Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.4s

Alternatives: 5

Speedup: 1.0×

• 60.4% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 85.7%

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

   2. *-un-lft-identity 85.7%

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

   3. log-prod 85.7%

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

   4. metadata-eval 85.7%

      \[\leadsto \left(\left(\color{blue}{0} + \log \left(e^{{\left(a \cdot a + b \cdot b\right)}^{2}}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   5. add-log-exp 99.9%

      \[\leadsto \left(\left(0 + \color{blue}{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   6. add-sqr-sqrt 99.9%

      \[\leadsto \left(\left(0 + {\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}}^{2}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   7. pow2 99.9%

      \[\leadsto \left(\left(0 + {\color{blue}{\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}\right)}}^{2}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   8. hypot-define 99.9%

      \[\leadsto \left(\left(0 + {\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{2}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(\color{blue}{\left(0 + {\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{2}\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
5. Step-by-step derivation

   1. +-lft-identity 99.9%

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

   2. unpow2 99.9%

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

   3. pow-sqr 100.0%

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

   4. hypot-undefine 100.0%

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

   5. unpow2 100.0%

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

   6. unpow2 100.0%

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

   7. +-commutative 100.0%

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

   8. unpow2 100.0%

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

   9. unpow2 100.0%

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

   10. hypot-define 100.0%

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

   11. metadata-eval 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 99.9%

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

Alternative 3: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 1.3 \cdot 10^{+34}:\\ \;\;\;\;\left(t\_0 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= a 1.3e+34)
     (+ (+ 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 <= 1.3e+34) {
		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 <= 1.3d+34) 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 <= 1.3e+34) {
		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 <= 1.3e+34:
		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 (a <= 1.3e+34)
		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 <= 1.3e+34)
		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[a, 1.3e+34], 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 a < 1.29999999999999999e34

   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. Add Preprocessing

   3. Taylor expanded in a around 0 82.3%

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

   if 1.29999999999999999e34 < 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. Add Preprocessing

   3. Taylor expanded in a around inf 95.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+34}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $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. Add Preprocessing

3. Taylor expanded in a around inf 83.7%

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

4. Final simplification 83.7%

   \[\leadsto \left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1 \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {b}^{2} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(4.0 * N[Power[b, 2.0], $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. Add Preprocessing

3. Taylor expanded in a around 0 69.9%

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

4. Taylor expanded in b around 0 50.6%

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

5. Final simplification 50.6%

   \[\leadsto 4 \cdot {b}^{2} + -1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))