Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 5.9s

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

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

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(b, Float64(b * 4.0), -1.0))
end

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision] + -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. +-commutative 99.9%

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

   3. unpow2 99.9%

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

   4. unpow1 99.9%

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

   5. sqr-pow 99.9%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 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]

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%

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

Alternative 3: 60.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.9) (fma b (* b 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.9) {
		tmp = fma(b, (b * 4.0), -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.9)
		tmp = fma(b, Float64(b * 4.0), -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.9], N[(b * N[(b * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8999999999999999

   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. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. unpow1 99.9%

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

      5. sqr-pow 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 68.6%

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

   5. Taylor expanded in b around 0 56.0%

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

      1. sub-neg 56.0%

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

      2. metadata-eval 56.0%

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

      3. *-commutative 56.0%

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

      4. unpow2 56.0%

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

      5. associate-*r* 56.0%

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

      6. fma-udef 56.0%

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

      7. *-commutative 56.0%

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

   7. Applied egg-rr 56.0%

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

   if 1.8999999999999999 < b

   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. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. unpow1 99.9%

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

      5. sqr-pow 99.9%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 89.6%

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

   5. Taylor expanded in b around inf 89.6%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} + {b}^{4}} \]

   6. Taylor expanded in b around inf 89.4%

      \[\leadsto \color{blue}{{b}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 4: 60.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.9) (* (+ 1.0 (* b 2.0)) (+ (* b 2.0) -1.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.9) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	} else {
		tmp = pow(b, 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 (b <= 1.9d0) then
        tmp = (1.0d0 + (b * 2.0d0)) * ((b * 2.0d0) + (-1.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.9) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.9:
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.9)
		tmp = Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(Float64(b * 2.0) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.9)
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.9], N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8999999999999999

   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. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. unpow1 99.9%

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

      5. sqr-pow 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 68.6%

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

   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%

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

      4. *-commutative 56.0%

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

      5. sqrt-prod 56.0%

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

      6. unpow2 56.0%

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

      7. sqrt-prod 14.2%

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

      8. add-sqr-sqrt 32.9%

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

      9. metadata-eval 32.9%

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

      10. *-commutative 32.9%

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

      11. sqrt-prod 32.9%

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

      12. unpow2 32.9%

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

      13. sqrt-prod 14.2%

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

      14. add-sqr-sqrt 56.0%

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

      15. metadata-eval 56.0%

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

   7. Applied egg-rr 56.0%

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

   if 1.8999999999999999 < b

   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. +-commutative 99.9%

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

      3. unpow2 99.9%

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

      4. unpow1 99.9%

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

      5. sqr-pow 99.9%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 89.6%

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

   5. Taylor expanded in b around inf 89.6%

      \[\leadsto \color{blue}{4 \cdot {b}^{2} + {b}^{4}} \]

   6. Taylor expanded in b around inf 89.4%

      \[\leadsto \color{blue}{{b}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 50.6% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] + -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. +-commutative 99.9%

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

   3. unpow2 99.9%

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

   4. unpow1 99.9%

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

   5. sqr-pow 99.9%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 73.9%

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

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%

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

   4. *-commutative 56.0%

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

   5. sqrt-prod 56.0%

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

   6. unpow2 56.0%

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

   7. sqrt-prod 24.6%

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

   8. add-sqr-sqrt 38.6%

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

   9. metadata-eval 38.6%

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

   10. *-commutative 38.6%

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

   11. sqrt-prod 38.6%

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

   12. unpow2 38.6%

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

   13. sqrt-prod 24.6%

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

   14. add-sqr-sqrt 56.0%

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

   15. metadata-eval 56.0%

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

7. Applied egg-rr 56.0%

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

8. Final simplification 56.0%

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

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. +-commutative 99.9%

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

   3. unpow2 99.9%

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

   4. unpow1 99.9%

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

   5. sqr-pow 99.9%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in a around 0 73.9%

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

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))