Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 5.4s

Alternatives: 6

Speedup: 1.0×

• 61.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(b * b))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(b * b))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ (pow (hypot a b) 4.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	return pow(hypot(a, b), 4.0) + fma((b * b), 4.0, -1.0);
}

Julia

function code(a, b)
	return Float64((hypot(a, b) ^ 4.0) + fma(Float64(b * b), 4.0, -1.0))
end

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 99.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

   2. unpow2 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. unpow1 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   4. sqr-pow 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   5. associate-*r* 100.0%

      \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

4. Final simplification 100.0%

   \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ -1.0 (+ (pow (+ (* b b) (* a a)) 2.0) (* 4.0 (* b b)))))

C

double code(double a, double b) {
	return -1.0 + (pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (b * b)))
end function

Java

public static double code(double a, double b) {
	return -1.0 + (Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * b)));
}

Python

def code(a, b):
	return -1.0 + (math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (b * b)))

Julia

function code(a, b)
	return Float64(-1.0 + Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(b * b))))
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((((b * b) + (a * a)) ^ 2.0) + (4.0 * (b * b)));
end

Wolfram

code[a_, b_] := N[(-1.0 + N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

2. Final simplification 99.9%

   \[\leadsto -1 + \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) \]

Alternative 3: 77.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(-1 + b \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e+153)
   (+ -1.0 (pow a 4.0))
   (* (+ 1.0 (* b 2.0)) (+ -1.0 (* b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+153) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.1d+153) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (1.0d0 + (b * 2.0d0)) * ((-1.0d0) + (b * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+153) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.1e+153:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e+153)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(-1.0 + Float64(b * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.1e+153)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e+153], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000017e153

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. unpow2 99.9%

         \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      3. unpow1 99.9%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      4. sqr-pow 99.9%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

   4. Taylor expanded in b around 0 99.4%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{-1} \]

   5. Taylor expanded in a around inf 71.1%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]

   if 2.10000000000000017e153 < b

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. unpow2 100.0%

         \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      3. unpow1 100.0%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      4. sqr-pow 100.0%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

   4. Taylor expanded in a around 0 100.0%

      \[\leadsto \color{blue}{{b}^{4}} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

   5. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

      2. difference-of-sqr-1 100.0%

         \[\leadsto \color{blue}{\left(\sqrt{4 \cdot {b}^{2}} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right)} \]

      3. *-commutative 100.0%

         \[\leadsto \left(\sqrt{\color{blue}{{b}^{2} \cdot 4}} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

      4. sqrt-prod 100.0%

         \[\leadsto \left(\color{blue}{\sqrt{{b}^{2}} \cdot \sqrt{4}} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

      5. unpow2 100.0%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b}} \cdot \sqrt{4} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

      6. sqrt-prod 100.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt{b} \cdot \sqrt{b}\right)} \cdot \sqrt{4} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

      7. add-sqr-sqrt 100.0%

         \[\leadsto \left(\color{blue}{b} \cdot \sqrt{4} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \left(b \cdot \color{blue}{2} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

      9. *-commutative 100.0%

         \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\sqrt{\color{blue}{{b}^{2} \cdot 4}} - 1\right) \]

      10. sqrt-prod 100.0%

          \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\color{blue}{\sqrt{{b}^{2}} \cdot \sqrt{4}} - 1\right) \]

      11. unpow2 100.0%

          \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\sqrt{\color{blue}{b \cdot b}} \cdot \sqrt{4} - 1\right) \]

      12. sqrt-prod 100.0%

          \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\color{blue}{\left(\sqrt{b} \cdot \sqrt{b}\right)} \cdot \sqrt{4} - 1\right) \]

      13. add-sqr-sqrt 100.0%

          \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\color{blue}{b} \cdot \sqrt{4} - 1\right) \]

      14. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(b \cdot 2 + 1\right) \cdot \left(b \cdot 2 - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(-1 + b \cdot 2\right)\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.75 \cdot 10^{+68}:\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.75e+68) (+ -1.0 (pow b 4.0)) (+ -1.0 (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.75e+68) {
		tmp = -1.0 + pow(b, 4.0);
	} else {
		tmp = -1.0 + pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 1.75d+68) then
        tmp = (-1.0d0) + (b ** 4.0d0)
    else
        tmp = (-1.0d0) + (a ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.75e+68) {
		tmp = -1.0 + Math.pow(b, 4.0);
	} else {
		tmp = -1.0 + Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.75e+68:
		tmp = -1.0 + math.pow(b, 4.0)
	else:
		tmp = -1.0 + math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.75e+68)
		tmp = Float64(-1.0 + (b ^ 4.0));
	else
		tmp = Float64(-1.0 + (a ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.75e+68)
		tmp = -1.0 + (b ^ 4.0);
	else
		tmp = -1.0 + (a ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.75e+68], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.74999999999999989e68

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. unpow2 99.9%

         \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      3. unpow1 99.9%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      4. sqr-pow 99.9%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

   4. Taylor expanded in a around 0 84.5%

      \[\leadsto \color{blue}{{b}^{4}} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

   5. Taylor expanded in b around 0 83.9%

      \[\leadsto {b}^{4} + \color{blue}{-1} \]

   if 1.74999999999999989e68 < a

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   2. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

      2. unpow2 100.0%

         \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      3. unpow1 100.0%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      4. sqr-pow 100.0%

         \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

   4. Taylor expanded in b around 0 100.0%

      \[\leadsto {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \color{blue}{-1} \]

   5. Taylor expanded in a around inf 100.0%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.75 \cdot 10^{+68}:\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {a}^{4}\\ \end{array} \]

Alternative 5: 50.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \left(1 + b \cdot 2\right) \cdot \left(-1 + b \cdot 2\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (+ 1.0 (* b 2.0)) (+ -1.0 (* b 2.0))))

C

double code(double a, double b) {
	return (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (1.0d0 + (b * 2.0d0)) * ((-1.0d0) + (b * 2.0d0))
end function

Java

public static double code(double a, double b) {
	return (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
}

Python

def code(a, b):
	return (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0))

Julia

function code(a, b)
	return Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(-1.0 + Float64(b * 2.0)))
end

MATLAB

function tmp = code(a, b)
	tmp = (1.0 + (b * 2.0)) * (-1.0 + (b * 2.0));
end

Wolfram

code[a_, b_] := N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 99.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

   2. unpow2 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. unpow1 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   4. sqr-pow 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   5. associate-*r* 100.0%

      \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

4. Taylor expanded in a around 0 73.9%

   \[\leadsto \color{blue}{{b}^{4}} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

5. Taylor expanded in b around 0 56.0%

   \[\leadsto \color{blue}{4 \cdot {b}^{2} - 1} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 56.0%

      \[\leadsto \color{blue}{\sqrt{4 \cdot {b}^{2}} \cdot \sqrt{4 \cdot {b}^{2}}} - 1 \]

   2. difference-of-sqr-1 56.0%

      \[\leadsto \color{blue}{\left(\sqrt{4 \cdot {b}^{2}} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right)} \]

   3. *-commutative 56.0%

      \[\leadsto \left(\sqrt{\color{blue}{{b}^{2} \cdot 4}} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

   4. sqrt-prod 56.0%

      \[\leadsto \left(\color{blue}{\sqrt{{b}^{2}} \cdot \sqrt{4}} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

   5. unpow2 56.0%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b}} \cdot \sqrt{4} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

   6. sqrt-prod 24.6%

      \[\leadsto \left(\color{blue}{\left(\sqrt{b} \cdot \sqrt{b}\right)} \cdot \sqrt{4} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

   7. add-sqr-sqrt 38.6%

      \[\leadsto \left(\color{blue}{b} \cdot \sqrt{4} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

   8. metadata-eval 38.6%

      \[\leadsto \left(b \cdot \color{blue}{2} + 1\right) \cdot \left(\sqrt{4 \cdot {b}^{2}} - 1\right) \]

   9. *-commutative 38.6%

      \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\sqrt{\color{blue}{{b}^{2} \cdot 4}} - 1\right) \]

   10. sqrt-prod 38.6%

       \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\color{blue}{\sqrt{{b}^{2}} \cdot \sqrt{4}} - 1\right) \]

   11. unpow2 38.6%

       \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\sqrt{\color{blue}{b \cdot b}} \cdot \sqrt{4} - 1\right) \]

   12. sqrt-prod 24.6%

       \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\color{blue}{\left(\sqrt{b} \cdot \sqrt{b}\right)} \cdot \sqrt{4} - 1\right) \]

   13. add-sqr-sqrt 56.0%

       \[\leadsto \left(b \cdot 2 + 1\right) \cdot \left(\color{blue}{b} \cdot \sqrt{4} - 1\right) \]

   14. metadata-eval 56.0%

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

7. Applied egg-rr 56.0%

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

8. Final simplification 56.0%

   \[\leadsto \left(1 + b \cdot 2\right) \cdot \left(-1 + b \cdot 2\right) \]

Alternative 6: 24.9% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

double code(double a, double b) {
	return -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0;
end

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Step-by-step derivation

   1. associate--l+ 99.9%

      \[\leadsto \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(4 \cdot \left(b \cdot b\right) - 1\right)} \]

   2. unpow2 99.9%

      \[\leadsto \color{blue}{\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   3. unpow1 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{1}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   4. sqr-pow 99.9%

      \[\leadsto \left(a \cdot a + b \cdot b\right) \cdot \color{blue}{\left({\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

   5. associate-*r* 100.0%

      \[\leadsto \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{\left(\frac{1}{2}\right)}} + \left(4 \cdot \left(b \cdot b\right) - 1\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(b \cdot b, 4, -1\right)} \]

4. Taylor expanded in a around 0 73.9%

   \[\leadsto \color{blue}{{b}^{4}} + \mathsf{fma}\left(b \cdot b, 4, -1\right) \]

5. Taylor expanded in b around 0 25.0%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 25.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))