Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.8s

Alternatives: 7

Speedup: 1.0×

• 61.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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. Step-by-step derivation

   1. associate--l+ 99.9%

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

   2. fma-define 99.9%

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

   3. *-commutative 99.9%

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

3. Simplified 99.9%

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

   1. fma-define 99.9%

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

   2. unpow2 99.9%

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

   3. +-commutative 99.9%

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

   4. distribute-lft-in 85.8%

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

   5. fma-define 85.8%

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

   6. add-sqr-sqrt 85.8%

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

   7. pow2 85.8%

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

   8. fma-define 85.8%

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

   9. hypot-define 85.8%

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

   10. pow2 85.8%

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

   11. fma-define 85.8%

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

   12. add-sqr-sqrt 85.8%

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

   13. pow2 85.8%

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

   14. fma-define 85.8%

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

   15. hypot-define 85.8%

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

   16. pow2 85.8%

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

6. Applied egg-rr 85.8%

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

   1. distribute-lft-out 99.9%

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

   2. rem-square-sqrt 99.9%

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

   3. +-commutative 99.9%

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

   4. unpow2 99.9%

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

   5. unpow2 99.9%

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

   6. hypot-undefine 99.9%

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

   7. +-commutative 99.9%

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

   8. unpow2 99.9%

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

   9. unpow2 99.9%

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

   10. hypot-undefine 99.9%

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

   11. unpow2 99.9%

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

   12. pow-sqr 100.0%

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

   13. metadata-eval 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((b * b) + (a * a)), 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 = ((((b * b) + (a * a)) ** 2.0d0) + (4.0d0 * (b * b))) + (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto \left({\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) + -1 \]
4. Add Preprocessing

Alternative 3: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 11000000000000:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + -1\right) + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 11000000000000.0)
   (+ (+ (* 4.0 (* b b)) -1.0) (pow b 4.0))
   (+ (pow a 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 11000000000000.0) {
		tmp = ((4.0 * (b * b)) + -1.0) + pow(b, 4.0);
	} else {
		tmp = 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 <= 11000000000000.0d0) then
        tmp = ((4.0d0 * (b * b)) + (-1.0d0)) + (b ** 4.0d0)
    else
        tmp = (a ** 4.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 11000000000000.0) {
		tmp = ((4.0 * (b * b)) + -1.0) + Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 11000000000000.0:
		tmp = ((4.0 * (b * b)) + -1.0) + math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 11000000000000.0)
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + -1.0) + (b ^ 4.0));
	else
		tmp = Float64((a ^ 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 11000000000000.0)
		tmp = ((4.0 * (b * b)) + -1.0) + (b ^ 4.0);
	else
		tmp = (a ^ 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 11000000000000.0], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.1e13

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-define 99.9%

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

      3. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 77.3%

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

   if 1.1e13 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-define 99.9%

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

      3. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 95.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 11000000000000:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + -1\right) + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5000000.0) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5000000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5000000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5000000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5000000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5000000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5000000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5000000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e6

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-define 99.9%

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

      3. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 99.3%

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

   if 5e6 < (*.f64 b b)

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-define 99.8%

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

      3. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around 0 89.9%

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

   6. Taylor expanded in b around inf 89.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5000000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-17) then
        tmp = (4.0d0 * (b * b)) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999999e-17

   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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. fma-define 99.9%

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

      3. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 45.1%

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

   6. Taylor expanded in b around 0 45.1%

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

      1. unpow2 45.1%

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

   8. Applied egg-rr 45.1%

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

   if 4.9999999999999999e-17 < (*.f64 b b)

   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. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-define 99.8%

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

      3. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around 0 88.5%

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

   6. Taylor expanded in b around inf 87.8%

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

4. Final simplification 64.3%

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

Alternative 6: 51.2% 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. Step-by-step derivation

   1. associate--l+ 99.9%

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

   2. fma-define 99.9%

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

   3. *-commutative 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in a around 0 64.6%

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

6. Taylor expanded in b around 0 46.8%

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

   1. unpow2 46.8%

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

8. Applied egg-rr 46.8%

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

9. Final simplification 46.8%

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

Alternative 7: 24.8% 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. Step-by-step derivation

   1. associate--l+ 99.9%

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

   2. fma-define 99.9%

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

   3. *-commutative 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in a around 0 64.6%

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

6. Taylor expanded in b around 0 25.2%

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

Reproduce

herbie shell --seed 2024148 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))