Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.3% → 98.5%

Time: 7.6s

Alternatives: 9

Speedup: 6.1×

• 60.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 75.6%

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

   2. sqr-pow 75.7%

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

   3. sqr-pow 75.6%

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

   4. sqr-neg 75.6%

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

   5. distribute-rgt-in 75.6%

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

   6. sqr-neg 75.6%

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

   7. distribute-rgt-in 75.6%

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

3. Simplified 77.2%

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

4. Taylor expanded in b around 0 81.2%

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

   1. associate-+r+ 81.2%

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

   2. fma-def 81.2%

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

   3. fma-def 81.2%

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

   4. unpow2 81.2%

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

   5. distribute-rgt-in 81.2%

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

   6. metadata-eval 81.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{12} + a \cdot 4\right), {b}^{2}, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

   7. unpow2 81.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), \color{blue}{b \cdot b}, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

   8. unpow2 81.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

   9. associate-*r* 81.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)}\right)\right) + -1 \]

6. Simplified 81.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right)} + -1 \]
7. Step-by-step derivation

   1. fma-udef 81.2%

      \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + {b}^{4}\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   2. metadata-eval 81.2%

      \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + {b}^{\color{blue}{\left(2 + 2\right)}}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   3. pow-prod-up 81.2%

      \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   4. pow2 81.2%

      \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right)} \cdot {b}^{2}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   5. pow2 81.2%

      \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   6. distribute-rgt-out 81.2%

      \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) + b \cdot b\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   7. +-commutative 81.2%

      \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{a \cdot 4 + 12}\right) + b \cdot b\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   8. fma-def 81.2%

      \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{\mathsf{fma}\left(a, 4, 12\right)}\right) + b \cdot b\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

8. Applied egg-rr 81.2%

   \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \mathsf{fma}\left(a, 4, 12\right)\right) + b \cdot b\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

9. Taylor expanded in a around inf 92.8%

   \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \mathsf{fma}\left(a, 4, 12\right)\right) + b \cdot b\right) + \color{blue}{{a}^{4}}\right) + -1 \]

10. Taylor expanded in b around inf 98.7%

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

    1. unpow2 98.7%

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

12. Simplified 98.7%

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

13. Final simplification 98.7%

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

Alternative 2: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+20}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e+20)
   (pow a 4.0)
   (if (<= a 3.8e+59) (+ -1.0 (+ (pow b 4.0) (* b (* b 12.0)))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6e+20) {
		tmp = pow(a, 4.0);
	} else if (a <= 3.8e+59) {
		tmp = -1.0 + (pow(b, 4.0) + (b * (b * 12.0)));
	} 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 <= (-8.6d+20)) then
        tmp = a ** 4.0d0
    else if (a <= 3.8d+59) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + (b * (b * 12.0d0)))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.6e+20) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 3.8e+59) {
		tmp = -1.0 + (Math.pow(b, 4.0) + (b * (b * 12.0)));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.6e+20:
		tmp = math.pow(a, 4.0)
	elif a <= 3.8e+59:
		tmp = -1.0 + (math.pow(b, 4.0) + (b * (b * 12.0)))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6e+20)
		tmp = a ^ 4.0;
	elseif (a <= 3.8e+59)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(b * Float64(b * 12.0))));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.6e+20)
		tmp = a ^ 4.0;
	elseif (a <= 3.8e+59)
		tmp = -1.0 + ((b ^ 4.0) + (b * (b * 12.0)));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6e+20], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 3.8e+59], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.6e20 or 3.8000000000000001e59 < a

   1. Initial program 43.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 43.0%

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

      2. sqr-pow 43.1%

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

      3. sqr-pow 43.0%

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

      4. sqr-neg 43.0%

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

      5. distribute-rgt-in 43.0%

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

      6. sqr-neg 43.0%

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

      7. distribute-rgt-in 43.0%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in a around inf 97.8%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
   5. Step-by-step derivation

      1. metadata-eval 97.8%

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

      2. pow-sqr 97.7%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} + -1 \]

      3. pow-prod-down 97.7%

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

      4. pow2 97.7%

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

      5. difference-of-sqr--1 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in a around inf 97.8%

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

   if -8.6e20 < a < 3.8000000000000001e59

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. sqr-pow 99.8%

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

      3. sqr-pow 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. sqr-neg 99.8%

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

      7. distribute-rgt-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around 0 89.8%

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + \left(4 \cdot \left(a \cdot {b}^{2}\right) + {b}^{4}\right)\right)} + -1 \]
   5. Step-by-step derivation

      1. associate-+r+ 89.8%

         \[\leadsto \color{blue}{\left(\left(12 \cdot {b}^{2} + 4 \cdot \left(a \cdot {b}^{2}\right)\right) + {b}^{4}\right)} + -1 \]

      2. associate-*r* 89.8%

         \[\leadsto \left(\left(12 \cdot {b}^{2} + \color{blue}{\left(4 \cdot a\right) \cdot {b}^{2}}\right) + {b}^{4}\right) + -1 \]

      3. distribute-rgt-out 95.9%

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot \left(12 + 4 \cdot a\right)} + {b}^{4}\right) + -1 \]

      4. metadata-eval 95.9%

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

      5. distribute-lft-in 95.9%

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

      6. +-commutative 95.9%

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

      7. unpow2 95.9%

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

      8. distribute-lft-in 95.9%

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

      9. metadata-eval 95.9%

         \[\leadsto \left({b}^{4} + \left(b \cdot b\right) \cdot \left(\color{blue}{12} + 4 \cdot a\right)\right) + -1 \]

   6. Simplified 95.9%

      \[\leadsto \color{blue}{\left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + 4 \cdot a\right)\right)} + -1 \]

   7. Taylor expanded in a around 0 95.9%

      \[\leadsto \left({b}^{4} + \color{blue}{12 \cdot {b}^{2}}\right) + -1 \]
   8. Step-by-step derivation

      1. unpow2 95.9%

         \[\leadsto \left({b}^{4} + 12 \cdot \color{blue}{\left(b \cdot b\right)}\right) + -1 \]

      2. associate-*r* 95.9%

         \[\leadsto \left({b}^{4} + \color{blue}{\left(12 \cdot b\right) \cdot b}\right) + -1 \]

      3. *-commutative 95.9%

         \[\leadsto \left({b}^{4} + \color{blue}{b \cdot \left(12 \cdot b\right)}\right) + -1 \]

   9. Simplified 95.9%

      \[\leadsto \left({b}^{4} + \color{blue}{b \cdot \left(12 \cdot b\right)}\right) + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+20}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+59}:\\ \;\;\;\;-1 + \left({b}^{4} + b \cdot \left(b \cdot 12\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 3: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+61}:\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.22e+24)
   (pow a 4.0)
   (if (<= a 1.05e+61) (+ -1.0 (pow b 4.0)) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.22e+24) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.05e+61) {
		tmp = -1.0 + pow(b, 4.0);
	} 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 <= (-1.22d+24)) then
        tmp = a ** 4.0d0
    else if (a <= 1.05d+61) then
        tmp = (-1.0d0) + (b ** 4.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.22e+24) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 1.05e+61) {
		tmp = -1.0 + Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.22e+24:
		tmp = math.pow(a, 4.0)
	elif a <= 1.05e+61:
		tmp = -1.0 + math.pow(b, 4.0)
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.22e+24)
		tmp = a ^ 4.0;
	elseif (a <= 1.05e+61)
		tmp = Float64(-1.0 + (b ^ 4.0));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.22e+24)
		tmp = a ^ 4.0;
	elseif (a <= 1.05e+61)
		tmp = -1.0 + (b ^ 4.0);
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.22e+24], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.05e+61], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.21999999999999996e24 or 1.0500000000000001e61 < a

   1. Initial program 43.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 43.0%

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

      2. sqr-pow 43.1%

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

      3. sqr-pow 43.0%

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

      4. sqr-neg 43.0%

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

      5. distribute-rgt-in 43.0%

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

      6. sqr-neg 43.0%

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

      7. distribute-rgt-in 43.0%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in a around inf 97.8%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
   5. Step-by-step derivation

      1. metadata-eval 97.8%

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

      2. pow-sqr 97.7%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} + -1 \]

      3. pow-prod-down 97.7%

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

      4. pow2 97.7%

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

      5. difference-of-sqr--1 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in a around inf 97.8%

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

   if -1.21999999999999996e24 < a < 1.0500000000000001e61

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. sqr-pow 99.8%

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

      3. sqr-pow 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. sqr-neg 99.8%

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

      7. distribute-rgt-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around inf 95.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+61}:\\ \;\;\;\;-1 + {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 7.3 \cdot 10^{+115}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right) + 4 \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 7.3e+115)
   (+ (pow a 4.0) -1.0)
   (+ -1.0 (+ (* (* a a) (* (* b b) 2.0)) (* 4.0 (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 7.3e+115) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)));
	}
	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) <= 7.3d+115) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (-1.0d0) + (((a * a) * ((b * b) * 2.0d0)) + (4.0d0 * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 7.3e+115) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 7.3e+115:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 7.3e+115)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(a * a) * Float64(Float64(b * b) * 2.0)) + Float64(4.0 * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 7.3e+115)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 7.3e+115], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[(N[(N[(a * a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 7.29999999999999968e115

   1. Initial program 81.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 81.9%

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

      2. sqr-pow 81.9%

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

      3. sqr-pow 81.9%

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

      4. sqr-neg 81.9%

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

      5. distribute-rgt-in 81.9%

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

      6. sqr-neg 81.9%

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

      7. distribute-rgt-in 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in a around inf 89.4%

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

   if 7.29999999999999968e115 < (*.f64 b b)

   1. Initial program 65.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 65.9%

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

      2. sqr-pow 65.9%

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

      3. sqr-pow 65.9%

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

      4. sqr-neg 65.9%

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

      5. distribute-rgt-in 65.9%

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

      6. sqr-neg 65.9%

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

      7. distribute-rgt-in 65.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around 0 86.0%

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

      1. associate-+r+ 86.0%

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

      2. fma-def 86.0%

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

      3. fma-def 86.0%

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

      4. unpow2 86.0%

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

      5. distribute-rgt-in 86.0%

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

      6. metadata-eval 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{12} + a \cdot 4\right), {b}^{2}, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      7. unpow2 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), \color{blue}{b \cdot b}, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      8. unpow2 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      9. associate-*r* 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)}\right)\right) + -1 \]

   6. Simplified 86.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right)} + -1 \]
   7. Step-by-step derivation

      1. fma-udef 86.0%

         \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + {b}^{4}\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      2. metadata-eval 86.0%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + {b}^{\color{blue}{\left(2 + 2\right)}}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      3. pow-prod-up 85.9%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      4. pow2 85.9%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right)} \cdot {b}^{2}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      5. pow2 85.9%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      6. distribute-rgt-out 85.9%

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) + b \cdot b\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      7. +-commutative 85.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{a \cdot 4 + 12}\right) + b \cdot b\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      8. fma-def 85.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{\mathsf{fma}\left(a, 4, 12\right)}\right) + b \cdot b\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   8. Applied egg-rr 85.9%

      \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \mathsf{fma}\left(a, 4, 12\right)\right) + b \cdot b\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   9. Taylor expanded in a around inf 45.6%

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

       1. unpow2 45.6%

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

       2. unpow2 45.6%

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

       3. associate-*r* 45.6%

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

       4. *-commutative 45.6%

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

       5. associate-*l* 45.6%

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

   11. Simplified 45.6%

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

   12. Taylor expanded in a around 0 59.6%

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

       1. unpow2 23.9%

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

   14. Simplified 59.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 7.3 \cdot 10^{+115}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right) + 4 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 5: 78.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.5 \cdot 10^{+116}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(-1 + a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right) + 4 \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 3.5e+116)
   (* (+ (* a a) 1.0) (+ -1.0 (* a a)))
   (+ -1.0 (+ (* (* a a) (* (* b b) 2.0)) (* 4.0 (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.5e+116) {
		tmp = ((a * a) + 1.0) * (-1.0 + (a * a));
	} else {
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)));
	}
	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) <= 3.5d+116) then
        tmp = ((a * a) + 1.0d0) * ((-1.0d0) + (a * a))
    else
        tmp = (-1.0d0) + (((a * a) * ((b * b) * 2.0d0)) + (4.0d0 * (a * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.5e+116) {
		tmp = ((a * a) + 1.0) * (-1.0 + (a * a));
	} else {
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 3.5e+116:
		tmp = ((a * a) + 1.0) * (-1.0 + (a * a))
	else:
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 3.5e+116)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(-1.0 + Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(a * a) * Float64(Float64(b * b) * 2.0)) + Float64(4.0 * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 3.5e+116)
		tmp = ((a * a) + 1.0) * (-1.0 + (a * a));
	else
		tmp = -1.0 + (((a * a) * ((b * b) * 2.0)) + (4.0 * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 3.5e+116], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(-1.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(a * a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.49999999999999997e116

   1. Initial program 81.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 81.9%

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

      2. sqr-pow 81.9%

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

      3. sqr-pow 81.9%

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

      4. sqr-neg 81.9%

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

      5. distribute-rgt-in 81.9%

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

      6. sqr-neg 81.9%

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

      7. distribute-rgt-in 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in a around inf 89.4%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
   5. Step-by-step derivation

      1. metadata-eval 89.4%

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

      2. pow-sqr 89.4%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} + -1 \]

      3. pow-prod-down 89.3%

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

      4. pow2 89.4%

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

      5. difference-of-sqr--1 89.4%

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

   6. Applied egg-rr 89.4%

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

   if 3.49999999999999997e116 < (*.f64 b b)

   1. Initial program 65.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 65.9%

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

      2. sqr-pow 65.9%

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

      3. sqr-pow 65.9%

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

      4. sqr-neg 65.9%

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

      5. distribute-rgt-in 65.9%

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

      6. sqr-neg 65.9%

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

      7. distribute-rgt-in 65.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in b around 0 86.0%

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

      1. associate-+r+ 86.0%

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

      2. fma-def 86.0%

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

      3. fma-def 86.0%

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

      4. unpow2 86.0%

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

      5. distribute-rgt-in 86.0%

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

      6. metadata-eval 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{12} + a \cdot 4\right), {b}^{2}, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      7. unpow2 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), \color{blue}{b \cdot b}, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      8. unpow2 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(1 - a\right)\right)\right)\right) + -1 \]

      9. associate-*r* 86.0%

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)}\right)\right) + -1 \]

   6. Simplified 86.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right), b \cdot b, {b}^{4}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right)} + -1 \]
   7. Step-by-step derivation

      1. fma-udef 86.0%

         \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + {b}^{4}\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      2. metadata-eval 86.0%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + {b}^{\color{blue}{\left(2 + 2\right)}}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      3. pow-prod-up 85.9%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \color{blue}{{b}^{2} \cdot {b}^{2}}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      4. pow2 85.9%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right)} \cdot {b}^{2}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      5. pow2 85.9%

         \[\leadsto \left(\left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      6. distribute-rgt-out 85.9%

         \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, 12 + a \cdot 4\right) + b \cdot b\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      7. +-commutative 85.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{a \cdot 4 + 12}\right) + b \cdot b\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

      8. fma-def 85.9%

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \color{blue}{\mathsf{fma}\left(a, 4, 12\right)}\right) + b \cdot b\right) + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   8. Applied egg-rr 85.9%

      \[\leadsto \left(\color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(2, a \cdot a, \mathsf{fma}\left(a, 4, 12\right)\right) + b \cdot b\right)} + \left({a}^{4} + 4 \cdot \left(a \cdot \left(a \cdot \left(1 - a\right)\right)\right)\right)\right) + -1 \]

   9. Taylor expanded in a around inf 45.6%

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

       1. unpow2 45.6%

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

       2. unpow2 45.6%

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

       3. associate-*r* 45.6%

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

       4. *-commutative 45.6%

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

       5. associate-*l* 45.6%

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

   11. Simplified 45.6%

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

   12. Taylor expanded in a around 0 59.6%

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

       1. unpow2 23.9%

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

   14. Simplified 59.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.5 \cdot 10^{+116}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(-1 + a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right) + 4 \cdot \left(a \cdot a\right)\right)\\ \end{array} \]

Alternative 6: 68.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-5} \lor \neg \left(a \leq 5.1\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(-1 + a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.65e-5) (not (<= a 5.1)))
   (* (* a a) (+ -1.0 (* a a)))
   (+ -1.0 (* 4.0 (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.65e-5) || !(a <= 5.1)) {
		tmp = (a * a) * (-1.0 + (a * a));
	} else {
		tmp = -1.0 + (4.0 * (a * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.65d-5)) .or. (.not. (a <= 5.1d0))) then
        tmp = (a * a) * ((-1.0d0) + (a * a))
    else
        tmp = (-1.0d0) + (4.0d0 * (a * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.65e-5) || !(a <= 5.1)) {
		tmp = (a * a) * (-1.0 + (a * a));
	} else {
		tmp = -1.0 + (4.0 * (a * a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.65e-5) or not (a <= 5.1):
		tmp = (a * a) * (-1.0 + (a * a))
	else:
		tmp = -1.0 + (4.0 * (a * a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.65e-5) || !(a <= 5.1))
		tmp = Float64(Float64(a * a) * Float64(-1.0 + Float64(a * a)));
	else
		tmp = Float64(-1.0 + Float64(4.0 * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.65e-5) || ~((a <= 5.1)))
		tmp = (a * a) * (-1.0 + (a * a));
	else
		tmp = -1.0 + (4.0 * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.65e-5], N[Not[LessEqual[a, 5.1]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(-1.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6500000000000001e-5 or 5.0999999999999996 < a

   1. Initial program 50.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 50.7%

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

      2. sqr-pow 50.7%

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

      3. sqr-pow 50.7%

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

      4. sqr-neg 50.7%

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

      5. distribute-rgt-in 50.7%

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

      6. sqr-neg 50.7%

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

      7. distribute-rgt-in 50.7%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in a around inf 88.1%

      \[\leadsto \color{blue}{{a}^{4}} + -1 \]
   5. Step-by-step derivation

      1. metadata-eval 88.1%

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

      2. pow-sqr 88.0%

         \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} + -1 \]

      3. pow-prod-down 88.0%

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

      4. pow2 88.0%

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

      5. difference-of-sqr--1 88.0%

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

   6. Applied egg-rr 88.0%

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

   7. Taylor expanded in a around inf 88.0%

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

      1. unpow2 88.0%

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

   9. Simplified 88.0%

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

   if -1.6500000000000001e-5 < a < 5.0999999999999996

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. sqr-pow 99.8%

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

      3. sqr-pow 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. sqr-neg 99.8%

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

      7. distribute-rgt-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 52.7%

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

      1. unpow2 52.7%

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

      2. associate-*r* 52.7%

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

   6. Simplified 52.7%

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

   7. Taylor expanded in a around 0 52.1%

      \[\leadsto \color{blue}{4 \cdot {a}^{2}} + -1 \]
   8. Step-by-step derivation

      1. unpow2 52.1%

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

   9. Simplified 52.1%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-5} \lor \neg \left(a \leq 5.1\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(-1 + a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + 4 \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 7: 68.8% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 75.6%

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

   2. sqr-pow 75.7%

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

   3. sqr-pow 75.6%

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

   4. sqr-neg 75.6%

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

   5. distribute-rgt-in 75.6%

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

   6. sqr-neg 75.6%

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

   7. distribute-rgt-in 75.6%

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

3. Simplified 77.2%

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

4. Taylor expanded in a around inf 69.7%

   \[\leadsto \color{blue}{{a}^{4}} + -1 \]
5. Step-by-step derivation

   1. metadata-eval 69.7%

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

   2. pow-sqr 69.7%

      \[\leadsto \color{blue}{{a}^{2} \cdot {a}^{2}} + -1 \]

   3. pow-prod-down 69.6%

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

   4. pow2 69.7%

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

   5. difference-of-sqr--1 69.7%

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

6. Applied egg-rr 69.7%

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

7. Final simplification 69.7%

   \[\leadsto \left(a \cdot a + 1\right) \cdot \left(-1 + a \cdot a\right) \]

Alternative 8: 50.9% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 75.6%

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

   2. sqr-pow 75.7%

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

   3. sqr-pow 75.6%

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

   4. sqr-neg 75.6%

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

   5. distribute-rgt-in 75.6%

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

   6. sqr-neg 75.6%

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

   7. distribute-rgt-in 75.6%

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

3. Simplified 77.2%

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

4. Taylor expanded in b around 0 54.5%

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

   1. unpow2 54.5%

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

   2. associate-*r* 54.5%

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

6. Simplified 54.5%

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

7. Taylor expanded in a around 0 51.5%

   \[\leadsto \color{blue}{4 \cdot {a}^{2}} + -1 \]
8. Step-by-step derivation

   1. unpow2 51.5%

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

9. Simplified 51.5%

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

10. Final simplification 51.5%

    \[\leadsto -1 + 4 \cdot \left(a \cdot a\right) \]

Alternative 9: 25.1% accurate, 128.0× 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 75.6%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 75.6%

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

   2. sqr-pow 75.7%

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

   3. sqr-pow 75.6%

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

   4. sqr-neg 75.6%

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

   5. distribute-rgt-in 75.6%

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

   6. sqr-neg 75.6%

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

   7. distribute-rgt-in 75.6%

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

3. Simplified 77.2%

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

4. Taylor expanded in a around inf 69.7%

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

5. Taylor expanded in a around 0 26.7%

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

6. Final simplification 26.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))