Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

Alternatives: 6

Speedup: 1.0×

• 60.2% 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-cbrt-cube 91.9%

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left({\left(a \cdot a + b \cdot b\right)}^{2} \cdot {\left(a \cdot a + b \cdot b\right)}^{2}\right) \cdot {\left(a \cdot a + b \cdot b\right)}^{2}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   2. pow3 91.9%

      \[\leadsto \left(\sqrt[3]{\color{blue}{{\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)}^{3}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   3. pow-pow 91.9%

      \[\leadsto \left(\sqrt[3]{\color{blue}{{\left(a \cdot a + b \cdot b\right)}^{\left(2 \cdot 3\right)}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   4. add-sqr-sqrt 91.9%

      \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(\sqrt{a \cdot a + b \cdot b} \cdot \sqrt{a \cdot a + b \cdot b}\right)}}^{\left(2 \cdot 3\right)}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   5. pow2 91.9%

      \[\leadsto \left(\sqrt[3]{{\color{blue}{\left({\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}\right)}}^{\left(2 \cdot 3\right)}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   6. hypot-define 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\color{blue}{\left(\mathsf{hypot}\left(a, b\right)\right)}}^{2}\right)}^{\left(2 \cdot 3\right)}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   7. metadata-eval 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

4. Applied egg-rr 91.9%

   \[\leadsto \left(\color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
5. Step-by-step derivation

   1. hypot-undefine 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\color{blue}{\left(\sqrt{a \cdot a + b \cdot b}\right)}}^{2}\right)}^{6}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   2. unpow2 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\left(\sqrt{\color{blue}{{a}^{2}} + b \cdot b}\right)}^{2}\right)}^{6}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   3. unpow2 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\left(\sqrt{{a}^{2} + \color{blue}{{b}^{2}}}\right)}^{2}\right)}^{6}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   4. +-commutative 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\left(\sqrt{\color{blue}{{b}^{2} + {a}^{2}}}\right)}^{2}\right)}^{6}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   5. unpow2 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\left(\sqrt{\color{blue}{b \cdot b} + {a}^{2}}\right)}^{2}\right)}^{6}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   6. unpow2 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\left(\sqrt{b \cdot b + \color{blue}{a \cdot a}}\right)}^{2}\right)}^{6}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   7. hypot-define 91.9%

      \[\leadsto \left(\sqrt[3]{{\left({\color{blue}{\left(\mathsf{hypot}\left(b, a\right)\right)}}^{2}\right)}^{6}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

6. Simplified 91.9%

   \[\leadsto \left(\color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{6}}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
7. Step-by-step derivation

   1. pow1/3 91.4%

      \[\leadsto \left(\color{blue}{{\left({\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{6}\right)}^{0.3333333333333333}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   2. pow-pow 99.9%

      \[\leadsto \left(\color{blue}{{\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{2}\right)}^{\left(6 \cdot 0.3333333333333333\right)}} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]

   3. metadata-eval 99.9%

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

   4. unpow2 99.9%

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

   5. hypot-undefine 99.9%

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

   6. unpow2 99.9%

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

   7. hypot-undefine 99.9%

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

   8. unpow2 99.9%

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

   9. add-sqr-sqrt 99.9%

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

   10. +-commutative 99.9%

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

   11. add-sqr-sqrt 99.9%

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

   12. unpow2 99.9%

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

   13. hypot-undefine 99.9%

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

   14. unpow2 99.9%

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

   15. hypot-undefine 99.9%

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

   16. pow-prod-down 99.9%

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

   17. unpow2 99.9%

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

   18. associate-*r* 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \left(\mathsf{hypot}\left(a, b\right) \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{3} + 4 \cdot \left(b \cdot b\right)\right) + -1 \]
10. 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: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;a \leq 5:\\ \;\;\;\;\left(t\_0 + {b}^{4}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + {a}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b b))))
   (if (<= a 5.0) (+ (+ 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 <= 5.0) {
		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 <= 5.0d0) 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 <= 5.0) {
		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 <= 5.0:
		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 <= 5.0)
		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 <= 5.0)
		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, 5.0], 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 < 5

   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 78.7%

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

   if 5 < 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 96.7%

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

4. Final simplification 82.6%

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

Alternative 4: 82.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5.0) (+ (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 <= 5.0) {
		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 <= 5.0d0) 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 <= 5.0) {
		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 <= 5.0:
		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 <= 5.0)
		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 <= 5.0)
		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, 5.0], 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 < 5

   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%

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

      2. pow2 99.9%

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

      3. unpow2 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. hypot-define 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. pow2 99.9%

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

      8. hypot-define 99.9%

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

      9. *-commutative 99.9%

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

      10. sqrt-prod 99.9%

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

      11. sqrt-prod 67.4%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around inf 78.3%

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

   if 5 < 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 96.7%

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

4. Final simplification 82.4%

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

Alternative 5: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4800000000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4800000000000.0) (+ (pow a 4.0) -1.0) (+ (pow b 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4800000000000.0) {
		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 <= 4800000000000.0d0) 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 <= 4800000000000.0) {
		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 <= 4800000000000.0:
		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 <= 4800000000000.0)
		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 <= 4800000000000.0)
		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, 4800000000000.0], 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 < 4.8e12

   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%

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

      2. pow2 99.9%

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

      3. unpow2 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. hypot-define 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. pow2 99.9%

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

      8. hypot-define 99.9%

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

      9. *-commutative 99.9%

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

      10. sqrt-prod 99.9%

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

      11. sqrt-prod 61.0%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf 82.6%

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

   if 4.8e12 < 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. Step-by-step derivation

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. unpow2 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. hypot-define 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. pow2 99.9%

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

      8. hypot-define 99.9%

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

      9. *-commutative 99.9%

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

      10. sqrt-prod 99.9%

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

      11. sqrt-prod 99.9%

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

      12. add-sqr-sqrt 99.9%

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

      13. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around inf 95.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4800000000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {a}^{4} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Power[a, 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. Step-by-step derivation

   1. add-sqr-sqrt 99.9%

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

   2. pow2 99.9%

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

   3. unpow2 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. hypot-define 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. pow2 99.9%

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

   8. hypot-define 99.9%

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

   9. *-commutative 99.9%

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

   10. sqrt-prod 99.9%

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

   11. sqrt-prod 71.0%

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

   12. add-sqr-sqrt 99.9%

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

   13. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in a around inf 71.4%

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

6. Final simplification 71.4%

   \[\leadsto {a}^{4} + -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))