Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.4s

Alternatives: 9

Speedup: 1.0×

• 60.9% 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. 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%

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

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: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+57}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 4e+57)
   (+ (* b (* b (fma b b 4.0))) -1.0)
   (+ (+ (* 4.0 (* b b)) (pow a 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 4e+57) {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + pow(a, 4.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 4e+57)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (a ^ 4.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 4e+57], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.00000000000000019e57

   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. Taylor expanded in a around 0 97.2%

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

      1. unpow2 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in b around 0 97.3%

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

      1. metadata-eval 97.3%

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

      2. pow-sqr 97.2%

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

      3. unpow2 97.2%

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

      4. unpow2 97.2%

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

      5. unpow2 97.2%

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

      6. distribute-rgt-in 97.2%

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

      7. fma-udef 97.2%

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

      8. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

   if 4.00000000000000019e57 < (*.f64 a a)

   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. Taylor expanded in a around inf 97.9%

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

      1. unpow2 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in a around 0 97.9%

      \[\leadsto \left(\color{blue}{{a}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+57}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]

Alternative 4: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+57}:\\ \;\;\;\;\left(t_0 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= (* a a) 4e+57)
     (+ (+ t_0 (pow b 4.0)) -1.0)
     (+ (+ t_0 (pow a 4.0)) -1.0))))

C

double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if ((a * a) <= 4e+57) {
		tmp = (t_0 + pow(b, 4.0)) + -1.0;
	} else {
		tmp = (t_0 + pow(a, 4.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (b * b)
    if ((a * a) <= 4d+57) then
        tmp = (t_0 + (b ** 4.0d0)) + (-1.0d0)
    else
        tmp = (t_0 + (a ** 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 4.0 * (b * b);
	double tmp;
	if ((a * a) <= 4e+57) {
		tmp = (t_0 + Math.pow(b, 4.0)) + -1.0;
	} else {
		tmp = (t_0 + Math.pow(a, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 4.0 * (b * b)
	tmp = 0
	if (a * a) <= 4e+57:
		tmp = (t_0 + math.pow(b, 4.0)) + -1.0
	else:
		tmp = (t_0 + math.pow(a, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(4.0 * Float64(b * b))
	tmp = 0.0
	if (Float64(a * a) <= 4e+57)
		tmp = Float64(Float64(t_0 + (b ^ 4.0)) + -1.0);
	else
		tmp = Float64(Float64(t_0 + (a ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 4.0 * (b * b);
	tmp = 0.0;
	if ((a * a) <= 4e+57)
		tmp = (t_0 + (b ^ 4.0)) + -1.0;
	else
		tmp = (t_0 + (a ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * a), $MachinePrecision], 4e+57], N[(N[(t$95$0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(t$95$0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.00000000000000019e57

   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. Taylor expanded in a around 0 97.2%

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

      1. unpow2 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in b around 0 97.3%

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

   if 4.00000000000000019e57 < (*.f64 a a)

   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. Taylor expanded in a around inf 97.9%

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

      1. unpow2 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in a around 0 97.9%

      \[\leadsto \left(\color{blue}{{a}^{4}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4 \cdot 10^{+57}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]

Alternative 5: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2.4 \cdot 10^{+57}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 2.4e+57)
   (+ (* b (* b (fma b b 4.0))) -1.0)
   (+ (+ (* 4.0 (* b b)) (* (* a a) (* a a))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 2.4e+57) {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 2.4e+57)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + Float64(Float64(a * a) * Float64(a * a))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 2.4e+57], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 2.40000000000000005e57

   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. Taylor expanded in a around 0 97.2%

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

      1. unpow2 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in b around 0 97.3%

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

      1. metadata-eval 97.3%

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

      2. pow-sqr 97.2%

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

      3. unpow2 97.2%

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

      4. unpow2 97.2%

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

      5. unpow2 97.2%

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

      6. distribute-rgt-in 97.2%

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

      7. fma-udef 97.2%

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

      8. associate-*l* 97.2%

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

   7. Simplified 97.2%

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

   if 2.40000000000000005e57 < (*.f64 a a)

   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. Taylor expanded in a around inf 97.9%

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

      1. unpow2 97.9%

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

   4. Simplified 97.9%

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

      1. unpow2 97.9%

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

   6. Applied egg-rr 97.9%

      \[\leadsto \left(\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 2.4 \cdot 10^{+57}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 6: 96.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 9.8 \cdot 10^{+57}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 9.8e+57)
   (+ (* (* b b) (+ 4.0 (* b b))) -1.0)
   (+ (+ (* 4.0 (* b b)) (* (* a a) (* a a))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 9.8e+57) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 9.8d+57) then
        tmp = ((b * b) * (4.0d0 + (b * b))) + (-1.0d0)
    else
        tmp = ((4.0d0 * (b * b)) + ((a * a) * (a * a))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 9.8e+57) {
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 9.8e+57:
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0
	else:
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 9.8e+57)
		tmp = Float64(Float64(Float64(b * b) * Float64(4.0 + Float64(b * b))) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + Float64(Float64(a * a) * Float64(a * a))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 9.8e+57)
		tmp = ((b * b) * (4.0 + (b * b))) + -1.0;
	else
		tmp = ((4.0 * (b * b)) + ((a * a) * (a * a))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 9.8e+57], N[(N[(N[(b * b), $MachinePrecision] * N[(4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 9.7999999999999998e57

   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. Taylor expanded in a around 0 97.2%

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

      1. unpow2 97.2%

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

   4. Simplified 97.2%

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

      1. unpow-prod-down 97.2%

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

      2. pow-sqr 97.3%

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

      3. metadata-eval 97.3%

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

      4. +-commutative 97.3%

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

      5. metadata-eval 97.3%

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

      6. pow-prod-up 97.2%

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

      7. pow2 97.2%

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

      8. pow2 97.2%

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

      9. distribute-rgt-out 97.2%

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

   6. Applied egg-rr 97.2%

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

   if 9.7999999999999998e57 < (*.f64 a a)

   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. Taylor expanded in a around inf 97.9%

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

      1. unpow2 97.9%

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

   4. Simplified 97.9%

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

      1. unpow2 97.9%

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

   6. Applied egg-rr 97.9%

      \[\leadsto \left(\color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 9.8 \cdot 10^{+57}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) + -1\\ \end{array} \]

Alternative 7: 69.5% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return ((b * b) * (4.0 + (b * b))) + -1.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $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. Taylor expanded in a around 0 72.7%

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

   1. unpow2 72.7%

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

4. Simplified 72.7%

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

   1. unpow-prod-down 72.7%

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

   2. pow-sqr 72.8%

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

   3. metadata-eval 72.8%

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

   4. +-commutative 72.8%

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

   5. metadata-eval 72.8%

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

   6. pow-prod-up 72.7%

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

   7. pow2 72.7%

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

   8. pow2 72.7%

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

   9. distribute-rgt-out 72.7%

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

6. Applied egg-rr 72.7%

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

7. Final simplification 72.7%

   \[\leadsto \left(b \cdot b\right) \cdot \left(4 + b \cdot b\right) + -1 \]

Alternative 8: 51.3% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return (4.0 * (b * b)) + -1.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(4.0 * N[(b * b), $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. Taylor expanded in a around 0 72.7%

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

   1. unpow2 72.7%

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

4. Simplified 72.7%

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

5. Taylor expanded in b around 0 52.3%

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

   1. unpow2 52.3%

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

7. Simplified 52.3%

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

8. Final simplification 52.3%

   \[\leadsto 4 \cdot \left(b \cdot b\right) + -1 \]

Alternative 9: 25.1% 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* 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%

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

4. Taylor expanded in a around 0 72.8%

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

5. Taylor expanded in b around 0 26.2%

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

6. Final simplification 26.2%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))