Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 98.3%

Time: 8.6s

Alternatives: 8

Speedup: 6.5×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY)
     (+ t_0 -1.0)
     (+ (* (* a a) (+ 4.0 (* a (+ a 4.0)))) -1.0))))

C

double code(double a, double b) {
	double t_0 = pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) < +inf.0

   1. Initial program 99.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a))))))

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

   3. Simplified 6.2%

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

   5. Taylor expanded in b around 0 32.8%

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

      1. add-cbrt-cube 38.5%

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

      2. pow3 38.5%

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

      3. fma-define 38.5%

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

      4. pow2 38.5%

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

      5. distribute-rgt-in 38.5%

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

      6. *-un-lft-identity 38.5%

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

      7. fma-define 38.5%

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

      8. cube-unmult 38.5%

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

   7. Applied egg-rr 38.5%

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

   8. Taylor expanded in a around 0 92.8%

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

      1. pow2 92.8%

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

   10. Applied egg-rr 92.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq \infty:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-5) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = (((b * b) * 4.0) + 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 * b) <= 5d-5) then
        tmp = ((a * a) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else
        tmp = (((b * b) * 4.0d0) + (b ** 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000024e-5

   1. Initial program 83.3%

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

      1. sub-neg 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 83.0%

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

      1. add-cbrt-cube 76.3%

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

      2. pow3 76.4%

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

      3. fma-define 76.4%

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

      4. pow2 76.4%

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

      5. distribute-rgt-in 76.4%

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

      6. *-un-lft-identity 76.4%

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

      7. fma-define 76.4%

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

      8. cube-unmult 76.4%

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

   7. Applied egg-rr 76.4%

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

   8. Taylor expanded in a around 0 99.4%

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

      1. pow2 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 5.00000000000000024e-5 < (*.f64 b b)

   1. Initial program 65.7%

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

      1. associate--l+ 65.7%

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

      2. +-commutative 65.7%

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

      3. +-commutative 65.7%

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

      4. sub-neg 65.7%

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

      5. associate-+l+ 65.7%

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

      6. +-commutative 65.7%

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

      7. associate-+l+ 65.7%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in a around 0 91.9%

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

      1. unpow2 91.9%

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

   7. Applied egg-rr 91.9%

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

4. Final simplification 95.6%

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

Alternative 3: 94.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e17

   1. Initial program 83.2%

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

      1. sub-neg 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in b around 0 81.1%

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

      1. add-cbrt-cube 74.6%

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

      2. pow3 74.7%

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

      3. fma-define 74.7%

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

      4. pow2 74.7%

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

      5. distribute-rgt-in 74.7%

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

      6. *-un-lft-identity 74.7%

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

      7. fma-define 74.7%

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

      8. cube-unmult 74.7%

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

   7. Applied egg-rr 74.7%

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

   8. Taylor expanded in a around 0 97.6%

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

      1. pow2 97.6%

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

   10. Applied egg-rr 97.6%

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

   if 5e17 < (*.f64 b b)

   1. Initial program 65.2%

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

      1. associate--l+ 65.2%

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

      2. +-commutative 65.2%

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

      3. +-commutative 65.2%

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

      4. sub-neg 65.2%

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

      5. associate-+l+ 65.2%

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

      6. +-commutative 65.2%

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

      7. associate-+l+ 65.2%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in a around 0 93.1%

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

      1. unpow2 93.1%

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

   7. Applied egg-rr 93.1%

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

   8. Taylor expanded in b around inf 93.1%

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

4. Final simplification 95.4%

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

Alternative 4: 84.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+303}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+303)
   (+ (* (* a a) (+ 4.0 (* a (+ a 4.0)))) -1.0)
   (+ (* (* b b) 4.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+303) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = ((b * 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 * b) <= 1d+303) then
        tmp = ((a * a) * (4.0d0 + (a * (a + 4.0d0)))) + (-1.0d0)
    else
        tmp = ((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 * b) <= 1e+303) {
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	} else {
		tmp = ((b * b) * 4.0) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+303:
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0
	else:
		tmp = ((b * b) * 4.0) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+303)
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * Float64(a + 4.0)))) + -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+303)
		tmp = ((a * a) * (4.0 + (a * (a + 4.0)))) + -1.0;
	else
		tmp = ((b * b) * 4.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+303], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1e303

   1. Initial program 78.9%

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

      1. sub-neg 78.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in b around 0 59.6%

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

      1. add-cbrt-cube 57.3%

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

      2. pow3 57.3%

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

      3. fma-define 57.3%

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

      4. pow2 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. *-un-lft-identity 57.3%

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

      7. fma-define 57.3%

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

      8. cube-unmult 57.3%

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

   7. Applied egg-rr 57.3%

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

   8. Taylor expanded in a around 0 75.4%

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

      1. pow2 75.4%

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

   10. Applied egg-rr 75.4%

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

   if 1e303 < (*.f64 b b)

   1. Initial program 58.2%

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

      1. associate--l+ 58.2%

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

      2. +-commutative 58.2%

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

      3. +-commutative 58.2%

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

      4. sub-neg 58.2%

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

      5. associate-+l+ 58.2%

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

      6. +-commutative 58.2%

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

      7. associate-+l+ 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. unpow2 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+303}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a + 4\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 8.2e+99)
   (* (+ 1.0 (* b 2.0)) (+ (* b 2.0) -1.0))
   (+ (* (* a a) (+ 4.0 (* a 4.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 8.2e+99) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	} else {
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 8.2d+99) then
        tmp = (1.0d0 + (b * 2.0d0)) * ((b * 2.0d0) + (-1.0d0))
    else
        tmp = ((a * a) * (4.0d0 + (a * 4.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 8.2e+99) {
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	} else {
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 8.2e+99:
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)
	else:
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 8.2e+99)
		tmp = Float64(Float64(1.0 + Float64(b * 2.0)) * Float64(Float64(b * 2.0) + -1.0));
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(4.0 + Float64(a * 4.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 8.2e+99)
		tmp = (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0);
	else
		tmp = ((a * a) * (4.0 + (a * 4.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 8.2e+99], N[(N[(1.0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.19999999999999959e99

   1. Initial program 78.0%

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

      1. associate--l+ 78.0%

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

      2. +-commutative 78.0%

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

      3. +-commutative 78.0%

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

      4. sub-neg 78.0%

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

      5. associate-+l+ 78.0%

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

      6. +-commutative 78.0%

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

      7. associate-+l+ 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in a around 0 78.6%

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

   6. Taylor expanded in b around 0 53.3%

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

      1. pow2 53.3%

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

      2. add-sqr-sqrt 53.3%

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

      3. difference-of-sqr-1 53.3%

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

      4. *-commutative 53.3%

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

      5. sqrt-prod 53.3%

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

      6. sqrt-prod 29.1%

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

      7. add-sqr-sqrt 44.1%

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

      8. metadata-eval 44.1%

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

      9. *-commutative 44.1%

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

      10. sqrt-prod 44.1%

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

      11. sqrt-prod 29.1%

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

      12. add-sqr-sqrt 53.3%

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

      13. metadata-eval 53.3%

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

   8. Applied egg-rr 53.3%

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

   if 8.19999999999999959e99 < a

   1. Initial program 57.8%

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

      1. sub-neg 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in b around 0 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

      3. fma-define 100.0%

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

      4. pow2 100.0%

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

      5. distribute-rgt-in 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. fma-define 100.0%

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

      8. cube-unmult 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in a around 0 98.0%

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

      1. pow2 100.0%

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

   10. Applied egg-rr 98.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;\left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot 4\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return (1.0 + (b * 2.0)) * ((b * 2.0) + -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate--l+ 74.5%

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

   2. +-commutative 74.5%

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

   3. +-commutative 74.5%

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

   4. sub-neg 74.5%

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

   5. associate-+l+ 74.5%

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

   6. +-commutative 74.5%

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

   7. associate-+l+ 74.5%

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

3. Simplified 77.2%

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

5. Taylor expanded in a around 0 71.2%

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

6. Taylor expanded in b around 0 48.4%

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

   1. pow2 48.4%

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

   2. add-sqr-sqrt 48.4%

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

   3. difference-of-sqr-1 48.4%

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

   4. *-commutative 48.4%

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

   5. sqrt-prod 48.4%

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

   6. sqrt-prod 25.6%

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

   7. add-sqr-sqrt 38.1%

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

   8. metadata-eval 38.1%

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

   9. *-commutative 38.1%

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

   10. sqrt-prod 38.1%

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

   11. sqrt-prod 25.6%

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

   12. add-sqr-sqrt 48.4%

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

   13. metadata-eval 48.4%

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

8. Applied egg-rr 48.4%

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

9. Final simplification 48.4%

   \[\leadsto \left(1 + b \cdot 2\right) \cdot \left(b \cdot 2 + -1\right) \]
10. Add Preprocessing

Alternative 7: 51.8% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. associate--l+ 74.5%

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

   2. +-commutative 74.5%

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

   3. +-commutative 74.5%

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

   4. sub-neg 74.5%

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

   5. associate-+l+ 74.5%

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

   6. +-commutative 74.5%

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

   7. associate-+l+ 74.5%

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

3. Simplified 77.2%

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

5. Taylor expanded in a around 0 71.2%

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

6. Taylor expanded in b around 0 48.4%

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

   1. unpow2 71.2%

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

8. Applied egg-rr 48.4%

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

9. Final simplification 48.4%

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

Alternative 8: 25.1% accurate, 130.0× 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 74.5%

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

   1. associate--l+ 74.5%

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

   2. +-commutative 74.5%

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

   3. +-commutative 74.5%

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

   4. sub-neg 74.5%

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

   5. associate-+l+ 74.5%

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

   6. +-commutative 74.5%

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

   7. associate-+l+ 74.5%

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

3. Simplified 77.2%

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

5. Taylor expanded in a around 0 71.2%

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

6. Taylor expanded in b around 0 25.2%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))