bug366, discussion (missed optimization)

Percentage Accurate: 53.6% → 99.3%

Time: 4.0s

Alternatives: 5

Speedup: 26.3×

Specification

Math

\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{a \cdot a - b \cdot b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return math.sqrt(((a * a) - (b * b)))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a}{b}}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-282)
   (- (/ 0.5 (/ (/ a b) b)) a)
   (* (sqrt (- a b)) (sqrt (+ a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e-282) {
		tmp = (0.5 / ((a / b) / b)) - a;
	} else {
		tmp = sqrt((a - b)) * sqrt((a + b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2d-282)) then
        tmp = (0.5d0 / ((a / b) / b)) - a
    else
        tmp = sqrt((a - b)) * sqrt((a + b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-282) {
		tmp = (0.5 / ((a / b) / b)) - a;
	} else {
		tmp = Math.sqrt((a - b)) * Math.sqrt((a + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e-282:
		tmp = (0.5 / ((a / b) / b)) - a
	else:
		tmp = math.sqrt((a - b)) * math.sqrt((a + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e-282)
		tmp = Float64(Float64(0.5 / Float64(Float64(a / b) / b)) - a);
	else
		tmp = Float64(sqrt(Float64(a - b)) * sqrt(Float64(a + b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-282)
		tmp = (0.5 / ((a / b) / b)) - a;
	else
		tmp = sqrt((a - b)) * sqrt((a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e-282], N[(N[(0.5 / N[(N[(a / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], N[(N[Sqrt[N[(a - b), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e-282

   1. Initial program 56.7%

      \[\sqrt{a \cdot a - b \cdot b} \]

   2. Taylor expanded in a around -inf 96.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a} \]
   3. Step-by-step derivation

      1. mul-1-neg 96.3%

         \[\leadsto 0.5 \cdot \frac{{b}^{2}}{a} + \color{blue}{\left(-a\right)} \]

      2. unsub-neg 96.3%

         \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} - a} \]

      3. unpow2 96.3%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{b \cdot b}}{a} - a \]

      4. associate-*r/ 96.3%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b \cdot b\right)}{a}} - a \]

      5. associate-/l* 96.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{b \cdot b}}} - a \]

      6. associate-/r* 99.7%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{a}{b}}{b}}} - a \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{a}{b}}{b}} - a} \]

   if -2e-282 < a

   1. Initial program 49.9%

      \[\sqrt{a \cdot a - b \cdot b} \]
   2. Step-by-step derivation

      1. difference-of-squares 50.2%

         \[\leadsto \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 50.2%

         \[\leadsto \sqrt{\color{blue}{\left(a - b\right) \cdot \left(a + b\right)}} \]

      2. sqrt-prod 99.2%

         \[\leadsto \color{blue}{\sqrt{a - b} \cdot \sqrt{a + b}} \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\sqrt{a - b} \cdot \sqrt{a + b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a}{b}}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a}{b}}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-282) (- (/ 0.5 (/ (/ a b) b)) a) (fma -0.5 (/ b (/ a b)) a)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e-282) {
		tmp = (0.5 / ((a / b) / b)) - a;
	} else {
		tmp = fma(-0.5, (b / (a / b)), a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e-282)
		tmp = Float64(Float64(0.5 / Float64(Float64(a / b) / b)) - a);
	else
		tmp = fma(-0.5, Float64(b / Float64(a / b)), a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e-282], N[(N[(0.5 / N[(N[(a / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], N[(-0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e-282

   1. Initial program 56.7%

      \[\sqrt{a \cdot a - b \cdot b} \]

   2. Taylor expanded in a around -inf 96.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a} \]
   3. Step-by-step derivation

      1. mul-1-neg 96.3%

         \[\leadsto 0.5 \cdot \frac{{b}^{2}}{a} + \color{blue}{\left(-a\right)} \]

      2. unsub-neg 96.3%

         \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} - a} \]

      3. unpow2 96.3%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{b \cdot b}}{a} - a \]

      4. associate-*r/ 96.3%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b \cdot b\right)}{a}} - a \]

      5. associate-/l* 96.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{b \cdot b}}} - a \]

      6. associate-/r* 99.7%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{a}{b}}{b}}} - a \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{a}{b}}{b}} - a} \]

   if -2e-282 < a

   1. Initial program 49.9%

      \[\sqrt{a \cdot a - b \cdot b} \]

   2. Taylor expanded in a around inf 95.3%

      \[\leadsto \color{blue}{a + -0.5 \cdot \frac{{b}^{2}}{a}} \]
   3. Step-by-step derivation

      1. +-commutative 95.3%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{{b}^{2}}{a} + a} \]

      2. fma-def 95.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{{b}^{2}}{a}, a\right)} \]

      3. unpow2 95.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{\color{blue}{b \cdot b}}{a}, a\right) \]

      4. associate-/l* 99.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{b}{\frac{a}{b}}}, a\right) \]

   4. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a}{b}}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a}{b}}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-282) (- (/ 0.5 (/ (/ a b) b)) a) a))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e-282) {
		tmp = (0.5 / ((a / b) / b)) - a;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2d-282)) then
        tmp = (0.5d0 / ((a / b) / b)) - a
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-282) {
		tmp = (0.5 / ((a / b) / b)) - a;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e-282:
		tmp = (0.5 / ((a / b) / b)) - a
	else:
		tmp = a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e-282)
		tmp = Float64(Float64(0.5 / Float64(Float64(a / b) / b)) - a);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-282)
		tmp = (0.5 / ((a / b) / b)) - a;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e-282], N[(N[(0.5 / N[(N[(a / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if a < -2e-282

   1. Initial program 56.7%

      \[\sqrt{a \cdot a - b \cdot b} \]

   2. Taylor expanded in a around -inf 96.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} + -1 \cdot a} \]
   3. Step-by-step derivation

      1. mul-1-neg 96.3%

         \[\leadsto 0.5 \cdot \frac{{b}^{2}}{a} + \color{blue}{\left(-a\right)} \]

      2. unsub-neg 96.3%

         \[\leadsto \color{blue}{0.5 \cdot \frac{{b}^{2}}{a} - a} \]

      3. unpow2 96.3%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{b \cdot b}}{a} - a \]

      4. associate-*r/ 96.3%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b \cdot b\right)}{a}} - a \]

      5. associate-/l* 96.3%

         \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{b \cdot b}}} - a \]

      6. associate-/r* 99.7%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{a}{b}}{b}}} - a \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{\frac{a}{b}}{b}} - a} \]

   if -2e-282 < a

   1. Initial program 49.9%

      \[\sqrt{a \cdot a - b \cdot b} \]
   2. Step-by-step derivation

      1. difference-of-squares 50.2%

         \[\leadsto \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \]

   4. Taylor expanded in a around inf 98.3%

      \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{a}{b}}{b}} - a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4: 99.0% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a -2e-282) (- a) a))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e-282) {
		tmp = -a;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2d-282)) then
        tmp = -a
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-282) {
		tmp = -a;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e-282:
		tmp = -a
	else:
		tmp = a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e-282)
		tmp = Float64(-a);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-282)
		tmp = -a;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e-282], (-a), a]

Derivation

1. Split input into 2 regimes

2. if a < -2e-282

   1. Initial program 56.7%

      \[\sqrt{a \cdot a - b \cdot b} \]

   2. Taylor expanded in a around -inf 99.4%

      \[\leadsto \color{blue}{-1 \cdot a} \]
   3. Step-by-step derivation

      1. mul-1-neg 99.4%

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

   4. Simplified 99.4%

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

   if -2e-282 < a

   1. Initial program 49.9%

      \[\sqrt{a \cdot a - b \cdot b} \]
   2. Step-by-step derivation

      1. difference-of-squares 50.2%

         \[\leadsto \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \]

   4. Taylor expanded in a around inf 98.3%

      \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-282}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 5: 50.5% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a b) :precision binary64 a)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return a

Julia

function code(a, b)
	return a
end

MATLAB

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

Wolfram

code[a_, b_] := a

Derivation

1. Initial program 53.1%

   \[\sqrt{a \cdot a - b \cdot b} \]
2. Step-by-step derivation

   1. difference-of-squares 53.4%

      \[\leadsto \sqrt{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

3. Simplified 53.4%

   \[\leadsto \color{blue}{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \]

4. Taylor expanded in a around inf 52.9%

   \[\leadsto \color{blue}{a} \]

5. Final simplification 52.9%

   \[\leadsto a \]

Developer target: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|a\right| + \left|b\right|} \cdot \sqrt{\left|a\right| - \left|b\right|} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (sqrt (+ (fabs a) (fabs b))) (sqrt (- (fabs a) (fabs b)))))

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return Math.sqrt((Math.abs(a) + Math.abs(b))) * Math.sqrt((Math.abs(a) - Math.abs(b)));
}

Python

def code(a, b):
	return math.sqrt((math.fabs(a) + math.fabs(b))) * math.sqrt((math.fabs(a) - math.fabs(b)))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(FPCore (a b)
  :name "bug366, discussion (missed optimization)"
  :precision binary64

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

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