Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 9

Speedup: 1.0×

• 61.3% 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: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.02:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left({b}^{3} + b \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.02)
   (+ (pow a 4.0) -1.0)
   (+ (* b (+ (pow b 3.0) (* b 4.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.02) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = (b * (pow(b, 3.0) + (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) :: tmp
    if ((b * b) <= 0.02d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b * ((b ** 3.0d0) + (b * 4.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.02) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = (b * (Math.pow(b, 3.0) + (b * 4.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.02:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = (b * (math.pow(b, 3.0) + (b * 4.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.02)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(b * Float64((b ^ 3.0) + Float64(b * 4.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.02)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b * ((b ^ 3.0) + (b * 4.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.02], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(N[Power[b, 3.0], $MachinePrecision] + N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0200000000000000004

   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 99.2%

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

   4. Taylor expanded in a around inf 98.4%

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

   if 0.0200000000000000004 < (*.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 94.8%

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

      1. metadata-eval 94.8%

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

      2. pow-sqr 94.7%

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

      3. pow2 94.7%

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

      4. pow2 94.7%

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

      5. associate-*r* 94.7%

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

      6. pow3 94.7%

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

   5. Applied egg-rr 94.7%

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

      1. associate-*r* 94.7%

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

      2. distribute-rgt-out 94.7%

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

      3. *-commutative 94.7%

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.02:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left({b}^{3} + b \cdot 4\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.02:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.02)
   (+ (pow a 4.0) -1.0)
   (+ (* b (* b (fma b b 4.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.02) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = (b * (b * fma(b, b, 4.0))) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.02)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.02], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.0200000000000000004

   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 99.2%

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

   4. Taylor expanded in a around inf 98.4%

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

   if 0.0200000000000000004 < (*.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 94.8%

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

      1. metadata-eval 94.8%

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

      2. pow-sqr 94.7%

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

      3. pow2 94.7%

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

      4. pow2 94.7%

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

      5. associate-*r* 94.7%

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

      6. pow3 94.7%

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

   5. Applied egg-rr 94.7%

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

      1. associate-*r* 94.7%

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

      2. distribute-rgt-out 94.7%

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

      3. *-commutative 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. cube-mult 94.7%

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

      2. distribute-lft-out 94.7%

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

      3. fma-define 94.7%

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

   9. Applied egg-rr 94.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.02:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \left({b}^{3} + b \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.3e-7)
   (+ (* b (+ (pow b 3.0) (* b 4.0))) -1.0)
   (+ (+ (* (* b b) 4.0) (pow a 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.3e-7) {
		tmp = (b * (pow(b, 3.0) + (b * 4.0))) + -1.0;
	} else {
		tmp = (((b * b) * 4.0) + 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) :: tmp
    if (a <= 1.3d-7) then
        tmp = (b * ((b ** 3.0d0) + (b * 4.0d0))) + (-1.0d0)
    else
        tmp = (((b * b) * 4.0d0) + (a ** 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.3e-7) {
		tmp = (b * (Math.pow(b, 3.0) + (b * 4.0))) + -1.0;
	} else {
		tmp = (((b * b) * 4.0) + Math.pow(a, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.3e-7:
		tmp = (b * (math.pow(b, 3.0) + (b * 4.0))) + -1.0
	else:
		tmp = (((b * b) * 4.0) + math.pow(a, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.3e-7)
		tmp = Float64(Float64(b * Float64((b ^ 3.0) + Float64(b * 4.0))) + -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(b * b) * 4.0) + (a ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.3e-7)
		tmp = (b * ((b ^ 3.0) + (b * 4.0))) + -1.0;
	else
		tmp = (((b * b) * 4.0) + (a ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.3e-7], N[(N[(b * N[(N[Power[b, 3.0], $MachinePrecision] + N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.29999999999999999e-7

   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 85.2%

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

      1. metadata-eval 85.2%

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

      2. pow-sqr 85.2%

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

      3. pow2 85.2%

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

      4. pow2 85.2%

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

      5. associate-*r* 85.2%

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

      6. pow3 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. associate-*r* 85.2%

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

      2. distribute-rgt-out 85.2%

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

      3. *-commutative 85.2%

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

   7. Applied egg-rr 85.2%

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

   if 1.29999999999999999e-7 < a

   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 92.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \left({b}^{3} + b \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot 4 + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+17) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+17) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = 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) :: tmp
    if ((b * b) <= 5d+17) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (b ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+17) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+17:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+17)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64((b ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+17)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+17], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e17

   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 97.0%

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

   4. Taylor expanded in a around inf 95.9%

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

   if 5e17 < (*.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 96.0%

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

   4. Taylor expanded in b around inf 96.0%

      \[\leadsto \color{blue}{{b}^{4}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+17}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+304}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+304) (+ (pow a 4.0) -1.0) (+ (* (* b b) 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+304) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = ((b * 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) :: tmp
    if ((b * b) <= 2d+304) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = ((b * b) * 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+304) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = ((b * b) * 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+304:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = ((b * b) * 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+304)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+304)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = ((b * b) * 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+304], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e304

   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 77.0%

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

   4. Taylor expanded in a around inf 75.1%

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

   if 1.9999999999999999e304 < (*.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 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+304}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5.5e+31) (+ (* (* b b) 4.0) -1.0) (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5.5e+31) {
		tmp = ((b * b) * 4.0) + -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 <= 5.5d+31) then
        tmp = ((b * b) * 4.0d0) + (-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 <= 5.5e+31) {
		tmp = ((b * b) * 4.0) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 5.5e+31:
		tmp = ((b * b) * 4.0) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5.5e+31)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 5.5e+31)
		tmp = ((b * b) * 4.0) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5.5e+31], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.50000000000000002e31

   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 80.2%

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

   4. Taylor expanded in a around 0 60.1%

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

      1. pow2 60.1%

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

   6. Applied egg-rr 60.1%

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

   if 5.50000000000000002e31 < a

   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 94.9%

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

   4. Taylor expanded in a around inf 91.4%

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

   5. Taylor expanded in a around inf 91.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.8% accurate, 16.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * 4.0), $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 83.3%

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

4. Taylor expanded in a around 0 52.8%

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

   1. pow2 52.8%

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

6. Applied egg-rr 52.8%

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

7. Final simplification 52.8%

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

Alternative 9: 24.9% 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 77.4%

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

4. Taylor expanded in b around 0 23.1%

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

Reproduce

herbie shell --seed 2024131 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))