Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

Alternatives: 8

Speedup: 1.0×

• 60.8% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $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. unpow2 99.9%

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

   2. add-sqr-sqrt 99.9%

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

   3. associate-*l* 99.9%

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

   4. hypot-define 99.9%

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

   5. hypot-define 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. pow2 99.9%

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

   8. hypot-define 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \left(\mathsf{hypot}\left(a, b\right) \cdot \left(\mathsf{hypot}\left(a, b\right) \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right) + 4 \cdot \left(b \cdot b\right)\right) + -1 \]
6. 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: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 26000000000000:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 26000000000000.0)
   (+ (pow b 4.0) -1.0)
   (+ (+ (* 4.0 (* b b)) (pow a 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 26000000000000.0) {
		tmp = pow(b, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + 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 <= 26000000000000.0d0) then
        tmp = (b ** 4.0d0) + (-1.0d0)
    else
        tmp = ((4.0d0 * (b * b)) + (a ** 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 26000000000000.0) {
		tmp = Math.pow(b, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + Math.pow(a, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 26000000000000.0:
		tmp = math.pow(b, 4.0) + -1.0
	else:
		tmp = ((4.0 * (b * b)) + math.pow(a, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 26000000000000.0)
		tmp = Float64((b ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (a ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 26000000000000.0)
		tmp = (b ^ 4.0) + -1.0;
	else
		tmp = ((4.0 * (b * b)) + (a ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 26000000000000.0], N[(N[Power[b, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.6e13

   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 80.4%

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

   4. Taylor expanded in b around inf 62.7%

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

      1. associate-*r/ 62.7%

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

      2. metadata-eval 62.7%

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

   6. Simplified 62.7%

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

   7. Taylor expanded in b around inf 79.5%

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

   if 2.6e13 < 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 98.2%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 26000000000000:\\ \;\;\;\;{b}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 26500000000000:\\ \;\;\;\;\left(t\_0 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= a 26500000000000.0)
     (+ (+ 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 <= 26500000000000.0) {
		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 <= 26500000000000.0d0) 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 <= 26500000000000.0) {
		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 <= 26500000000000.0:
		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 <= 26500000000000.0)
		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 <= 26500000000000.0)
		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, 26500000000000.0], 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 < 2.65e13

   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 80.4%

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

   if 2.65e13 < 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 98.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 26500000000000:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {a}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0132:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 0.0132) -1.0 (* 4.0 (pow b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.0132) {
		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 <= 0.0132d0) 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 <= 0.0132) {
		tmp = -1.0;
	} else {
		tmp = 4.0 * Math.pow(b, 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.0132:
		tmp = -1.0
	else:
		tmp = 4.0 * math.pow(b, 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.0132)
		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 <= 0.0132)
		tmp = -1.0;
	else
		tmp = 4.0 * (b ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.0132], -1.0, N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0132

   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 89.2%

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

   4. Taylor expanded in a around 0 50.9%

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

   5. Taylor expanded in b around 0 33.4%

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

   if 0.0132 < 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 66.2%

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

   4. Taylor expanded in b around inf 49.0%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0132:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.6e+153) (+ (pow a 4.0) -1.0) (* 4.0 (pow b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e+153) {
		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 <= 6.6d+153) 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 <= 6.6e+153) {
		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 <= 6.6e+153:
		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 <= 6.6e+153)
		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 <= 6.6e+153)
		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, 6.6e+153], 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 < 6.59999999999999989e153

   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 81.0%

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

   4. Taylor expanded in b around 0 71.9%

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

   if 6.59999999999999989e153 < 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 b around inf 100.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {b}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2400000000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2400000000000.0) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2400000000000.0) {
		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 <= 2400000000000.0d0) 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 <= 2400000000000.0) {
		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 <= 2400000000000.0:
		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 (b <= 2400000000000.0)
		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 <= 2400000000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = (b ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2400000000000.0], 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 b < 2.4e12

   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 88.6%

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

   4. Taylor expanded in b around 0 78.6%

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

   if 2.4e12 < 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 95.4%

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

   4. Taylor expanded in b around inf 95.4%

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

      1. associate-*r/ 95.4%

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

      2. metadata-eval 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in b around inf 95.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2400000000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.5% 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 83.3%

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

4. Taylor expanded in a around 0 50.4%

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

5. Taylor expanded in b around 0 25.0%

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

6. Final simplification 25.0%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024056 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))