Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.1% → 98.3%

Time: 7.7s

Alternatives: 11

Speedup: 9.7×

• 61.0% 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: 74.1% 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: 98.3% accurate, 0.2× speedup

Math

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

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) (+ (* 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) * ((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(a * a) * 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[(a * a), $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 99.9%

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

      1. associate--l+ 99.9%

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

   3. Simplified 100.0%

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

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

   1. Initial program 0.0%

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

      1. 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 6.3%

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

      4. +-commutative 6.3%

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

      5. metadata-eval 6.3%

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

   3. Simplified 6.3%

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

   4. Taylor expanded in a around inf 92.6%

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

      1. sqr-pow 92.6%

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

      2. metadata-eval 92.6%

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

      3. pow2 92.6%

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

      4. metadata-eval 92.6%

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

      5. pow2 92.6%

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

      6. difference-of-sqr--1 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in a around inf 92.6%

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

      1. unpow2 92.6%

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

   9. Simplified 92.6%

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

4. Final simplification 98.2%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

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) (+ (* 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) * ((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) * ((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) * ((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(a * a) * 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) * ((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[(a * a), $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 99.9%

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

   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 6.3%

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

      4. +-commutative 6.3%

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

      5. metadata-eval 6.3%

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

   3. Simplified 6.3%

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

   4. Taylor expanded in a around inf 92.6%

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

      1. sqr-pow 92.6%

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

      2. metadata-eval 92.6%

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

      3. pow2 92.6%

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

      4. metadata-eval 92.6%

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

      5. pow2 92.6%

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

      6. difference-of-sqr--1 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in a around inf 92.6%

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

      1. unpow2 92.6%

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

   9. Simplified 92.6%

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

4. Final simplification 98.1%

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

Alternative 3: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+57}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+57) (+ -1.0 (pow a 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+57) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+57) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+57) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+57:
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+57)
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+57)
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+57], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999972e57

   1. Initial program 84.1%

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

      1. sub-neg 84.1%

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

      2. fma-def 84.1%

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

      3. fma-def 84.1%

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

      4. +-commutative 84.1%

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

      5. metadata-eval 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in a around inf 95.4%

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

   if 4.99999999999999972e57 < (*.f64 b b)

   1. Initial program 60.1%

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

      1. sub-neg 60.1%

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

      2. fma-def 60.1%

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

      3. fma-def 64.2%

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

      4. +-commutative 64.2%

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

      5. metadata-eval 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in b around inf 96.1%

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

      1. sqr-pow 96.1%

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

      2. metadata-eval 96.1%

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

      3. pow2 96.1%

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

      4. metadata-eval 96.1%

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

      5. pow2 96.1%

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

      6. difference-of-sqr--1 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in b around inf 96.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+57}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 4: 93.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+57) (* (+ (* a a) 1.0) (+ (* a a) -1.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+57) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d+57) then
        tmp = ((a * a) + 1.0d0) * ((a * a) + (-1.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+57) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+57:
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+57)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+57)
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+57], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999972e57

   1. Initial program 84.1%

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

      1. sub-neg 84.1%

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

      2. fma-def 84.1%

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

      3. fma-def 84.1%

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

      4. +-commutative 84.1%

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

      5. metadata-eval 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in a around inf 95.4%

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

      1. sqr-pow 95.3%

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

      2. metadata-eval 95.3%

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

      3. pow2 95.3%

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

      4. metadata-eval 95.3%

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

      5. pow2 95.3%

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

      6. difference-of-sqr--1 95.3%

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

   6. Applied egg-rr 95.3%

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

   if 4.99999999999999972e57 < (*.f64 b b)

   1. Initial program 60.1%

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

      1. sub-neg 60.1%

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

      2. fma-def 60.1%

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

      3. fma-def 64.2%

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

      4. +-commutative 64.2%

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

      5. metadata-eval 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in b around inf 96.1%

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

      1. sqr-pow 96.1%

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

      2. metadata-eval 96.1%

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

      3. pow2 96.1%

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

      4. metadata-eval 96.1%

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

      5. pow2 96.1%

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

      6. difference-of-sqr--1 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in b around inf 96.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 93.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.4 \cdot 10^{+64}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4.4e+64)
   (* (+ (* a a) 1.0) (+ (* a a) -1.0))
   (* (* b b) (+ (* b b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.4e+64) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = (b * b) * ((b * b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4.4d+64) then
        tmp = ((a * a) + 1.0d0) * ((a * a) + (-1.0d0))
    else
        tmp = (b * b) * ((b * b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.4e+64) {
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	} else {
		tmp = (b * b) * ((b * b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4.4e+64:
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0)
	else:
		tmp = (b * b) * ((b * b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4.4e+64)
		tmp = Float64(Float64(Float64(a * a) + 1.0) * Float64(Float64(a * a) + -1.0));
	else
		tmp = Float64(Float64(b * b) * Float64(Float64(b * b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4.4e+64)
		tmp = ((a * a) + 1.0) * ((a * a) + -1.0);
	else
		tmp = (b * b) * ((b * b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4.4e+64], N[(N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.40000000000000004e64

   1. Initial program 84.1%

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

      1. sub-neg 84.1%

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

      2. fma-def 84.1%

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

      3. fma-def 84.1%

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

      4. +-commutative 84.1%

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

      5. metadata-eval 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in a around inf 95.4%

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

      1. sqr-pow 95.3%

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

      2. metadata-eval 95.3%

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

      3. pow2 95.3%

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

      4. metadata-eval 95.3%

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

      5. pow2 95.3%

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

      6. difference-of-sqr--1 95.3%

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

   6. Applied egg-rr 95.3%

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

   if 4.40000000000000004e64 < (*.f64 b b)

   1. Initial program 60.1%

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

      1. sub-neg 60.1%

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

      2. fma-def 60.1%

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

      3. fma-def 64.2%

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

      4. +-commutative 64.2%

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

      5. metadata-eval 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in b around inf 96.1%

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

      1. sqr-pow 96.1%

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

      2. metadata-eval 96.1%

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

      3. pow2 96.1%

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

      4. metadata-eval 96.1%

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

      5. pow2 96.1%

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

      6. difference-of-sqr--1 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in b around inf 96.1%

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

      1. unpow2 96.1%

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

   9. Simplified 96.1%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.4 \cdot 10^{+64}:\\ \;\;\;\;\left(a \cdot a + 1\right) \cdot \left(a \cdot a + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + 1\right)\\ \end{array} \]

Alternative 6: 56.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;a \leq -8.8 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq -1.52 \cdot 10^{-12}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+149}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (* a 4.0))))
   (if (<= a -8.8e+152)
     t_0
     (if (<= a -1.52e-12)
       (* b b)
       (if (<= a 1.7e-8) -1.0 (if (<= a 1.75e+149) (* b b) t_0))))))

C

double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double tmp;
	if (a <= -8.8e+152) {
		tmp = t_0;
	} else if (a <= -1.52e-12) {
		tmp = b * b;
	} else if (a <= 1.7e-8) {
		tmp = -1.0;
	} else if (a <= 1.75e+149) {
		tmp = b * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a * 4.0d0)
    if (a <= (-8.8d+152)) then
        tmp = t_0
    else if (a <= (-1.52d-12)) then
        tmp = b * b
    else if (a <= 1.7d-8) then
        tmp = -1.0d0
    else if (a <= 1.75d+149) then
        tmp = b * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a * 4.0);
	double tmp;
	if (a <= -8.8e+152) {
		tmp = t_0;
	} else if (a <= -1.52e-12) {
		tmp = b * b;
	} else if (a <= 1.7e-8) {
		tmp = -1.0;
	} else if (a <= 1.75e+149) {
		tmp = b * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a * 4.0)
	tmp = 0
	if a <= -8.8e+152:
		tmp = t_0
	elif a <= -1.52e-12:
		tmp = b * b
	elif a <= 1.7e-8:
		tmp = -1.0
	elif a <= 1.75e+149:
		tmp = b * b
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a * 4.0))
	tmp = 0.0
	if (a <= -8.8e+152)
		tmp = t_0;
	elseif (a <= -1.52e-12)
		tmp = Float64(b * b);
	elseif (a <= 1.7e-8)
		tmp = -1.0;
	elseif (a <= 1.75e+149)
		tmp = Float64(b * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a * 4.0);
	tmp = 0.0;
	if (a <= -8.8e+152)
		tmp = t_0;
	elseif (a <= -1.52e-12)
		tmp = b * b;
	elseif (a <= 1.7e-8)
		tmp = -1.0;
	elseif (a <= 1.75e+149)
		tmp = b * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -8.8e+152], t$95$0, If[LessEqual[a, -1.52e-12], N[(b * b), $MachinePrecision], If[LessEqual[a, 1.7e-8], -1.0, If[LessEqual[a, 1.75e+149], N[(b * b), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.7999999999999991e152 or 1.75000000000000006e149 < a

   1. Initial program 30.9%

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

      1. sub-neg 30.9%

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

      2. fma-def 30.9%

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

      3. fma-def 32.4%

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

      4. +-commutative 32.4%

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

      5. metadata-eval 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in b around 0 50.0%

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

      1. associate-*r* 50.0%

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

      2. unpow2 50.0%

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

   6. Simplified 50.0%

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

   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%

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

   9. Simplified 97.5%

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

   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%

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

       2. *-commutative 97.5%

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

       3. associate-*r* 97.5%

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

   12. Simplified 97.5%

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

   if -8.7999999999999991e152 < a < -1.52e-12 or 1.7e-8 < a < 1.75000000000000006e149

   1. Initial program 77.2%

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

      1. sub-neg 77.2%

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

      2. fma-def 77.2%

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

      3. fma-def 81.2%

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

      4. +-commutative 81.2%

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

      5. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in b around inf 43.3%

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

      1. sqr-pow 43.3%

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

      2. metadata-eval 43.3%

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

      3. pow2 43.3%

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

      4. metadata-eval 43.3%

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

      5. pow2 43.3%

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

      6. difference-of-sqr--1 43.3%

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

   6. Applied egg-rr 43.3%

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

   7. Taylor expanded in b around inf 42.9%

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

      1. unpow2 42.9%

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

   9. Simplified 42.9%

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

   10. Taylor expanded in b around 0 25.4%

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

       1. unpow2 25.4%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 25.4%

       \[\leadsto \color{blue}{b \cdot b} \]

   if -1.52e-12 < a < 1.7e-8

   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 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. unpow2 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in a around 0 60.2%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.8 \cdot 10^{+152}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \mathbf{elif}\;a \leq -1.52 \cdot 10^{-12}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+149}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 4\right)\\ \end{array} \]

Alternative 7: 82.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+14} \lor \neg \left(a \leq 3000000000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.6e+14) (not (<= a 3000000000000.0)))
   (* (* a a) (+ (* a a) 1.0))
   (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.6e+14) || !(a <= 3000000000000.0)) {
		tmp = (a * a) * ((a * a) + 1.0);
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-2.6d+14)) .or. (.not. (a <= 3000000000000.0d0))) then
        tmp = (a * a) * ((a * a) + 1.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.6e+14) || !(a <= 3000000000000.0)) {
		tmp = (a * a) * ((a * a) + 1.0);
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.6e+14) or not (a <= 3000000000000.0):
		tmp = (a * a) * ((a * a) + 1.0)
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.6e+14) || !(a <= 3000000000000.0))
		tmp = Float64(Float64(a * a) * Float64(Float64(a * a) + 1.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.6e+14) || ~((a <= 3000000000000.0)))
		tmp = (a * a) * ((a * a) + 1.0);
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.6e+14], N[Not[LessEqual[a, 3000000000000.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6e14 or 3e12 < a

   1. Initial program 52.2%

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

      1. sub-neg 52.2%

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

      2. fma-def 52.2%

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

      3. fma-def 55.2%

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

      4. +-commutative 55.2%

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

      5. metadata-eval 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in a around inf 91.5%

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

      1. sqr-pow 91.5%

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

      2. metadata-eval 91.5%

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

      3. pow2 91.5%

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

      4. metadata-eval 91.5%

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

      5. pow2 91.5%

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

      6. difference-of-sqr--1 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in a around inf 91.5%

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

      1. unpow2 91.5%

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

   9. Simplified 91.5%

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

   if -2.6e14 < a < 3e12

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. fma-def 99.1%

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

      3. fma-def 99.1%

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

      4. +-commutative 99.1%

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

      5. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in a around 0 91.0%

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

      1. associate-+r+ 91.0%

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

      2. associate-*r* 91.0%

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

      3. distribute-rgt-out 99.0%

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

      4. metadata-eval 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. +-commutative 99.0%

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

      7. unpow2 99.0%

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

      8. distribute-lft-in 99.0%

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

      9. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in a around 0 99.8%

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

      1. unpow2 99.8%

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

   9. Simplified 99.8%

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

       1. sqr-pow 99.7%

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

       2. metadata-eval 99.7%

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

       3. pow2 99.7%

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

       4. metadata-eval 99.7%

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

       5. pow2 99.7%

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

       6. distribute-rgt-out 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in b around 0 78.2%

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

       1. unpow2 78.2%

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

   14. Simplified 78.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+14} \lor \neg \left(a \leq 3000000000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a + 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 8: 56.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.95 \cdot 10^{+153}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;a \leq -1.52 \cdot 10^{-12}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+149}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.95e+153)
   (* a a)
   (if (<= a -1.52e-12)
     (* b b)
     (if (<= a 2.7e-9) -1.0 (if (<= a 3e+149) (* b b) (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.95e+153) {
		tmp = a * a;
	} else if (a <= -1.52e-12) {
		tmp = b * b;
	} else if (a <= 2.7e-9) {
		tmp = -1.0;
	} else if (a <= 3e+149) {
		tmp = b * b;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.95d+153)) then
        tmp = a * a
    else if (a <= (-1.52d-12)) then
        tmp = b * b
    else if (a <= 2.7d-9) then
        tmp = -1.0d0
    else if (a <= 3d+149) then
        tmp = b * b
    else
        tmp = a * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.95e+153) {
		tmp = a * a;
	} else if (a <= -1.52e-12) {
		tmp = b * b;
	} else if (a <= 2.7e-9) {
		tmp = -1.0;
	} else if (a <= 3e+149) {
		tmp = b * b;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.95e+153:
		tmp = a * a
	elif a <= -1.52e-12:
		tmp = b * b
	elif a <= 2.7e-9:
		tmp = -1.0
	elif a <= 3e+149:
		tmp = b * b
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.95e+153)
		tmp = Float64(a * a);
	elseif (a <= -1.52e-12)
		tmp = Float64(b * b);
	elseif (a <= 2.7e-9)
		tmp = -1.0;
	elseif (a <= 3e+149)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.95e+153)
		tmp = a * a;
	elseif (a <= -1.52e-12)
		tmp = b * b;
	elseif (a <= 2.7e-9)
		tmp = -1.0;
	elseif (a <= 3e+149)
		tmp = b * b;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.95e+153], N[(a * a), $MachinePrecision], If[LessEqual[a, -1.52e-12], N[(b * b), $MachinePrecision], If[LessEqual[a, 2.7e-9], -1.0, If[LessEqual[a, 3e+149], N[(b * b), $MachinePrecision], N[(a * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.94999999999999992e153 or 3.00000000000000003e149 < a

   1. Initial program 30.9%

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

      1. sub-neg 30.9%

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

      2. fma-def 30.9%

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

      3. fma-def 32.4%

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

      4. +-commutative 32.4%

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

      5. metadata-eval 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in a around inf 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

      6. difference-of-sqr--1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in a around 0 97.4%

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

       1. unpow2 97.4%

          \[\leadsto \color{blue}{a \cdot a} \]

   12. Simplified 97.4%

       \[\leadsto \color{blue}{a \cdot a} \]

   if -1.94999999999999992e153 < a < -1.52e-12 or 2.7000000000000002e-9 < a < 3.00000000000000003e149

   1. Initial program 77.2%

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

      1. sub-neg 77.2%

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

      2. fma-def 77.2%

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

      3. fma-def 81.2%

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

      4. +-commutative 81.2%

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

      5. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in b around inf 43.3%

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

      1. sqr-pow 43.3%

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

      2. metadata-eval 43.3%

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

      3. pow2 43.3%

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

      4. metadata-eval 43.3%

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

      5. pow2 43.3%

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

      6. difference-of-sqr--1 43.3%

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

   6. Applied egg-rr 43.3%

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

   7. Taylor expanded in b around inf 42.9%

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

      1. unpow2 42.9%

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

   9. Simplified 42.9%

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

   10. Taylor expanded in b around 0 25.4%

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

       1. unpow2 25.4%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 25.4%

       \[\leadsto \color{blue}{b \cdot b} \]

   if -1.52e-12 < a < 2.7000000000000002e-9

   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 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. unpow2 60.2%

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

   6. Simplified 60.2%

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

   7. Taylor expanded in a around 0 60.2%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.95 \cdot 10^{+153}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;a \leq -1.52 \cdot 10^{-12}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+149}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 9: 69.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.5 \cdot 10^{+306}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 3.5e+306) (+ -1.0 (* (* a a) 4.0)) (* b b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.5e+306) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = b * b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 3.5d+306) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = b * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.5e+306) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = b * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 3.5e+306:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = b * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 3.5e+306)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 3.5e+306)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = b * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 3.5e+306], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.5000000000000001e306

   1. Initial program 79.9%

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

      1. sub-neg 79.9%

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

      2. fma-def 79.9%

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

      3. fma-def 80.9%

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

      4. +-commutative 80.9%

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

      5. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in b around 0 66.2%

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

      1. associate-*r* 66.2%

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

      2. unpow2 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in a around 0 60.7%

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

      1. unpow2 60.7%

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

   9. Simplified 60.7%

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

   if 3.5000000000000001e306 < (*.f64 b b)

   1. Initial program 54.9%

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

      1. sub-neg 54.9%

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

      2. fma-def 54.9%

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

      3. fma-def 58.8%

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

      4. +-commutative 58.8%

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

      5. metadata-eval 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

      6. difference-of-sqr--1 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 100.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.5 \cdot 10^{+306}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]

Alternative 10: 50.4% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.41:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.41) (* a a) (if (<= a 1.7e-8) -1.0 (* a a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.41) {
		tmp = a * a;
	} else if (a <= 1.7e-8) {
		tmp = -1.0;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.41d0)) then
        tmp = a * a
    else if (a <= 1.7d-8) then
        tmp = -1.0d0
    else
        tmp = a * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.41) {
		tmp = a * a;
	} else if (a <= 1.7e-8) {
		tmp = -1.0;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.41:
		tmp = a * a
	elif a <= 1.7e-8:
		tmp = -1.0
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.41)
		tmp = Float64(a * a);
	elseif (a <= 1.7e-8)
		tmp = -1.0;
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.41)
		tmp = a * a;
	elseif (a <= 1.7e-8)
		tmp = -1.0;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.41], N[(a * a), $MachinePrecision], If[LessEqual[a, 1.7e-8], -1.0, N[(a * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.409999999999999976 or 1.7e-8 < a

   1. Initial program 53.6%

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

      1. sub-neg 53.6%

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

      2. fma-def 53.6%

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

      3. fma-def 56.5%

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

      4. +-commutative 56.5%

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

      5. metadata-eval 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in a around inf 87.7%

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

      1. sqr-pow 87.6%

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

      2. metadata-eval 87.6%

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

      3. pow2 87.6%

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

      4. metadata-eval 87.6%

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

      5. pow2 87.6%

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

      6. difference-of-sqr--1 87.6%

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

   6. Applied egg-rr 87.6%

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

   7. Taylor expanded in a around inf 87.7%

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

      1. unpow2 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in a around 0 51.0%

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

       1. unpow2 51.0%

          \[\leadsto \color{blue}{a \cdot a} \]

   12. Simplified 51.0%

       \[\leadsto \color{blue}{a \cdot a} \]

   if -0.409999999999999976 < a < 1.7e-8

   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 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 58.4%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.41:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 11: 24.9% 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 74.9%

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

   1. sub-neg 74.9%

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

   2. fma-def 74.9%

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

   3. fma-def 76.5%

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

   4. +-commutative 76.5%

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

   5. metadata-eval 76.5%

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

3. Simplified 76.5%

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

4. Taylor expanded in b around 0 57.9%

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

   1. associate-*r* 57.9%

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

   2. unpow2 57.9%

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

6. Simplified 57.9%

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

7. Taylor expanded in a around 0 27.3%

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

8. Final simplification 27.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023252 
(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))