Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.8% → 98.6%

Time: 8.0s

Alternatives: 11

Speedup: 8.5×

• 60.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.8% 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: 98.6% accurate, 0.2× speedup

Math

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

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))
   (* (+ (* a a) 1.0) (+ (* a a) -1.0))))

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 = ((a * a) + 1.0) * ((a * a) + -1.0);
	}
	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 = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -1.0));
	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[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $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 99.9%

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

   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. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 9.7%

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

      4. +-commutative 9.7%

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

      5. metadata-eval 9.7%

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

   3. Simplified 9.7%

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

   4. Taylor expanded in a around inf 96.9%

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

      1. metadata-eval 96.9%

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

      2. pow-prod-up 96.9%

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

      3. pow2 96.9%

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

      4. pow2 96.9%

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

      5. difference-of-sqr--1 96.9%

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

   6. Applied egg-rr 96.9%

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

4. Final simplification 99.2%

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

Alternative 2: 98.5% accurate, 0.5× speedup

Math

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

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) (* (+ (* a a) 1.0) (+ (* a a) -1.0)))))

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 = ((a * a) + 1.0) * ((a * a) + -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) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	}
	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 = ((a * a) + 1.0) * ((a * a) + -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(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 = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -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) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = ((a * a) + 1.0) * ((a * a) + -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[(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[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $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. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 9.7%

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

      4. +-commutative 9.7%

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

      5. metadata-eval 9.7%

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

   3. Simplified 9.7%

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

   4. Taylor expanded in a around inf 96.9%

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

      1. metadata-eval 96.9%

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

      2. pow-prod-up 96.9%

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

      3. pow2 96.9%

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

      4. pow2 96.9%

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

      5. difference-of-sqr--1 96.9%

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

   6. Applied egg-rr 96.9%

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

4. Final simplification 99.1%

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

Alternative 3: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* (* b b) (+ (* b b) (* (* a a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)));
	}
	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) <= 400000000.0d0) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + ((a * a) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   1. Initial program 84.9%

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

      1. sub-neg 84.9%

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

      2. fma-def 84.9%

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

      3. fma-def 84.9%

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

      4. +-commutative 84.9%

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

      5. metadata-eval 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in a around inf 96.7%

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

   if 4e8 < (*.f64 b b)

   1. Initial program 65.5%

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

      1. sub-neg 65.5%

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

      2. fma-def 65.5%

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

      3. fma-def 70.4%

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

      4. +-commutative 70.4%

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

      5. metadata-eval 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. fma-def 98.1%

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

      2. fma-def 98.1%

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

      3. unpow2 98.1%

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

      4. distribute-lft-in 98.1%

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

      5. metadata-eval 98.1%

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

      6. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around inf 86.5%

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

      1. unpow2 86.5%

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

   9. Simplified 86.5%

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

       1. fma-udef 86.5%

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

       2. metadata-eval 86.5%

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

       3. pow-prod-up 86.4%

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

       4. pow2 86.4%

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

       5. pow2 86.4%

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

       6. distribute-rgt-out 97.9%

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

   11. Applied egg-rr 97.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 4: 96.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 400000000.0)
   (* (+ (* a a) 1.0) (+ (* a a) -1.0))
   (+ -1.0 (* (* b b) (+ (* b b) (* (* a a) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)));
	}
	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) <= 400000000.0d0) then
        tmp = ((a * a) + 1.0d0) * ((a * a) + (-1.0d0))
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + ((a * a) * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 400000000.0) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 400000000.0:
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 400000000.0)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -1.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + Float64(Float64(a * a) * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 400000000.0)
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + ((a * a) * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 400000000.0], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e8

   1. Initial program 84.9%

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

      1. sub-neg 84.9%

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

      2. fma-def 84.9%

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

      3. fma-def 84.9%

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

      4. +-commutative 84.9%

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

      5. metadata-eval 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in a around inf 96.7%

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

      1. metadata-eval 96.7%

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

      2. pow-prod-up 96.6%

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

      3. pow2 96.6%

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

      4. pow2 96.6%

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

      5. difference-of-sqr--1 96.6%

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

   6. Applied egg-rr 96.6%

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

   if 4e8 < (*.f64 b b)

   1. Initial program 65.5%

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

      1. sub-neg 65.5%

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

      2. fma-def 65.5%

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

      3. fma-def 70.4%

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

      4. +-commutative 70.4%

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

      5. metadata-eval 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in b around inf 98.1%

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

      1. fma-def 98.1%

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

      2. fma-def 98.1%

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

      3. unpow2 98.1%

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

      4. distribute-lft-in 98.1%

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

      5. metadata-eval 98.1%

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

      6. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in a around inf 86.5%

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

      1. unpow2 86.5%

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

   9. Simplified 86.5%

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

       1. fma-udef 86.5%

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

       2. metadata-eval 86.5%

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

       3. pow-prod-up 86.4%

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

       4. pow2 86.4%

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

       5. pow2 86.4%

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

       6. distribute-rgt-out 97.9%

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

   11. Applied egg-rr 97.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 400000000:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\\ \end{array} \]

Alternative 5: 71.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-309} \lor \neg \left(a \leq 580\right):\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1e-309) (not (<= a 580.0)))
   (* (+ (* a a) 1.0) (+ (* a a) -1.0))
   (+ -1.0 (* b (* b (* a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1e-309) || !(a <= 580.0)) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = -1.0 + (b * (b * (a * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1d-309)) .or. (.not. (a <= 580.0d0))) then
        tmp = ((a * a) + 1.0d0) * ((a * a) + (-1.0d0))
    else
        tmp = (-1.0d0) + (b * (b * (a * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1e-309) || !(a <= 580.0)) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = -1.0 + (b * (b * (a * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1e-309) or not (a <= 580.0):
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0)
	else:
		tmp = -1.0 + (b * (b * (a * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1e-309) || !(a <= 580.0))
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -1.0));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(a * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1e-309) || ~((a <= 580.0)))
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	else
		tmp = -1.0 + (b * (b * (a * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1e-309], N[Not[LessEqual[a, 580.0]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.000000000000002e-309 or 580 < a

   1. Initial program 68.0%

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

      1. sub-neg 68.0%

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

      2. fma-def 68.0%

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

      3. fma-def 71.1%

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

      4. +-commutative 71.1%

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

      5. metadata-eval 71.1%

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

   3. Simplified 71.1%

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

   4. Taylor expanded in a around inf 76.4%

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

      1. metadata-eval 76.4%

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

      2. pow-prod-up 76.3%

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

      3. pow2 76.3%

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

      4. pow2 76.3%

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

      5. difference-of-sqr--1 76.3%

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

   6. Applied egg-rr 76.3%

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

   if -1.000000000000002e-309 < a < 580

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. fma-def 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around inf 99.3%

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

      1. fma-def 99.3%

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

      2. fma-def 99.3%

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

      3. unpow2 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. metadata-eval 99.3%

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

      6. unpow2 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in a around inf 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. associate-*r* 59.5%

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

      4. associate-*l* 59.5%

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

      5. unpow2 59.5%

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

      6. associate-*r* 59.5%

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

      7. *-commutative 59.5%

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

      8. distribute-rgt-in 70.9%

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

      9. associate-*l* 70.9%

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

      10. unpow2 70.9%

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

      11. *-commutative 70.9%

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

      12. +-commutative 70.9%

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

      13. associate-*l* 57.0%

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

      14. +-commutative 57.0%

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

      15. unpow2 57.0%

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

      16. associate-*l* 57.0%

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

      17. distribute-rgt-out 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in a around 0 57.0%

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

       1. associate-*r* 57.0%

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

       2. *-commutative 57.0%

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

       3. *-commutative 57.0%

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

       4. *-commutative 57.0%

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

   12. Simplified 57.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-309} \lor \neg \left(a \leq 580\right):\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \end{array} \]

Alternative 6: 83.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+61)
   (* (+ (* a a) 1.0) (+ (* a a) -1.0))
   (+ -1.0 (* a (* a (* b (* b 2.0)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+61) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = -1.0 + (a * (a * (b * (b * 2.0))));
	}
	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+61) then
        tmp = ((a * a) + 1.0d0) * ((a * a) + (-1.0d0))
    else
        tmp = (-1.0d0) + (a * (a * (b * (b * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+61) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = -1.0 + (a * (a * (b * (b * 2.0))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+61:
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0)
	else:
		tmp = -1.0 + (a * (a * (b * (b * 2.0))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+61)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -1.0));
	else
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(b * Float64(b * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+61)
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	else
		tmp = -1.0 + (a * (a * (b * (b * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+61], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(a * N[(a * N[(b * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5.00000000000000018e61

   1. Initial program 85.2%

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

      1. sub-neg 85.2%

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

      2. fma-def 85.2%

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

      3. fma-def 85.2%

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

      4. +-commutative 85.2%

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

      5. metadata-eval 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in a around inf 92.2%

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

      1. metadata-eval 92.2%

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

      2. pow-prod-up 92.0%

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

      3. pow2 92.0%

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

      4. pow2 92.0%

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

      5. difference-of-sqr--1 92.0%

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

   6. Applied egg-rr 92.0%

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

   if 5.00000000000000018e61 < (*.f64 b b)

   1. Initial program 63.3%

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

      1. sub-neg 63.3%

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

      2. fma-def 63.3%

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

      3. fma-def 68.7%

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

      4. +-commutative 68.7%

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

      5. metadata-eval 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. fma-def 100.0%

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

      2. fma-def 100.0%

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

      3. unpow2 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 87.5%

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

      1. unpow2 87.5%

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

   9. Simplified 87.5%

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

   10. Taylor expanded in a around inf 60.5%

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

       1. *-commutative 60.5%

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

       2. unpow2 60.5%

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

       3. unpow2 60.5%

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

       4. associate-*l* 73.0%

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

       5. associate-*l* 73.0%

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

       6. *-commutative 73.0%

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

       7. associate-*r* 73.0%

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

       8. *-commutative 73.0%

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

       9. *-commutative 73.0%

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

       10. *-commutative 73.0%

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

       11. associate-*l* 73.0%

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

       12. *-commutative 73.0%

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

       13. associate-*l* 73.0%

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

   12. Simplified 73.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 7: 61.6% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+150}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e-310)
   (+ -1.0 (* a (* a (* a -4.0))))
   (if (<= a 6.8e+150) (+ -1.0 (* b (* b (* a 4.0)))) (* (* a a) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e-310) {
		tmp = -1.0 + (a * (a * (a * -4.0)));
	} else if (a <= 6.8e+150) {
		tmp = -1.0 + (b * (b * (a * 4.0)));
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5d-310)) then
        tmp = (-1.0d0) + (a * (a * (a * (-4.0d0))))
    else if (a <= 6.8d+150) then
        tmp = (-1.0d0) + (b * (b * (a * 4.0d0)))
    else
        tmp = (a * a) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e-310) {
		tmp = -1.0 + (a * (a * (a * -4.0)));
	} else if (a <= 6.8e+150) {
		tmp = -1.0 + (b * (b * (a * 4.0)));
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e-310:
		tmp = -1.0 + (a * (a * (a * -4.0)))
	elif a <= 6.8e+150:
		tmp = -1.0 + (b * (b * (a * 4.0)))
	else:
		tmp = (a * a) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e-310)
		tmp = Float64(-1.0 + Float64(a * Float64(a * Float64(a * -4.0))));
	elseif (a <= 6.8e+150)
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(a * 4.0))));
	else
		tmp = Float64(Float64(a * a) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e-310)
		tmp = -1.0 + (a * (a * (a * -4.0)));
	elseif (a <= 6.8e+150)
		tmp = -1.0 + (b * (b * (a * 4.0)));
	else
		tmp = (a * a) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e-310], N[(-1.0 + N[(a * N[(a * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.8e+150], N[(-1.0 + N[(b * N[(b * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.999999999999985e-310

   1. Initial program 85.7%

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

      1. sub-neg 85.7%

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

      2. fma-def 85.7%

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

      3. fma-def 85.7%

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

      4. +-commutative 85.7%

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

      5. metadata-eval 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in b around 0 72.6%

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

      1. associate-*r* 72.6%

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

      2. unpow2 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in a around 0 59.9%

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

      1. unpow2 59.9%

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

      2. *-commutative 59.9%

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

      3. associate-*r* 59.9%

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

      4. cube-mult 59.9%

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

      5. associate-*r* 59.9%

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

      6. metadata-eval 59.9%

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

      7. distribute-lft-neg-in 59.9%

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

      8. *-commutative 59.9%

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

      9. distribute-lft-neg-in 59.9%

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

      10. associate-*r* 59.9%

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

      11. *-commutative 59.9%

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

      12. associate-*r* 59.9%

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

      13. distribute-rgt1-in 59.9%

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

      14. +-commutative 59.9%

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

      15. sub-neg 59.9%

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

      16. *-commutative 59.9%

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

      17. associate-*l* 59.9%

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

      18. *-commutative 59.9%

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

      19. associate-*r* 59.9%

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

   9. Simplified 59.9%

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

   10. Taylor expanded in a around inf 59.3%

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

       1. unpow2 59.3%

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

       2. mul-1-neg 59.3%

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

       3. distribute-rgt-neg-out 59.3%

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

   12. Simplified 59.3%

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

   13. Taylor expanded in a around 0 59.3%

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

       1. unpow2 59.3%

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

       2. associate-*r* 59.3%

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

   15. Simplified 59.3%

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

   if -4.999999999999985e-310 < a < 6.79999999999999965e150

   1. Initial program 90.2%

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

      1. sub-neg 90.2%

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

      2. fma-def 90.2%

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

      3. fma-def 96.6%

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

      4. +-commutative 96.6%

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

      5. metadata-eval 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in b around inf 83.4%

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

      1. fma-def 83.4%

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

      2. fma-def 83.4%

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

      3. unpow2 83.4%

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

      4. distribute-lft-in 83.4%

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

      5. metadata-eval 83.4%

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

      6. unpow2 83.4%

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

   6. Simplified 83.4%

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

   7. Taylor expanded in a around inf 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. associate-*r* 55.2%

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

      4. associate-*l* 55.2%

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

      5. unpow2 55.2%

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

      6. associate-*r* 55.2%

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

      7. *-commutative 55.2%

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

      8. distribute-rgt-in 62.6%

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

      9. associate-*l* 62.6%

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

      10. unpow2 62.6%

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

      11. *-commutative 62.6%

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

      12. +-commutative 62.6%

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

      13. associate-*l* 53.6%

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

      14. +-commutative 53.6%

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

      15. unpow2 53.6%

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

      16. associate-*l* 53.6%

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

      17. distribute-rgt-out 53.6%

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

   9. Simplified 53.6%

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

   10. Taylor expanded in a around 0 51.5%

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

       1. associate-*r* 51.5%

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

       2. *-commutative 51.5%

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

       3. *-commutative 51.5%

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

       4. *-commutative 51.5%

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

   12. Simplified 51.5%

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

   if 6.79999999999999965e150 < 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. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 0.0%

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

      4. +-commutative 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in a around 0 97.5%

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

      1. unpow2 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 97.5%

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

      4. *-commutative 97.5%

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

   9. Simplified 97.5%

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

   10. Taylor expanded in a around inf 97.5%

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

       1. unpow2 97.5%

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

   12. Simplified 97.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + a \cdot \left(a \cdot \left(a \cdot -4\right)\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+150}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \]

Alternative 8: 54.4% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 72.2%

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

      1. sub-neg 72.2%

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

      2. fma-def 72.2%

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

      3. fma-def 72.2%

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

      4. +-commutative 72.2%

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

      5. metadata-eval 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in b around 0 95.3%

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

      1. associate-*r* 95.3%

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

      2. unpow2 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in a around 0 71.8%

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

      1. unpow2 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r* 71.8%

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

      4. cube-mult 71.8%

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

      5. associate-*r* 71.8%

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

      6. metadata-eval 71.8%

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

      7. distribute-lft-neg-in 71.8%

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

      8. *-commutative 71.8%

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

      9. distribute-lft-neg-in 71.8%

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

      10. associate-*r* 71.8%

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

      11. *-commutative 71.8%

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

      12. associate-*r* 71.8%

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

      13. distribute-rgt1-in 71.8%

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

      14. +-commutative 71.8%

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

      15. sub-neg 71.8%

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

      16. *-commutative 71.8%

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

      17. associate-*l* 71.8%

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

      18. *-commutative 71.8%

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

      19. associate-*r* 71.8%

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

   9. Simplified 71.8%

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

   10. Taylor expanded in a around inf 71.8%

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

       1. unpow2 71.8%

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

       2. mul-1-neg 71.8%

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

       3. distribute-rgt-neg-out 71.8%

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

   12. Simplified 71.8%

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

   13. Taylor expanded in a around 0 71.8%

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

       1. unpow2 71.8%

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

       2. associate-*r* 71.8%

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

   15. Simplified 71.8%

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

   if -1 < a

   1. Initial program 76.8%

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

      1. sub-neg 76.8%

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

      2. fma-def 76.8%

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

      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 b around 0 39.3%

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

      1. associate-*r* 39.3%

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

      2. unpow2 39.3%

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

   6. Simplified 39.3%

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

   7. Taylor expanded in a around 0 48.8%

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

      1. unpow2 48.8%

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

      2. *-commutative 48.8%

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

      3. associate-*r* 48.8%

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

      4. *-commutative 48.8%

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

   9. Simplified 48.8%

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

4. Final simplification 54.6%

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

Alternative 9: 50.6% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.42 \lor \neg \left(a \leq 0.08\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.42) (not (<= a 0.08))) (* (* a a) 4.0) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.42) || !(a <= 0.08)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -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 <= (-0.42d0)) .or. (.not. (a <= 0.08d0))) then
        tmp = (a * a) * 4.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -0.42) || !(a <= 0.08)) {
		tmp = (a * a) * 4.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -0.42) or not (a <= 0.08):
		tmp = (a * a) * 4.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.42) || !(a <= 0.08))
		tmp = Float64(Float64(a * a) * 4.0);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -0.42) || ~((a <= 0.08)))
		tmp = (a * a) * 4.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.42], N[Not[LessEqual[a, 0.08]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -0.419999999999999984 or 0.0800000000000000017 < a

   1. Initial program 53.5%

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

      1. sub-neg 53.5%

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

      2. fma-def 53.5%

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

      3. fma-def 58.0%

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

      4. +-commutative 58.0%

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

      5. metadata-eval 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in b around 0 58.6%

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

      1. associate-*r* 58.6%

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

      2. unpow2 58.6%

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

   6. Simplified 58.6%

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

   7. Taylor expanded in a around 0 54.8%

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

      1. unpow2 54.8%

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

      2. *-commutative 54.8%

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

      3. associate-*r* 54.8%

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

      4. *-commutative 54.8%

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

   9. Simplified 54.8%

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

   10. Taylor expanded in a around inf 54.8%

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

       1. unpow2 54.8%

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

   12. Simplified 54.8%

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

   if -0.419999999999999984 < a < 0.0800000000000000017

   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 b around inf 98.3%

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

      1. fma-def 98.3%

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

      2. fma-def 98.3%

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

      3. unpow2 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. metadata-eval 98.3%

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

      6. unpow2 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in b around 0 46.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.42 \lor \neg \left(a \leq 0.08\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 50.7% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

   1. sub-neg 75.6%

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

   2. fma-def 75.6%

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

   3. fma-def 78.0%

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

   4. +-commutative 78.0%

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

   5. metadata-eval 78.0%

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

3. Simplified 78.0%

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

4. Taylor expanded in b around 0 53.5%

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

   1. associate-*r* 53.5%

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

   2. unpow2 53.5%

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

6. Simplified 53.5%

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

7. Taylor expanded in a around 0 50.9%

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

   1. unpow2 50.9%

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

   2. *-commutative 50.9%

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

   3. associate-*r* 50.9%

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

   4. *-commutative 50.9%

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

9. Simplified 50.9%

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

10. Final simplification 50.9%

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

Alternative 11: 25.1% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 75.6%

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

   1. sub-neg 75.6%

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

   2. fma-def 75.6%

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

   3. fma-def 78.0%

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

   4. +-commutative 78.0%

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

   5. metadata-eval 78.0%

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

3. Simplified 78.0%

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

4. Taylor expanded in b around inf 76.7%

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

   1. fma-def 76.7%

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

   2. fma-def 76.7%

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

   3. unpow2 76.7%

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

   4. distribute-lft-in 76.7%

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

   5. metadata-eval 76.7%

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

   6. unpow2 76.7%

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

6. Simplified 76.7%

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

7. Taylor expanded in b around 0 22.4%

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

8. Final simplification 22.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023256 
(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))