Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 3.8s

Alternatives: 6

Speedup: 1.0×

• 61.4% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + ((b * b) * 4.0)) + -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) + ((b * b) * 4.0d0)) + (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + ((b * b) * 4.0)) + -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[(N[(b * b), $MachinePrecision] * 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. Final simplification 99.9%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(t\_0 + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + {b}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) 4.0)))
   (if (<= (* b b) 2e+14)
     (+ (+ t_0 (pow a 4.0)) -1.0)
     (+ (+ t_0 (pow b 4.0)) -1.0))))

C

double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if ((b * b) <= 2e+14) {
		tmp = (t_0 + pow(a, 4.0)) + -1.0;
	} else {
		tmp = (t_0 + pow(b, 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 = (b * b) * 4.0d0
    if ((b * b) <= 2d+14) then
        tmp = (t_0 + (a ** 4.0d0)) + (-1.0d0)
    else
        tmp = (t_0 + (b ** 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if ((b * b) <= 2e+14) {
		tmp = (t_0 + Math.pow(a, 4.0)) + -1.0;
	} else {
		tmp = (t_0 + Math.pow(b, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * 4.0
	tmp = 0
	if (b * b) <= 2e+14:
		tmp = (t_0 + math.pow(a, 4.0)) + -1.0
	else:
		tmp = (t_0 + math.pow(b, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (Float64(b * b) <= 2e+14)
		tmp = Float64(Float64(t_0 + (a ^ 4.0)) + -1.0);
	else
		tmp = Float64(Float64(t_0 + (b ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * 4.0;
	tmp = 0.0;
	if ((b * b) <= 2e+14)
		tmp = (t_0 + (a ^ 4.0)) + -1.0;
	else
		tmp = (t_0 + (b ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 2e+14], N[(N[(t$95$0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(t$95$0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e14

   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 99.7%

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

   if 2e14 < (*.f64 b 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. Add Preprocessing

   3. Taylor expanded in a around 0 95.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + {b}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $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 88.1%

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

4. Final simplification 88.1%

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

Alternative 4: 77.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{+148}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e+148) (+ (pow a 4.0) -1.0) (* 4.0 (pow b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e+148) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = 4.0 * pow(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 <= 1d+148) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = 4.0d0 * (b ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1e+148) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = 4.0 * Math.pow(b, 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e+148:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = 4.0 * math.pow(b, 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e+148)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(4.0 * (b ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1e+148)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = 4.0 * (b ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e+148], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1e148

   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 86.4%

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

   4. Taylor expanded in a around inf 72.5%

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

   if 1e148 < 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. Add Preprocessing

   3. Taylor expanded in a around inf 97.7%

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

   4. Taylor expanded in a around 0 97.7%

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

   5. Taylor expanded in b around inf 97.7%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{+148}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.056:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.056) -1.0 (* 4.0 (pow b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.056) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * pow(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 <= 0.056d0) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.056) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * Math.pow(b, 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.056:
		tmp = -1.0
	else:
		tmp = 4.0 * math.pow(b, 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.056)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * (b ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.056)
		tmp = -1.0;
	else
		tmp = 4.0 * (b ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.056], -1.0, N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0560000000000000012

   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 91.8%

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

   4. Taylor expanded in a around 0 55.2%

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

   5. Taylor expanded in b around 0 29.1%

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

   if 0.0560000000000000012 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf 77.5%

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

   4. Taylor expanded in a around 0 58.7%

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

   5. Taylor expanded in b around inf 58.7%

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

Alternative 6: 25.0% 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. Add Preprocessing

3. Taylor expanded in a around inf 88.1%

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

4. Taylor expanded in a around 0 56.1%

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

5. Taylor expanded in b around 0 21.8%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))