Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 5.6s

Alternatives: 7

Speedup: 1.0×

• 60.6% 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

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

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(Float64((hypot(b, a) ^ 4.0) + 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[(N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $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. Taylor expanded in a around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. unpow2 99.9%

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

   3. fma-undefine 99.9%

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

5. Simplified 99.9%

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

   1. unpow2 99.9%

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

   2. fma-undefine 99.9%

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

   3. pow2 99.9%

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

   4. metadata-eval 99.9%

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

   5. distribute-lft-in 90.5%

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

   6. add-sqr-sqrt 90.5%

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

   7. pow2 90.5%

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

   8. fma-undefine 90.5%

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

   9. unpow2 90.5%

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

   10. hypot-define 90.5%

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

   11. metadata-eval 90.5%

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

   12. add-sqr-sqrt 90.5%

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

   13. pow2 90.5%

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

   14. fma-undefine 90.5%

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

   15. unpow2 90.5%

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

   16. hypot-define 90.5%

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

7. Applied egg-rr 90.5%

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

   1. distribute-lft-out 99.9%

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

   2. rem-square-sqrt 99.9%

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

   3. unpow2 99.9%

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

   4. unpow2 99.9%

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

   5. hypot-undefine 99.9%

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

   6. unpow2 99.9%

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

   7. unpow2 99.9%

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

   8. hypot-undefine 99.9%

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

   9. unpow2 99.9%

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

   10. pow-sqr 100.0%

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

   11. metadata-eval 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 14200:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.5e-149)
   -1.0
   (if (<= b 9.5e-76)
     (pow a 4.0)
     (if (<= b 4.8e-28) -1.0 (if (<= b 14200.0) (pow a 4.0) (pow b 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.5e-149) {
		tmp = -1.0;
	} else if (b <= 9.5e-76) {
		tmp = pow(a, 4.0);
	} else if (b <= 4.8e-28) {
		tmp = -1.0;
	} else if (b <= 14200.0) {
		tmp = pow(a, 4.0);
	} 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 <= 5.5d-149) then
        tmp = -1.0d0
    else if (b <= 9.5d-76) then
        tmp = a ** 4.0d0
    else if (b <= 4.8d-28) then
        tmp = -1.0d0
    else if (b <= 14200.0d0) then
        tmp = a ** 4.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.5e-149) {
		tmp = -1.0;
	} else if (b <= 9.5e-76) {
		tmp = Math.pow(a, 4.0);
	} else if (b <= 4.8e-28) {
		tmp = -1.0;
	} else if (b <= 14200.0) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.5e-149:
		tmp = -1.0
	elif b <= 9.5e-76:
		tmp = math.pow(a, 4.0)
	elif b <= 4.8e-28:
		tmp = -1.0
	elif b <= 14200.0:
		tmp = math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.5e-149)
		tmp = -1.0;
	elseif (b <= 9.5e-76)
		tmp = a ^ 4.0;
	elseif (b <= 4.8e-28)
		tmp = -1.0;
	elseif (b <= 14200.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.5e-149)
		tmp = -1.0;
	elseif (b <= 9.5e-76)
		tmp = a ^ 4.0;
	elseif (b <= 4.8e-28)
		tmp = -1.0;
	elseif (b <= 14200.0)
		tmp = a ^ 4.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.5e-149], -1.0, If[LessEqual[b, 9.5e-76], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[b, 4.8e-28], -1.0, If[LessEqual[b, 14200.0], N[Power[a, 4.0], $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 5.50000000000000043e-149 or 9.49999999999999984e-76 < b < 4.8000000000000004e-28

   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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 67.0%

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

   6. Taylor expanded in b around 0 33.5%

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

   if 5.50000000000000043e-149 < b < 9.49999999999999984e-76 or 4.8000000000000004e-28 < b < 14200

   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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

   6. Taylor expanded in a around inf 77.1%

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

   if 14200 < 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. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 94.8%

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

   6. Taylor expanded in b around inf 94.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-149}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-76}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 14200:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 12000.0)
   (+ (pow a 4.0) -1.0)
   (+ (+ (* 4.0 (* b b)) (pow b 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 12000.0) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + 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 <= 12000.0d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = ((4.0d0 * (b * 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 <= 12000.0) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (b * b)) + Math.pow(b, 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 12000.0:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = ((4.0 * (b * b)) + math.pow(b, 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 12000.0)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(b * b)) + (b ^ 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 12000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = ((4.0 * (b * b)) + (b ^ 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 12000.0], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 12000

   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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 88.9%

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

   6. Taylor expanded in b around 0 78.6%

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

   if 12000 < 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. Add Preprocessing

   3. Taylor expanded in a around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. fma-undefine 99.8%

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

   5. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. fma-undefine 99.8%

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

      3. pow2 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-lft-in 90.8%

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

      6. add-sqr-sqrt 90.8%

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

      7. pow2 90.8%

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

      8. fma-undefine 90.8%

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

      9. unpow2 90.8%

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

      10. hypot-define 90.8%

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

      11. metadata-eval 90.8%

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

      12. add-sqr-sqrt 90.8%

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

      13. pow2 90.8%

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

      14. fma-undefine 90.8%

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

      15. unpow2 90.8%

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

      16. hypot-define 90.8%

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

   7. Applied egg-rr 90.8%

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

      1. distribute-lft-out 99.8%

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

      2. rem-square-sqrt 99.8%

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

      3. unpow2 99.8%

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

      4. unpow2 99.8%

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

      5. hypot-undefine 99.8%

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

      6. unpow2 99.8%

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

      7. unpow2 99.8%

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

      8. hypot-undefine 99.8%

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

      9. unpow2 99.8%

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

      10. pow-sqr 100.0%

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

      11. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around inf 94.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 12000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(b \cdot b\right) + {b}^{4}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 14000.0) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 14000.0) {
		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 <= 14000.0d0) 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 <= 14000.0) {
		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 <= 14000.0:
		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 (b <= 14000.0)
		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 <= 14000.0)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 14000.0], 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 b < 14000

   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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 88.9%

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

   6. Taylor expanded in b around 0 78.6%

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

   if 14000 < 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. unpow2 99.8%

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

      3. unpow1 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*r* 99.8%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 94.8%

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

   6. Taylor expanded in b around inf 94.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14000:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.085:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= a 0.085) -1.0 (pow a 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.085) {
		tmp = -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 <= 0.085d0) then
        tmp = -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 <= 0.085) {
		tmp = -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 0.085:
		tmp = -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.085)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 0.085)
		tmp = -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.085], -1.0, N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 0.0850000000000000061

   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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 76.8%

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

   6. Taylor expanded in b around 0 34.2%

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

   if 0.0850000000000000061 < 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. unpow2 99.9%

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

      3. unpow1 99.9%

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

      4. sqr-pow 99.9%

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

      5. associate-*r* 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around inf 98.6%

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

   6. Taylor expanded in a around inf 94.3%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 0.085:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 25.1% 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. unpow2 99.9%

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

   3. unpow1 99.9%

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

   4. sqr-pow 99.9%

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

   5. associate-*r* 99.9%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0 69.3%

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

6. Taylor expanded in b around 0 25.5%

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

7. Final simplification 25.5%

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

Reproduce

herbie shell --seed 2024052 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))