bug366, discussion (missed optimization)

Percentage Accurate: 54.0% → 99.2%

Time: 4.9s

Alternatives: 6

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: 54.0% 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: 99.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + -0.5 \cdot \frac{b}{\frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e-294) (- a) (+ a (* -0.5 (/ b (/ a b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e-294) {
		tmp = -a;
	} else {
		tmp = a + (-0.5 * (b / (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-294)) then
        tmp = -a
    else
        tmp = a + ((-0.5d0) * (b / (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2e-294) {
		tmp = -a;
	} else {
		tmp = a + (-0.5 * (b / (a / b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e-294:
		tmp = -a
	else:
		tmp = a + (-0.5 * (b / (a / b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e-294)
		tmp = Float64(-a);
	else
		tmp = Float64(a + Float64(-0.5 * Float64(b / Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-294)
		tmp = -a;
	else
		tmp = a + (-0.5 * (b / (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e-294], (-a), N[(a + N[(-0.5 * N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000003e-294

   1. Initial program 59.3%

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

      1. difference-of-squares 59.7%

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

   3. Simplified 59.7%

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

      1. add-cbrt-cube 38.9%

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

      2. pow1/3 36.3%

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

      3. add-sqr-sqrt 36.3%

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

      4. add-exp-log 36.3%

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

      5. add-exp-log 36.4%

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

      6. prod-exp 36.4%

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

      7. pow-exp 54.7%

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

   5. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

   6. Taylor expanded in a around -inf 99.5%

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

      1. neg-mul-1 99.5%

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

   8. Simplified 99.5%

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

   if -2.00000000000000003e-294 < a

   1. Initial program 51.6%

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

      1. add-cbrt-cube 36.7%

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

      2. pow1/3 34.4%

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

      3. add-sqr-sqrt 34.4%

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

      4. add-exp-log 34.4%

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

      5. add-exp-log 34.3%

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

      6. prod-exp 34.3%

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

      7. pow-exp 48.2%

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

   5. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

   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%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in b around 0 90.6%

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

       1. unpow2 90.6%

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

       2. associate-/l* 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a + -0.5 \cdot \frac{b}{\frac{a}{b}}\\ \end{array} \]

Alternative 2: 98.9% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a -2e-294) (- a) a))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2e-294) {
		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-294)) 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-294) {
		tmp = -a;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2e-294:
		tmp = -a
	else:
		tmp = a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2e-294)
		tmp = Float64(-a);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2e-294)
		tmp = -a;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2e-294], (-a), a]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000003e-294

   1. Initial program 59.3%

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

      1. difference-of-squares 59.7%

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

   3. Simplified 59.7%

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

      1. add-cbrt-cube 38.9%

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

      2. pow1/3 36.3%

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

      3. add-sqr-sqrt 36.3%

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

      4. add-exp-log 36.3%

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

      5. add-exp-log 36.4%

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

      6. prod-exp 36.4%

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

      7. pow-exp 54.7%

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

   5. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

   6. Taylor expanded in a around -inf 99.5%

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

      1. neg-mul-1 99.5%

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

   8. Simplified 99.5%

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

   if -2.00000000000000003e-294 < a

   1. Initial program 51.6%

      \[\sqrt{a \cdot a - b \cdot b} \]
   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 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-294}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 3: 52.6% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;729\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a -4.5e-205) 729.0 a))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-205) {
		tmp = 729.0;
	} 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 <= (-4.5d-205)) then
        tmp = 729.0d0
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5e-205) {
		tmp = 729.0;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.5e-205:
		tmp = 729.0
	else:
		tmp = a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.5e-205)
		tmp = 729.0;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5e-205)
		tmp = 729.0;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.5e-205], 729.0, a]

Derivation

1. Split input into 2 regimes

2. if a < -4.49999999999999956e-205

   1. Initial program 61.0%

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

      1. difference-of-squares 61.5%

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

   3. Simplified 61.5%

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

      1. add-cbrt-cube 40.0%

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

      2. pow1/3 37.3%

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

      3. add-sqr-sqrt 37.3%

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

      4. add-exp-log 37.2%

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

      5. add-exp-log 37.3%

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

      6. prod-exp 37.3%

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

      7. pow-exp 56.3%

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

   5. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

   6. Taylor expanded in a around -inf 90.7%

      \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{-1}{a}\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg 90.7%

         \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

   8. Simplified 90.7%

      \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

   10. Applied egg-rr 5.7%

       \[\leadsto \color{blue}{729} \]

   if -4.49999999999999956e-205 < a

   1. Initial program 50.3%

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

      1. difference-of-squares 51.0%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in a around inf 96.4%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;729\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 4: 2.7% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.0)

C

double code(double a, double b) {
	return 0.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.0d0
end function

Java

public static double code(double a, double b) {
	return 0.0;
}

Python

def code(a, b):
	return 0.0

Julia

function code(a, b)
	return 0.0
end

MATLAB

function tmp = code(a, b)
	tmp = 0.0;
end

Wolfram

code[a_, b_] := 0.0

Derivation

1. Initial program 55.4%

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

   1. difference-of-squares 56.0%

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

3. Simplified 56.0%

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

   1. add-cbrt-cube 37.8%

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

   2. pow1/3 35.3%

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

   3. add-sqr-sqrt 35.3%

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

   4. add-exp-log 35.3%

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

   5. add-exp-log 35.3%

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

   6. prod-exp 35.3%

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

   7. pow-exp 51.4%

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

5. Applied egg-rr 50.9%

   \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

6. Taylor expanded in a around -inf 45.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{-1}{a}\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg 45.0%

      \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

8. Simplified 45.0%

   \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

10. Applied egg-rr 2.8%

    \[\leadsto \color{blue}{0} \]

11. Final simplification 2.8%

    \[\leadsto 0 \]

Alternative 5: 5.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0.037037037037037035 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.037037037037037035)

C

double code(double a, double b) {
	return 0.037037037037037035;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.037037037037037035d0
end function

Java

public static double code(double a, double b) {
	return 0.037037037037037035;
}

Python

def code(a, b):
	return 0.037037037037037035

Julia

function code(a, b)
	return 0.037037037037037035
end

MATLAB

function tmp = code(a, b)
	tmp = 0.037037037037037035;
end

Wolfram

code[a_, b_] := 0.037037037037037035

Derivation

1. Initial program 55.4%

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

   1. difference-of-squares 56.0%

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

3. Simplified 56.0%

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

   1. add-cbrt-cube 37.8%

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

   2. pow1/3 35.3%

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

   3. add-sqr-sqrt 35.3%

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

   4. add-exp-log 35.3%

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

   5. add-exp-log 35.3%

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

   6. prod-exp 35.3%

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

   7. pow-exp 51.4%

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

5. Applied egg-rr 50.9%

   \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

6. Taylor expanded in a around -inf 45.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{-1}{a}\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg 45.0%

      \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

8. Simplified 45.0%

   \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

10. Applied egg-rr 5.3%

    \[\leadsto \color{blue}{0.037037037037037035} \]

11. Final simplification 5.3%

    \[\leadsto 0.037037037037037035 \]

Alternative 6: 5.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 729 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 729.0)

C

double code(double a, double b) {
	return 729.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 729.0d0
end function

Java

public static double code(double a, double b) {
	return 729.0;
}

Python

def code(a, b):
	return 729.0

Julia

function code(a, b)
	return 729.0
end

MATLAB

function tmp = code(a, b)
	tmp = 729.0;
end

Wolfram

code[a_, b_] := 729.0

Derivation

1. Initial program 55.4%

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

   1. difference-of-squares 56.0%

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

3. Simplified 56.0%

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

   1. add-cbrt-cube 37.8%

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

   2. pow1/3 35.3%

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

   3. add-sqr-sqrt 35.3%

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

   4. add-exp-log 35.3%

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

   5. add-exp-log 35.3%

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

   6. prod-exp 35.3%

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

   7. pow-exp 51.4%

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

5. Applied egg-rr 50.9%

   \[\leadsto \color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{fma}\left(a, a, -b \cdot b\right)\right)\right) \cdot 0.3333333333333333}} \]

6. Taylor expanded in a around -inf 45.0%

   \[\leadsto e^{\color{blue}{-1 \cdot \log \left(\frac{-1}{a}\right)}} \]
7. Step-by-step derivation

   1. mul-1-neg 45.0%

      \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

8. Simplified 45.0%

   \[\leadsto e^{\color{blue}{-\log \left(\frac{-1}{a}\right)}} \]

10. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{729} \]

11. Final simplification 5.4%

    \[\leadsto 729 \]

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 2023185 
(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))))