Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 6.0s

Alternatives: 8

Speedup: 1.0×

• 60.1% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{hypot}\left(a, b\right) \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+ (+ (* (hypot a b) (pow (hypot a b) 3.0)) (* 4.0 (* b b))) -1.0))

C

double code(double a, double b) {
	return ((hypot(a, b) * pow(hypot(a, b), 3.0)) + (4.0 * (b * b))) + -1.0;
}

Java

public static double code(double a, double b) {
	return ((Math.hypot(a, b) * Math.pow(Math.hypot(a, b), 3.0)) + (4.0 * (b * b))) + -1.0;
}

Python

def code(a, b):
	return ((math.hypot(a, b) * math.pow(math.hypot(a, b), 3.0)) + (4.0 * (b * b))) + -1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(hypot(a, b) * (hypot(a, b) ^ 3.0)) + Float64(4.0 * Float64(b * b))) + -1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((hypot(a, b) * (hypot(a, b) ^ 3.0)) + (4.0 * (b * b))) + -1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 3.0], $MachinePrecision]), $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-sqr-sqrt 99.9%

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

   2. sqrt-unprod 93.4%

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

   3. pow-prod-up 93.4%

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

   4. add-sqr-sqrt 93.4%

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

   5. pow2 93.4%

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

   6. hypot-define 93.4%

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

   7. metadata-eval 93.4%

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

4. Applied egg-rr 93.4%

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

   1. sqrt-pow1 99.9%

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

   2. metadata-eval 99.9%

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

   3. pow2 99.9%

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

   4. unpow2 99.9%

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

   5. associate-*r* 99.9%

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

   6. unpow2 99.9%

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

   7. pow3 99.9%

      \[\leadsto \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{3}} \cdot \mathsf{hypot}\left(a, b\right) + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

6. Applied egg-rr 99.9%

   \[\leadsto \left(\color{blue}{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{3} \cdot \mathsf{hypot}\left(a, b\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

7. Final simplification 99.9%

   \[\leadsto \left(\mathsf{hypot}\left(a, b\right) \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) + -1 \]
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: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 0.43:\\ \;\;\;\;\left(t\_0 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= a 0.43) (+ (+ 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 <= 0.43) {
		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 <= 0.43d0) 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 <= 0.43) {
		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 <= 0.43:
		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 <= 0.43)
		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 <= 0.43)
		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, 0.43], 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 < 0.429999999999999993

   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 74.1%

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

   if 0.429999999999999993 < a

   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 96.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.43:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+283}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+283) (+ (pow a 4.0) -1.0) (+ (* 4.0 (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+283) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = (4.0 * (b * b)) + -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 ((b * b) <= 1d+283) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (4.0d0 * (b * b)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+283) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = (4.0 * (b * b)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+283:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = (4.0 * (b * b)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+283)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(4.0 * Float64(b * b)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+283)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (4.0 * (b * b)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+283], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999955e282

   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 84.8%

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

   4. Taylor expanded in b around 0 83.9%

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

   if 9.99999999999999955e282 < (*.f64 b 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 96.8%

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

   4. Taylor expanded in a around 0 96.8%

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

      1. pow2 96.8%

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

   6. Applied egg-rr 96.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+283}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% 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 87.4%

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

4. Final simplification 87.4%

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

Alternative 6: 51.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 8.6 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 8.6e-9) -1.0 (* 4.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 8.6e-9) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	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) <= 8.6d-9) then
        tmp = -1.0d0
    else
        tmp = 4.0d0 * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 8.6e-9) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 8.6e-9:
		tmp = -1.0
	else:
		tmp = 4.0 * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 8.6e-9)
		tmp = -1.0;
	else
		tmp = Float64(4.0 * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 8.6e-9)
		tmp = -1.0;
	else
		tmp = 4.0 * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 8.6e-9], -1.0, N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 8.59999999999999925e-9

   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 47.5%

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

   4. Taylor expanded in b around 0 47.5%

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

   if 8.59999999999999925e-9 < (*.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 inf 72.7%

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

   4. Taylor expanded in b around inf 49.0%

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

      1. pow2 49.0%

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

   6. Applied egg-rr 49.0%

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

Alternative 7: 51.4% 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 87.4%

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

4. Taylor expanded in a around 0 48.2%

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

   1. pow2 48.2%

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

6. Applied egg-rr 48.2%

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

7. Final simplification 48.2%

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

Alternative 8: 25.8% 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 65.5%

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

4. Taylor expanded in b around 0 25.9%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))