bug366, discussion (missed optimization)

Percentage Accurate: 53.5% → 100.0%

Time: 4.3s

Alternatives: 3

Speedup: 107.0×

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: 53.5% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \frac{a_m - b_m}{\frac{\sqrt{a_m - b_m}}{\sqrt{a_m + b_m}}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (/ (- a_m b_m) (/ (sqrt (- a_m b_m)) (sqrt (+ a_m b_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	return (a_m - b_m) / (sqrt((a_m - b_m)) / sqrt((a_m + b_m)));
}

Fortran

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

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m) {
	return (a_m - b_m) / (Math.sqrt((a_m - b_m)) / Math.sqrt((a_m + b_m)));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	return (a_m - b_m) / (math.sqrt((a_m - b_m)) / math.sqrt((a_m + b_m)))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	return Float64(Float64(a_m - b_m) / Float64(sqrt(Float64(a_m - b_m)) / sqrt(Float64(a_m + b_m))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = (a_m - b_m) / (sqrt((a_m - b_m)) / sqrt((a_m + b_m)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := N[(N[(a$95$m - b$95$m), $MachinePrecision] / N[(N[Sqrt[N[(a$95$m - b$95$m), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(a$95$m + b$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. difference-of-squares 57.3%

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

   2. sqrt-prod 49.6%

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

   3. *-commutative 49.6%

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

3. Applied egg-rr 49.6%

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

   1. flip-+ 26.4%

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

   2. sqrt-div 26.5%

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

   3. difference-of-squares 26.8%

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

   4. sqrt-unprod 49.6%

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

   5. *-commutative 49.6%

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

   6. associate-*r/ 33.5%

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

5. Applied egg-rr 33.7%

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

   1. associate-/l* 50.0%

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

   2. +-commutative 50.0%

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

7. Simplified 50.0%

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

8. Final simplification 50.0%

   \[\leadsto \frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{a + b}}} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \left(a_m - b_m\right) \cdot \sqrt{\frac{a_m + b_m}{a_m - b_m}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (* (- a_m b_m) (sqrt (/ (+ a_m b_m) (- a_m b_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	return Float64(Float64(a_m - b_m) * sqrt(Float64(Float64(a_m + b_m) / Float64(a_m - b_m))))
end

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := N[(N[(a$95$m - b$95$m), $MachinePrecision] * N[Sqrt[N[(N[(a$95$m + b$95$m), $MachinePrecision] / N[(a$95$m - b$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. difference-of-squares 57.3%

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

   2. sqrt-prod 49.6%

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

   3. *-commutative 49.6%

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

3. Applied egg-rr 49.6%

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

   1. flip-+ 26.4%

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

   2. sqrt-div 26.5%

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

   3. difference-of-squares 26.8%

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

   4. sqrt-unprod 49.6%

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

   5. *-commutative 49.6%

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

   6. associate-*r/ 33.5%

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

5. Applied egg-rr 33.7%

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

   1. associate-/l* 50.0%

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

   2. +-commutative 50.0%

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

7. Simplified 50.0%

   \[\leadsto \color{blue}{\frac{a - b}{\frac{\sqrt{a - b}}{\sqrt{b + a}}}} \]
8. Step-by-step derivation

   1. div-inv 50.0%

      \[\leadsto \color{blue}{\left(a - b\right) \cdot \frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}}} \]

   2. *-commutative 50.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}} \cdot \left(a - b\right)} \]

   3. sub-neg 50.0%

      \[\leadsto \frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}} \cdot \color{blue}{\left(a + \left(-b\right)\right)} \]

   4. distribute-rgt-in 50.0%

      \[\leadsto \color{blue}{a \cdot \frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}} + \left(-b\right) \cdot \frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}}} \]

   5. clear-num 50.0%

      \[\leadsto a \cdot \color{blue}{\frac{\sqrt{b + a}}{\sqrt{a - b}}} + \left(-b\right) \cdot \frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}} \]

   6. sqrt-undiv 50.0%

      \[\leadsto a \cdot \color{blue}{\sqrt{\frac{b + a}{a - b}}} + \left(-b\right) \cdot \frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}} \]

   7. +-commutative 50.0%

      \[\leadsto a \cdot \sqrt{\frac{\color{blue}{a + b}}{a - b}} + \left(-b\right) \cdot \frac{1}{\frac{\sqrt{a - b}}{\sqrt{b + a}}} \]

   8. clear-num 50.0%

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

   9. sqrt-undiv 50.6%

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

   10. +-commutative 50.6%

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

9. Applied egg-rr 50.6%

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

    1. cancel-sign-sub-inv 50.6%

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

    2. distribute-rgt-out-- 50.6%

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

    3. *-commutative 50.6%

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

11. Simplified 50.6%

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

12. Final simplification 50.6%

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

Alternative 3: 99.1% accurate, 107.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ a_m \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m) :precision binary64 a_m)

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	return a_m;
}

Fortran

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

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m) {
	return a_m;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	return a_m

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	return a_m
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = a_m;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := a$95$m

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in a around inf 50.3%

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

3. Final simplification 50.3%

   \[\leadsto a \]

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 2023336 
(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))))