bug366, discussion (missed optimization)

Percentage Accurate: 53.2% → 98.9%

Time: 3.8s

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: 53.2% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{\frac{a}{b}}\\ \mathbf{if}\;a \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\mathsf{fma}\left(0.5, t_0, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + t_0 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ b (/ a b))))
   (if (<= a -1e-248) (fma 0.5 t_0 (- a)) (+ a (* t_0 -0.5)))))

C

double code(double a, double b) {
	double t_0 = b / (a / b);
	double tmp;
	if (a <= -1e-248) {
		tmp = fma(0.5, t_0, -a);
	} else {
		tmp = a + (t_0 * -0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b / Float64(a / b))
	tmp = 0.0
	if (a <= -1e-248)
		tmp = fma(0.5, t_0, Float64(-a));
	else
		tmp = Float64(a + Float64(t_0 * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1e-248], N[(0.5 * t$95$0 + (-a)), $MachinePrecision], N[(a + N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999998e-249

   1. Initial program 57.3%

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

      1. difference-of-squares 57.4%

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

   3. Simplified 57.4%

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

      1. add-cube-cbrt 56.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      2. associate-*l* 56.3%

         \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right)} \]

      3. pow1/3 53.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      4. pow1/2 53.8%

         \[\leadsto {\color{blue}{\left({\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      5. difference-of-squares 53.7%

         \[\leadsto {\left({\color{blue}{\left(a \cdot a - b \cdot b\right)}}^{0.5}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      6. pow-pow 53.7%

         \[\leadsto \color{blue}{{\left(a \cdot a - b \cdot b\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      7. fma-neg 53.8%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      8. metadata-eval 53.8%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      9. cbrt-prod 53.8%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)} \cdot \sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      10. add-sqr-sqrt 53.8%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

      11. difference-of-squares 53.6%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{a \cdot a - b \cdot b}} \]

      12. fma-neg 53.7%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   5. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   6. Taylor expanded in a around -inf 97.7%

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

      1. fma-def 97.7%

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

      2. unpow2 97.7%

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

      3. associate-*l/ 100.0%

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

      4. associate-/r/ 100.0%

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

      5. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   if -9.9999999999999998e-249 < a

   1. Initial program 46.6%

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

      1. difference-of-squares 47.3%

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

   3. Simplified 47.3%

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

      1. add-cube-cbrt 46.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      2. associate-*l* 46.5%

         \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right)} \]

      3. pow1/3 44.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      4. pow1/2 44.6%

         \[\leadsto {\color{blue}{\left({\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      5. difference-of-squares 43.8%

         \[\leadsto {\left({\color{blue}{\left(a \cdot a - b \cdot b\right)}}^{0.5}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      6. pow-pow 43.8%

         \[\leadsto \color{blue}{{\left(a \cdot a - b \cdot b\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      7. fma-neg 44.6%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      8. metadata-eval 44.6%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      9. cbrt-prod 44.6%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)} \cdot \sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      10. add-sqr-sqrt 44.6%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

      11. difference-of-squares 43.8%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{a \cdot a - b \cdot b}} \]

      12. fma-neg 43.8%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   5. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   6. Taylor expanded in a around inf 90.6%

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

      1. unpow2 90.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-248}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{b}{\frac{a}{b}}, -a\right)\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b}{\frac{a}{b}} \cdot -0.5\\ \end{array} \]

Alternative 2: 98.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-248}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b}{\frac{a}{b}} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e-248) (- a) (+ a (* (/ b (/ a b)) -0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e-248) {
		tmp = -a;
	} else {
		tmp = a + ((b / (a / b)) * -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 <= (-1d-248)) then
        tmp = -a
    else
        tmp = a + ((b / (a / b)) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e-248) {
		tmp = -a;
	} else {
		tmp = a + ((b / (a / b)) * -0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e-248:
		tmp = -a
	else:
		tmp = a + ((b / (a / b)) * -0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e-248)
		tmp = Float64(-a);
	else
		tmp = Float64(a + Float64(Float64(b / Float64(a / b)) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e-248)
		tmp = -a;
	else
		tmp = a + ((b / (a / b)) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e-248], (-a), N[(a + N[(N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999998e-249

   1. Initial program 57.3%

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

      1. difference-of-squares 57.4%

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

   3. Simplified 57.4%

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

      1. add-cube-cbrt 56.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      2. associate-*l* 56.3%

         \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right)} \]

      3. pow1/3 53.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      4. pow1/2 53.8%

         \[\leadsto {\color{blue}{\left({\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      5. difference-of-squares 53.7%

         \[\leadsto {\left({\color{blue}{\left(a \cdot a - b \cdot b\right)}}^{0.5}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      6. pow-pow 53.7%

         \[\leadsto \color{blue}{{\left(a \cdot a - b \cdot b\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      7. fma-neg 53.8%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      8. metadata-eval 53.8%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      9. cbrt-prod 53.8%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)} \cdot \sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      10. add-sqr-sqrt 53.8%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

      11. difference-of-squares 53.6%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{a \cdot a - b \cdot b}} \]

      12. fma-neg 53.7%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   5. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   6. Taylor expanded in a around -inf 99.2%

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

      1. neg-mul-1 99.2%

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

   8. Simplified 99.2%

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

   if -9.9999999999999998e-249 < a

   1. Initial program 46.6%

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

      1. difference-of-squares 47.3%

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

   3. Simplified 47.3%

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

      1. add-cube-cbrt 46.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      2. associate-*l* 46.5%

         \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right)} \]

      3. pow1/3 44.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      4. pow1/2 44.6%

         \[\leadsto {\color{blue}{\left({\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      5. difference-of-squares 43.8%

         \[\leadsto {\left({\color{blue}{\left(a \cdot a - b \cdot b\right)}}^{0.5}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      6. pow-pow 43.8%

         \[\leadsto \color{blue}{{\left(a \cdot a - b \cdot b\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      7. fma-neg 44.6%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      8. metadata-eval 44.6%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      9. cbrt-prod 44.6%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)} \cdot \sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      10. add-sqr-sqrt 44.6%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

      11. difference-of-squares 43.8%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{a \cdot a - b \cdot b}} \]

      12. fma-neg 43.8%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   5. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   6. Taylor expanded in a around inf 90.6%

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

      1. unpow2 90.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-248}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + \frac{b}{\frac{a}{b}} \cdot -0.5\\ \end{array} \]

Alternative 3: 98.4% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-248}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a -1e-248) (- a) a))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e-248) {
		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 <= (-1d-248)) 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 <= -1e-248) {
		tmp = -a;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e-248:
		tmp = -a
	else:
		tmp = a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e-248)
		tmp = Float64(-a);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e-248)
		tmp = -a;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e-248], (-a), a]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999998e-249

   1. Initial program 57.3%

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

      1. difference-of-squares 57.4%

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

   3. Simplified 57.4%

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

      1. add-cube-cbrt 56.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      2. associate-*l* 56.3%

         \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right)} \]

      3. pow1/3 53.8%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(a + b\right) \cdot \left(a - b\right)}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      4. pow1/2 53.8%

         \[\leadsto {\color{blue}{\left({\left(\left(a + b\right) \cdot \left(a - b\right)\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      5. difference-of-squares 53.7%

         \[\leadsto {\left({\color{blue}{\left(a \cdot a - b \cdot b\right)}}^{0.5}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      6. pow-pow 53.7%

         \[\leadsto \color{blue}{{\left(a \cdot a - b \cdot b\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      7. fma-neg 53.8%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      8. metadata-eval 53.8%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}} \cdot \sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)}}\right) \]

      9. cbrt-prod 53.8%

         \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{\left(a + b\right) \cdot \left(a - b\right)} \cdot \sqrt{\left(a + b\right) \cdot \left(a - b\right)}}} \]

      10. add-sqr-sqrt 53.8%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}} \]

      11. difference-of-squares 53.6%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{a \cdot a - b \cdot b}} \]

      12. fma-neg 53.7%

          \[\leadsto {\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   5. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\mathsf{fma}\left(a, a, -b \cdot b\right)}} \]

   6. Taylor expanded in a around -inf 99.2%

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

      1. neg-mul-1 99.2%

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

   8. Simplified 99.2%

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

   if -9.9999999999999998e-249 < a

   1. Initial program 46.6%

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

      1. difference-of-squares 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in a around inf 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-248}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4: 49.9% 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 52.0%

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

   1. difference-of-squares 52.4%

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

3. Simplified 52.4%

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

4. Taylor expanded in a around inf 49.4%

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

5. Final simplification 49.4%

   \[\leadsto a \]

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 2023178 
(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))))