Difference of squares

Percentage Accurate: 93.4% → 96.6%

Time: 1.9s

Precision: binary64

Cost: 708

• 43.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[a \cdot a - b \cdot b \]

↓

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+304}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))

↓

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+304) (- (* a a) (* b b)) (* b (- b))))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

↓

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+304) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) - (b * b)
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 1d+304) then
        tmp = (a * a) - (b * b)
    else
        tmp = b * -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	return (a * a) - (b * b);
}

↓

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+304) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Python

def code(a, b):
	return (a * a) - (b * b)

↓

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+304:
		tmp = (a * a) - (b * b)
	else:
		tmp = b * -b
	return tmp

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

↓

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+304)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64(b * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) - (b * b);
end

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+304)
		tmp = (a * a) - (b * b);
	else
		tmp = b * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+304], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * (-b)), $MachinePrecision]]

Target

Original	93.4%
Target	100.0%
Herbie	96.6%

\[\left(a + b\right) \cdot \left(a - b\right) \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.9999999999999994e303

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]

   if 9.9999999999999994e303 < (*.f64 b b)

   1. Initial program 79.0%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around 0 90.3%

      \[\leadsto \color{blue}{-1 \cdot {b}^{2}} \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{b \cdot \left(-b\right)} \]
      Step-by-step derivation

      [Start] 90.3%	\[ -1 \cdot {b}^{2} \]
      unpow2 [=>] 90.3%	\[ -1 \cdot \color{blue}{\left(b \cdot b\right)} \]
      mul-1-neg [=>] 90.3%	\[ \color{blue}{-b \cdot b} \]
      distribute-rgt-neg-in [=>] 90.3%	\[ \color{blue}{b \cdot \left(-b\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+304}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+304}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]

Alternative 2
Accuracy	96.7%
Cost	6784

\[\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right) \]

Alternative 3
Accuracy	75.4%
Cost	521

\[\begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+109} \lor \neg \left(b \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 4
Accuracy	53.8%
Cost	192

\[a \cdot a \]

Reproduce

herbie shell --seed 2023178 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

  (- (* a a) (* b b)))