Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 7.6s

Alternatives: 7

Speedup: 1.0×

• 61.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[Power[N[Sqrt[a ^ 2 + b ^ 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 87.8%

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

   2. *-un-lft-identity 87.8%

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

   3. log-prod 87.8%

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

   4. metadata-eval 87.8%

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

   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. metadata-eval 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+62) (+ (+ (* 4.0 (* b b)) (pow a 4.0)) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+62) {
		tmp = ((4.0 * (b * b)) + pow(a, 4.0)) + -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 * b) <= 5d+62) then
        tmp = ((4.0d0 * (b * b)) + (a ** 4.0d0)) + (-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 * b) <= 5e+62) {
		tmp = ((4.0 * (b * b)) + Math.pow(a, 4.0)) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+62:
		tmp = ((4.0 * (b * b)) + math.pow(a, 4.0)) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+62)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (a ^ 4.0)) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+62)
		tmp = ((4.0 * (b * b)) + (a ^ 4.0)) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+62], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000029e62

   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 96.4%

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

   if 5.00000000000000029e62 < (*.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 95.1%

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

   4. Taylor expanded in b around inf 95.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 4.1 \cdot 10^{+16}:\\ \;\;\;\;\left(t\_0 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= a 4.1e+16)
     (+ (+ 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 <= 4.1e+16) {
		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 <= 4.1d+16) 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 <= 4.1e+16) {
		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 <= 4.1e+16:
		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 <= 4.1e+16)
		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 <= 4.1e+16)
		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, 4.1e+16], 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 < 4.1e16

   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 84.9%

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

   if 4.1e16 < 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 97.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+16}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{+20}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 4.6e+20) (+ (* 4.0 (* b b)) -1.0) (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 4.6e+20) {
		tmp = (4.0 * (b * b)) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 4.6d+20) then
        tmp = (4.0d0 * (b * b)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 4.6e+20) {
		tmp = (4.0 * (b * b)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 4.6e+20:
		tmp = (4.0 * (b * b)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 4.6e+20)
		tmp = Float64(Float64(4.0 * Float64(b * b)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 4.6e+20)
		tmp = (4.0 * (b * b)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 4.6e+20], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.6e20

   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 83.6%

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

   4. Taylor expanded in a around 0 64.5%

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

      1. pow2 64.5%

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

   6. Applied egg-rr 64.5%

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

   if 4.6e20 < 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 97.4%

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

   4. Taylor expanded in a around inf 92.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{+20}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% 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. Add Preprocessing

3. Taylor expanded in a around inf 87.8%

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

4. Taylor expanded in a around 0 52.1%

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

   1. pow2 52.1%

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

6. Applied egg-rr 52.1%

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

7. Final simplification 52.1%

   \[\leadsto 4 \cdot \left(b \cdot b\right) + -1 \]
8. Add Preprocessing

Alternative 7: 25.3% 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 0 70.3%

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

4. Taylor expanded in b around 0 22.9%

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

Reproduce

herbie shell --seed 2024136 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))