bug366, discussion (missed optimization)

Percentage Accurate: 54.2% → 99.1%

Time: 7.3s

Alternatives: 4

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: 54.2% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e-271) (- a) (* (sqrt (- a b)) (sqrt (+ a b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e-271) {
		tmp = -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 <= (-5d-271)) then
        tmp = -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 <= -5e-271) {
		tmp = -a;
	} else {
		tmp = Math.sqrt((a - b)) * Math.sqrt((a + b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e-271:
		tmp = -a
	else:
		tmp = math.sqrt((a - b)) * math.sqrt((a + b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e-271)
		tmp = Float64(-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 <= -5e-271)
		tmp = -a;
	else
		tmp = sqrt((a - b)) * sqrt((a + b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e-271], (-a), 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 < -5.0000000000000002e-271

   1. Initial program 57.9%

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

      1. difference-of-squares 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in a around inf 1.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(b + -1 \cdot b\right) + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
   5. Step-by-step derivation

      1. fma-def 1.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, b + -1 \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      2. distribute-rgt1-in 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(-1 + 1\right) \cdot b}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      3. metadata-eval 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0} \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      4. mul0-lft 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      5. fma-udef 1.6%

         \[\leadsto \color{blue}{0.5 \cdot 0 + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      6. metadata-eval 1.6%

         \[\leadsto \color{blue}{0} + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      7. +-lft-identity 1.6%

         \[\leadsto \color{blue}{0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a} \]

   6. Simplified 1.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)} \]
   7. Step-by-step derivation

      1. frac-2neg 1.1%

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

      2. div-inv 1.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(-b\right) \cdot \frac{1}{-\frac{a}{b}}}, a\right) \]

      3. distribute-neg-frac 1.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \left(-b\right) \cdot \frac{1}{\color{blue}{\frac{-a}{b}}}, a\right) \]

   8. Applied egg-rr 1.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(-b\right) \cdot \frac{1}{\frac{-a}{b}}}, a\right) \]
   9. Step-by-step derivation

      1. un-div-inv 1.1%

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

      2. associate-/r/ 1.1%

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

      3. frac-2neg 1.1%

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

      4. *-commutative 1.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 57.5%

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

      7. sqr-neg 57.5%

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

      8. sqrt-unprod 98.5%

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

      9. add-sqr-sqrt 99.1%

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

      10. fma-neg 99.1%

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

      11. *-commutative 99.1%

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

      12. *-commutative 99.1%

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

      13. associate-*l* 99.1%

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

   10. Applied egg-rr 99.1%

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

   11. Taylor expanded in b around 0 99.2%

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

       1. neg-mul-1 99.2%

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

   13. Simplified 99.2%

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

   if -5.0000000000000002e-271 < a

   1. Initial program 55.6%

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

      1. difference-of-squares 56.2%

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

   3. Simplified 56.2%

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

      1. *-commutative 56.2%

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

      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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{a - b} \cdot \sqrt{a + b}\\ \end{array} \]

Alternative 2: 99.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \left(b \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e-271) (- a) (+ a (* (* b (/ b a)) -0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e-271) {
		tmp = -a;
	} else {
		tmp = a + ((b * (b / a)) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5d-271)) then
        tmp = -a
    else
        tmp = a + ((b * (b / a)) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e-271) {
		tmp = -a;
	} else {
		tmp = a + ((b * (b / a)) * -0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e-271:
		tmp = -a
	else:
		tmp = a + ((b * (b / a)) * -0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e-271)
		tmp = Float64(-a);
	else
		tmp = Float64(a + Float64(Float64(b * Float64(b / a)) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e-271)
		tmp = -a;
	else
		tmp = a + ((b * (b / a)) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e-271], (-a), N[(a + N[(N[(b * N[(b / a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.0000000000000002e-271

   1. Initial program 57.9%

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

      1. difference-of-squares 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in a around inf 1.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(b + -1 \cdot b\right) + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
   5. Step-by-step derivation

      1. fma-def 1.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, b + -1 \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      2. distribute-rgt1-in 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(-1 + 1\right) \cdot b}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      3. metadata-eval 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0} \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      4. mul0-lft 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      5. fma-udef 1.6%

         \[\leadsto \color{blue}{0.5 \cdot 0 + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      6. metadata-eval 1.6%

         \[\leadsto \color{blue}{0} + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      7. +-lft-identity 1.6%

         \[\leadsto \color{blue}{0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a} \]

   6. Simplified 1.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)} \]
   7. Step-by-step derivation

      1. frac-2neg 1.1%

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

      2. div-inv 1.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(-b\right) \cdot \frac{1}{-\frac{a}{b}}}, a\right) \]

      3. distribute-neg-frac 1.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \left(-b\right) \cdot \frac{1}{\color{blue}{\frac{-a}{b}}}, a\right) \]

   8. Applied egg-rr 1.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(-b\right) \cdot \frac{1}{\frac{-a}{b}}}, a\right) \]
   9. Step-by-step derivation

      1. un-div-inv 1.1%

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

      2. associate-/r/ 1.1%

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

      3. frac-2neg 1.1%

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

      4. *-commutative 1.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 57.5%

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

      7. sqr-neg 57.5%

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

      8. sqrt-unprod 98.5%

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

      9. add-sqr-sqrt 99.1%

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

      10. fma-neg 99.1%

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

      11. *-commutative 99.1%

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

      12. *-commutative 99.1%

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

      13. associate-*l* 99.1%

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

   10. Applied egg-rr 99.1%

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

   11. Taylor expanded in b around 0 99.2%

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

       1. neg-mul-1 99.2%

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

   13. Simplified 99.2%

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

   if -5.0000000000000002e-271 < a

   1. Initial program 55.6%

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

      1. difference-of-squares 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in a around inf 91.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(b + -1 \cdot b\right) + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
   5. Step-by-step derivation

      1. fma-def 91.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, b + -1 \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      2. distribute-rgt1-in 91.7%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(-1 + 1\right) \cdot b}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      3. metadata-eval 91.7%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0} \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      4. mul0-lft 91.7%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      5. fma-udef 91.7%

         \[\leadsto \color{blue}{0.5 \cdot 0 + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      6. metadata-eval 91.7%

         \[\leadsto \color{blue}{0} + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      7. +-lft-identity 91.7%

         \[\leadsto \color{blue}{0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a} \]

   6. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)} \]
   7. Step-by-step derivation

      1. fma-udef 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-/r/ 99.1%

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

      4. *-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \left(b \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array} \]

Alternative 3: 99.0% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a -5e-271) (- a) a))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e-271) {
		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 <= (-5d-271)) 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 <= -5e-271) {
		tmp = -a;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e-271:
		tmp = -a
	else:
		tmp = a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e-271)
		tmp = Float64(-a);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e-271)
		tmp = -a;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e-271], (-a), a]

Derivation

1. Split input into 2 regimes

2. if a < -5.0000000000000002e-271

   1. Initial program 57.9%

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

      1. difference-of-squares 58.4%

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

   3. Simplified 58.4%

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

   4. Taylor expanded in a around inf 1.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(b + -1 \cdot b\right) + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]
   5. Step-by-step derivation

      1. fma-def 1.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, b + -1 \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      2. distribute-rgt1-in 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(-1 + 1\right) \cdot b}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      3. metadata-eval 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0} \cdot b, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      4. mul0-lft 1.6%

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{0}, 0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      5. fma-udef 1.6%

         \[\leadsto \color{blue}{0.5 \cdot 0 + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right)} \]

      6. metadata-eval 1.6%

         \[\leadsto \color{blue}{0} + \left(0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a\right) \]

      7. +-lft-identity 1.6%

         \[\leadsto \color{blue}{0.5 \cdot \frac{-1 \cdot {b}^{2} - {\left(0.5 \cdot \left(b + -1 \cdot b\right)\right)}^{2}}{a} + a} \]

   6. Simplified 1.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{\frac{a}{b}}, a\right)} \]
   7. Step-by-step derivation

      1. frac-2neg 1.1%

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

      2. div-inv 1.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(-b\right) \cdot \frac{1}{-\frac{a}{b}}}, a\right) \]

      3. distribute-neg-frac 1.1%

         \[\leadsto \mathsf{fma}\left(-0.5, \left(-b\right) \cdot \frac{1}{\color{blue}{\frac{-a}{b}}}, a\right) \]

   8. Applied egg-rr 1.1%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(-b\right) \cdot \frac{1}{\frac{-a}{b}}}, a\right) \]
   9. Step-by-step derivation

      1. un-div-inv 1.1%

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

      2. associate-/r/ 1.1%

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

      3. frac-2neg 1.1%

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

      4. *-commutative 1.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 57.5%

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

      7. sqr-neg 57.5%

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

      8. sqrt-unprod 98.5%

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

      9. add-sqr-sqrt 99.1%

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

      10. fma-neg 99.1%

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

      11. *-commutative 99.1%

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

      12. *-commutative 99.1%

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

      13. associate-*l* 99.1%

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

   10. Applied egg-rr 99.1%

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

   11. Taylor expanded in b around 0 99.2%

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

       1. neg-mul-1 99.2%

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

   13. Simplified 99.2%

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

   if -5.0000000000000002e-271 < a

   1. Initial program 55.6%

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

      1. difference-of-squares 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in a around inf 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-271}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4: 49.0% 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 56.7%

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

   1. difference-of-squares 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in a around inf 49.0%

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

5. Final simplification 49.0%

   \[\leadsto a \]

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 2023187 
(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))))