Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.7% → 98.0%

Time: 5.7s

Alternatives: 10

Speedup: 1.2×

• 61.5% 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: 73.7% 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: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[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[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

   3. Simplified 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. fma-neg 0.0%

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

      4. associate-*l* 0.0%

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

      5. fma-def 12.1%

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

      6. +-commutative 12.1%

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

      7. metadata-eval 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in b around 0 31.3%

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

      1. associate--l+ 31.3%

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

      2. unpow2 31.3%

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

   6. Simplified 31.3%

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

   7. Taylor expanded in a around inf 95.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) \leq \infty:\\ \;\;\;\;{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, a + 3, a \cdot a\right) - {a}^{3}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(a * N[(a * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

      3. fma-neg 99.8%

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

      4. associate-*l* 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. fma-neg 0.0%

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

      4. associate-*l* 0.0%

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

      5. fma-def 12.1%

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

      6. +-commutative 12.1%

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

      7. metadata-eval 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in b around 0 31.3%

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

      1. associate--l+ 31.3%

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

      2. unpow2 31.3%

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

   6. Simplified 31.3%

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

   7. Taylor expanded in a around inf 95.1%

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

4. Final simplification 98.8%

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

Alternative 3: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	t_0 = math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = math.pow(a, 4.0)
	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(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = a ^ 4.0;
	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[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. fma-neg 0.0%

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

      4. associate-*l* 0.0%

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

      5. fma-def 12.1%

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

      6. +-commutative 12.1%

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

      7. metadata-eval 12.1%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in b around 0 31.3%

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

      1. associate--l+ 31.3%

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

      2. unpow2 31.3%

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

   6. Simplified 31.3%

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

   7. Taylor expanded in a around inf 95.1%

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

4. Final simplification 98.8%

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

Alternative 4: 93.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+58}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+58)
   (+ (pow a 4.0) (+ -1.0 (* (* a a) 4.0)))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+58) {
		tmp = pow(a, 4.0) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = -1.0 + 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) <= 1d+58) then
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a * a) * 4.0d0))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+58) {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+58:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a * a) * 4.0))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+58)
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a * a) * 4.0)));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+58)
		tmp = (a ^ 4.0) + (-1.0 + ((a * a) * 4.0));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+58], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999944e57

   1. Initial program 84.3%

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

      1. associate--l+ 84.3%

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

      2. fma-def 84.3%

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

      3. fma-neg 84.3%

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

      4. associate-*l* 84.3%

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

      5. fma-def 84.3%

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

      6. +-commutative 84.3%

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

      7. metadata-eval 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in b around 0 80.7%

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

      1. associate--l+ 80.7%

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

      2. unpow2 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in a around 0 95.5%

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

      1. unpow2 95.5%

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

   9. Simplified 95.5%

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

   if 9.99999999999999944e57 < (*.f64 b b)

   1. Initial program 69.3%

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

      1. sub-neg 69.3%

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

      2. fma-def 69.3%

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

      3. fma-def 75.1%

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

      4. +-commutative 75.1%

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

      5. metadata-eval 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in b around inf 94.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+58}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 5: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+58}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+58)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+58) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	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+58) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+58) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+58:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+58)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+58)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+58], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999944e57

   1. Initial program 84.3%

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

      1. sub-neg 84.3%

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

      2. fma-def 84.3%

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

      3. fma-def 84.3%

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

      4. +-commutative 84.3%

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

      5. metadata-eval 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in a around inf 95.5%

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

   if 9.99999999999999944e57 < (*.f64 b b)

   1. Initial program 69.3%

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

      1. sub-neg 69.3%

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

      2. fma-def 69.3%

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

      3. fma-def 75.1%

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

      4. +-commutative 75.1%

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

      5. metadata-eval 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in a around 0 65.7%

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

      1. associate-+r+ 65.7%

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

      2. associate-*r* 65.7%

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

      3. distribute-rgt-out 79.8%

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

      4. metadata-eval 79.8%

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

      5. distribute-lft-in 79.8%

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

      6. +-commutative 79.8%

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

      7. unpow2 79.8%

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

      8. distribute-lft-in 79.8%

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

      9. metadata-eval 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in a around 0 94.1%

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

      1. unpow2 94.1%

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

   9. Simplified 94.1%

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

       1. +-commutative 94.1%

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

       2. sqr-pow 93.9%

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

       3. metadata-eval 93.9%

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

       4. pow2 93.9%

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

       5. metadata-eval 93.9%

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

       6. pow2 93.9%

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

       7. distribute-rgt-out 93.9%

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

   11. Applied egg-rr 93.9%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+58}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 6: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+58}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+58) (+ -1.0 (pow a 4.0)) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+58) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + 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) <= 1d+58) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+58) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+58:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+58)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 1e+58)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+58], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 9.99999999999999944e57

   1. Initial program 84.3%

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

      1. sub-neg 84.3%

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

      2. fma-def 84.3%

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

      3. fma-def 84.3%

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

      4. +-commutative 84.3%

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

      5. metadata-eval 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in a around inf 95.5%

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

   if 9.99999999999999944e57 < (*.f64 b b)

   1. Initial program 69.3%

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

      1. sub-neg 69.3%

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

      2. fma-def 69.3%

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

      3. fma-def 75.1%

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

      4. +-commutative 75.1%

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

      5. metadata-eval 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in b around inf 94.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+58}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 7: 93.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -26 \lor \neg \left(a \leq 1.65 \cdot 10^{+20}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -26.0) (not (<= a 1.65e+20)))
   (pow a 4.0)
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -26.0) || !(a <= 1.65e+20)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-26.0d0)) .or. (.not. (a <= 1.65d+20))) then
        tmp = a ** 4.0d0
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -26.0) || !(a <= 1.65e+20)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -26.0) or not (a <= 1.65e+20):
		tmp = math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -26.0) || !(a <= 1.65e+20))
		tmp = a ^ 4.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -26.0) || ~((a <= 1.65e+20)))
		tmp = a ^ 4.0;
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -26.0], N[Not[LessEqual[a, 1.65e+20]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -26 or 1.65e20 < a

   1. Initial program 50.8%

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

      1. associate--l+ 50.8%

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

      2. fma-def 50.8%

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

      3. fma-neg 50.8%

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

      4. associate-*l* 50.8%

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

      5. fma-def 56.7%

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

      6. +-commutative 56.7%

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

      7. metadata-eval 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in b around 0 61.5%

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

      1. associate--l+ 61.5%

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

      2. unpow2 61.5%

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

   6. Simplified 61.5%

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

   7. Taylor expanded in a around inf 92.0%

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

   if -26 < a < 1.65e20

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. fma-def 99.9%

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

      3. fma-def 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 86.3%

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

      1. associate-+r+ 86.3%

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

      2. associate-*r* 86.3%

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

      3. distribute-rgt-out 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-lft-in 98.6%

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

      6. +-commutative 98.6%

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

      7. unpow2 98.6%

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

      8. distribute-lft-in 98.6%

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

      9. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in a around 0 98.6%

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

      1. unpow2 98.6%

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

   9. Simplified 98.6%

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

       1. +-commutative 98.6%

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

       2. sqr-pow 98.4%

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

       3. metadata-eval 98.4%

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

       4. pow2 98.4%

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

       5. metadata-eval 98.4%

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

       6. pow2 98.4%

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

       7. distribute-rgt-out 98.4%

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

   11. Applied egg-rr 98.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -26 \lor \neg \left(a \leq 1.65 \cdot 10^{+20}\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 8: 69.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ -1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ -1.0 (* (* b b) (+ (* b b) 12.0))))

C

double code(double a, double b) {
	return -1.0 + ((b * b) * ((b * b) + 12.0));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
end function

Java

public static double code(double a, double b) {
	return -1.0 + ((b * b) * ((b * b) + 12.0));
}

Python

def code(a, b):
	return -1.0 + ((b * b) * ((b * b) + 12.0))

Julia

function code(a, b)
	return Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)))
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
end

Wolfram

code[a_, b_] := N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. sub-neg 77.2%

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

   2. fma-def 77.2%

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

   3. fma-def 80.0%

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

   4. +-commutative 80.0%

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

   5. metadata-eval 80.0%

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

3. Simplified 80.0%

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

4. Taylor expanded in a around 0 55.2%

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

   1. associate-+r+ 55.2%

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

   2. associate-*r* 55.2%

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

   3. distribute-rgt-out 61.9%

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

   4. metadata-eval 61.9%

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

   5. distribute-lft-in 61.9%

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

   6. +-commutative 61.9%

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

   7. unpow2 61.9%

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

   8. distribute-lft-in 61.9%

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

   9. metadata-eval 61.9%

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

6. Simplified 61.9%

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

7. Taylor expanded in a around 0 68.6%

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

   1. unpow2 68.6%

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

9. Simplified 68.6%

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

    1. +-commutative 68.6%

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

    2. sqr-pow 68.5%

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

    3. metadata-eval 68.5%

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

    4. pow2 68.5%

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

    5. metadata-eval 68.5%

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

    6. pow2 68.5%

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

    7. distribute-rgt-out 68.5%

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

11. Applied egg-rr 68.5%

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

12. Final simplification 68.5%

    \[\leadsto -1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) \]

Alternative 9: 50.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ -1 + \left(b \cdot b\right) \cdot 12 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ -1.0 (* (* b b) 12.0)))

C

double code(double a, double b) {
	return -1.0 + ((b * b) * 12.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (-1.0d0) + ((b * b) * 12.0d0)
end function

Java

public static double code(double a, double b) {
	return -1.0 + ((b * b) * 12.0);
}

Python

def code(a, b):
	return -1.0 + ((b * b) * 12.0)

Julia

function code(a, b)
	return Float64(-1.0 + Float64(Float64(b * b) * 12.0))
end

MATLAB

function tmp = code(a, b)
	tmp = -1.0 + ((b * b) * 12.0);
end

Wolfram

code[a_, b_] := N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. sub-neg 77.2%

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

   2. fma-def 77.2%

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

   3. fma-def 80.0%

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

   4. +-commutative 80.0%

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

   5. metadata-eval 80.0%

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

3. Simplified 80.0%

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

4. Taylor expanded in a around 0 55.2%

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

   1. associate-+r+ 55.2%

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

   2. associate-*r* 55.2%

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

   3. distribute-rgt-out 61.9%

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

   4. metadata-eval 61.9%

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

   5. distribute-lft-in 61.9%

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

   6. +-commutative 61.9%

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

   7. unpow2 61.9%

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

   8. distribute-lft-in 61.9%

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

   9. metadata-eval 61.9%

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

6. Simplified 61.9%

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

7. Taylor expanded in a around 0 68.6%

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

   1. unpow2 68.6%

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

9. Simplified 68.6%

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

    1. +-commutative 68.6%

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

    2. sqr-pow 68.5%

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

    3. metadata-eval 68.5%

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

    4. pow2 68.5%

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

    5. metadata-eval 68.5%

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

    6. pow2 68.5%

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

    7. distribute-rgt-out 68.5%

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

11. Applied egg-rr 68.5%

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

12. Taylor expanded in b around 0 48.9%

    \[\leadsto \color{blue}{12 \cdot {b}^{2}} + -1 \]
13. Step-by-step derivation

    1. unpow2 48.9%

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

14. Simplified 48.9%

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

15. Final simplification 48.9%

    \[\leadsto -1 + \left(b \cdot b\right) \cdot 12 \]

Alternative 10: 24.4% 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 77.2%

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

   1. associate--l+ 77.2%

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

   2. fma-def 77.2%

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

   3. fma-neg 77.2%

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

   4. associate-*l* 77.2%

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

   5. fma-def 80.0%

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

   6. +-commutative 80.0%

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

   7. metadata-eval 80.0%

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

3. Simplified 80.0%

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

4. Taylor expanded in b around 0 51.2%

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

   1. associate--l+ 51.2%

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

   2. unpow2 51.2%

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

6. Simplified 51.2%

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

7. Taylor expanded in a around 0 22.4%

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

8. Final simplification 22.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023240 
(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))