac{x + y}{y}} + x \cdot \frac{x}{x + y}} \]

Alternatives

Alternative 1
Accuracy	91.7%
Cost	1620

\[\begin{array}{l} t_0 := 1 + \frac{\frac{y}{\frac{x}{y}} \cdot -2}{x}\\ t_1 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{x - y}{\left(y - x\right) + \frac{x \cdot x}{y}}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	82.6%
Cost	1362

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-111} \lor \neg \left(y \leq 8.5 \cdot 10^{-198} \lor \neg \left(y \leq 3.2 \cdot 10^{-175}\right) \land y \leq 4 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{x - y}{\left(y - x\right) + \frac{x \cdot x}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{y}{\frac{x}{y}} \cdot -2}{x}\\ \end{array} \]

Alternative 3
Accuracy	82.3%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-110}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-198} \lor \neg \left(y \leq 3.6 \cdot 10^{-175}\right) \land y \leq 3.3 \cdot 10^{-131}:\\ \;\;\;\;1 + \frac{\frac{y}{\frac{x}{y}} \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	1216

\[\frac{x - y}{x \cdot \frac{x}{x + y} + \frac{y}{1 + \frac{x}{y}}} \]

Alternative 5
Accuracy	82.1%
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-137}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-198}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-175}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6
Accuracy	66.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023137 
(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))))