Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.3% → 98.3%

Time: 7.2s

Alternatives: 9

Speedup: 8.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ a -2.0))))
   (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))
     (+ -1.0 (* t_0 t_0)))))

C

double code(double a, double b) {
	double t_0 = a * (a + -2.0);
	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 = -1.0 + (t_0 * t_0);
	}
	return tmp;
}

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + 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[(-1.0 + N[(t$95$0 * t$95$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 3.6%

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

      4. +-commutative 3.6%

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

      5. metadata-eval 3.6%

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

   3. Simplified 3.6%

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

      1. fma-def 3.6%

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

      2. fma-udef 0.0%

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

      3. +-commutative 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. pow2 0.0%

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

   5. Applied egg-rr 3.6%

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

   6. Taylor expanded in a around inf 93.2%

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

      1. +-commutative 93.2%

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

      2. unpow2 93.2%

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

      3. distribute-rgt-out 93.2%

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

   8. Simplified 93.2%

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

      1. unpow2 93.2%

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

   10. Applied egg-rr 93.2%

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

4. Final simplification 98.5%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ t_1 := a \cdot \left(a + -2\right)\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + t_1 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))))
        (t_1 (* a (+ a -2.0))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (+ -1.0 (* t_1 t_1)))))

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 t_1 = a * (a + -2.0);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = -1.0 + (t_1 * t_1);
	}
	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 t_1 = a * (a + -2.0);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = -1.0 + (t_1 * t_1);
	}
	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))))
	t_1 = a * (a + -2.0)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = -1.0 + (t_1 * t_1)
	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)))))
	t_1 = Float64(a * Float64(a + -2.0))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(-1.0 + Float64(t_1 * t_1));
	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))));
	t_1 = a * (a + -2.0);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = -1.0 + (t_1 * t_1);
	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]}, Block[{t$95$1 = N[(a * N[(a + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(-1.0 + N[(t$95$1 * t$95$1), $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 3.6%

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

      4. +-commutative 3.6%

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

      5. metadata-eval 3.6%

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

   3. Simplified 3.6%

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

      1. fma-def 3.6%

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

      2. fma-udef 0.0%

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

      3. +-commutative 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. pow2 0.0%

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

   5. Applied egg-rr 3.6%

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

   6. Taylor expanded in a around inf 93.2%

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

      1. +-commutative 93.2%

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

      2. unpow2 93.2%

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

      3. distribute-rgt-out 93.2%

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

   8. Simplified 93.2%

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

      1. unpow2 93.2%

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

   10. Applied egg-rr 93.2%

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

4. Final simplification 98.4%

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

Alternative 3: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;-1 + \left(a + -2\right) \cdot \left(a \cdot \left(a \cdot \left(a + -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+96)
   (+ -1.0 (* (+ a -2.0) (* a (* a (+ a -2.0)))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+96) {
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))));
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+96) then
        tmp = (-1.0d0) + ((a + (-2.0d0)) * (a * (a * (a + (-2.0d0)))))
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+96) {
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+96:
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+96)
		tmp = Float64(-1.0 + Float64(Float64(a + -2.0) * Float64(a * Float64(a * Float64(a + -2.0)))));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+96)
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+96], N[(-1.0 + N[(N[(a + -2.0), $MachinePrecision] * N[(a * N[(a * N[(a + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e96

   1. Initial program 87.1%

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

      1. sub-neg 87.1%

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

      2. fma-def 87.1%

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

      3. fma-def 87.1%

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

      4. +-commutative 87.1%

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

      5. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

      1. fma-def 87.1%

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

      2. fma-udef 87.1%

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

      3. +-commutative 87.1%

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

      4. add-sqr-sqrt 87.1%

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

      5. pow2 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in a around inf 96.5%

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

      1. +-commutative 96.5%

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

      2. unpow2 96.5%

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

      3. distribute-rgt-out 96.6%

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

   8. Simplified 96.6%

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

      1. unpow2 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*l* 96.6%

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

   10. Applied egg-rr 96.6%

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

   if 2.0000000000000001e96 < (*.f64 b b)

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. fma-def 65.4%

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

      3. fma-def 67.2%

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

      4. +-commutative 67.2%

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

      5. metadata-eval 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in b around inf 96.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;-1 + \left(a + -2\right) \cdot \left(a \cdot \left(a \cdot \left(a + -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 4: 93.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a + -2\right)\\ \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;-1 + t_0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (+ a -2.0))))
   (if (<= (* b b) 2e+96)
     (+ -1.0 (* t_0 t_0))
     (+ -1.0 (* (* b b) (+ (* b b) 12.0))))))

C

double code(double a, double b) {
	double t_0 = a * (a + -2.0);
	double tmp;
	if ((b * b) <= 2e+96) {
		tmp = -1.0 + (t_0 * t_0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (a + (-2.0d0))
    if ((b * b) <= 2d+96) then
        tmp = (-1.0d0) + (t_0 * t_0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (a + -2.0);
	double tmp;
	if ((b * b) <= 2e+96) {
		tmp = -1.0 + (t_0 * t_0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (a + -2.0)
	tmp = 0
	if (b * b) <= 2e+96:
		tmp = -1.0 + (t_0 * t_0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(a + -2.0))
	tmp = 0.0
	if (Float64(b * b) <= 2e+96)
		tmp = Float64(-1.0 + Float64(t_0 * t_0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (a + -2.0);
	tmp = 0.0;
	if ((b * b) <= 2e+96)
		tmp = -1.0 + (t_0 * t_0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a + -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 2e+96], N[(-1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e96

   1. Initial program 87.1%

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

      1. sub-neg 87.1%

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

      2. fma-def 87.1%

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

      3. fma-def 87.1%

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

      4. +-commutative 87.1%

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

      5. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

      1. fma-def 87.1%

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

      2. fma-udef 87.1%

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

      3. +-commutative 87.1%

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

      4. add-sqr-sqrt 87.1%

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

      5. pow2 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in a around inf 96.5%

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

      1. +-commutative 96.5%

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

      2. unpow2 96.5%

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

      3. distribute-rgt-out 96.6%

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

   8. Simplified 96.6%

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

      1. unpow2 96.6%

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

   10. Applied egg-rr 96.6%

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

   if 2.0000000000000001e96 < (*.f64 b b)

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. fma-def 65.4%

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

      3. fma-def 67.2%

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

      4. +-commutative 67.2%

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

      5. metadata-eval 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in a around 0 65.4%

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

      1. associate-+r+ 65.4%

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

      2. associate-*r* 65.4%

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

      3. distribute-rgt-out 79.4%

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

      4. metadata-eval 79.4%

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

      5. distribute-lft-in 79.4%

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

      6. unpow2 79.4%

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

      7. distribute-rgt-in 79.4%

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

      8. metadata-eval 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in a around 0 96.6%

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

      1. unpow2 96.6%

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

   9. Simplified 96.6%

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

       1. sqr-pow 96.6%

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

       2. metadata-eval 96.6%

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

       3. pow2 96.6%

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

       4. metadata-eval 96.6%

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

       5. pow2 96.6%

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

       6. distribute-rgt-out 96.6%

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

   11. Applied egg-rr 96.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;-1 + \left(a \cdot \left(a + -2\right)\right) \cdot \left(a \cdot \left(a + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 5: 93.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;-1 + \left(a + -2\right) \cdot \left(a \cdot \left(a \cdot \left(a + -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+96)
   (+ -1.0 (* (+ a -2.0) (* a (* a (+ a -2.0)))))
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+96) {
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))));
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+96) then
        tmp = (-1.0d0) + ((a + (-2.0d0)) * (a * (a * (a + (-2.0d0)))))
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+96) {
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))));
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+96:
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))))
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+96)
		tmp = Float64(-1.0 + Float64(Float64(a + -2.0) * Float64(a * Float64(a * Float64(a + -2.0)))));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+96)
		tmp = -1.0 + ((a + -2.0) * (a * (a * (a + -2.0))));
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+96], N[(-1.0 + N[(N[(a + -2.0), $MachinePrecision] * N[(a * N[(a * N[(a + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e96

   1. Initial program 87.1%

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

      1. sub-neg 87.1%

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

      2. fma-def 87.1%

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

      3. fma-def 87.1%

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

      4. +-commutative 87.1%

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

      5. metadata-eval 87.1%

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

   3. Simplified 87.1%

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

      1. fma-def 87.1%

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

      2. fma-udef 87.1%

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

      3. +-commutative 87.1%

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

      4. add-sqr-sqrt 87.1%

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

      5. pow2 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in a around inf 96.5%

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

      1. +-commutative 96.5%

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

      2. unpow2 96.5%

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

      3. distribute-rgt-out 96.6%

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

   8. Simplified 96.6%

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

      1. unpow2 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*l* 96.6%

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

   10. Applied egg-rr 96.6%

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

   if 2.0000000000000001e96 < (*.f64 b b)

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. fma-def 65.4%

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

      3. fma-def 67.2%

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

      4. +-commutative 67.2%

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

      5. metadata-eval 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in a around 0 65.4%

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

      1. associate-+r+ 65.4%

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

      2. associate-*r* 65.4%

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

      3. distribute-rgt-out 79.4%

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

      4. metadata-eval 79.4%

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

      5. distribute-lft-in 79.4%

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

      6. unpow2 79.4%

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

      7. distribute-rgt-in 79.4%

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

      8. metadata-eval 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in a around 0 96.6%

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

      1. unpow2 96.6%

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

   9. Simplified 96.6%

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

       1. sqr-pow 96.6%

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

       2. metadata-eval 96.6%

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

       3. pow2 96.6%

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

       4. metadata-eval 96.6%

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

       5. pow2 96.6%

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

       6. distribute-rgt-out 96.6%

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

   11. Applied egg-rr 96.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+96}:\\ \;\;\;\;-1 + \left(a + -2\right) \cdot \left(a \cdot \left(a \cdot \left(a + -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 6: 86.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+90}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 \cdot \left(1 - a\right)\right)\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+141}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.05e+90)
   (+ -1.0 (* (* a a) (* 4.0 (- 1.0 a))))
   (if (<= a 8e+141)
     (+ -1.0 (* (* b b) (+ (* b b) 12.0)))
     (+ -1.0 (* (* a a) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.05e+90) {
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)));
	} else if (a <= 8e+141) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.05d+90)) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 * (1.0d0 - a)))
    else if (a <= 8d+141) then
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    else
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.05e+90) {
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)));
	} else if (a <= 8e+141) {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.05e+90:
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)))
	elif a <= 8e+141:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	else:
		tmp = -1.0 + ((a * a) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.05e+90)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 * Float64(1.0 - a))));
	elseif (a <= 8e+141)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.05e+90)
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)));
	elseif (a <= 8e+141)
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	else
		tmp = -1.0 + ((a * a) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.05e+90], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8e+141], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.0499999999999999e90

   1. Initial program 53.8%

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

      1. sub-neg 53.8%

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

      2. fma-def 53.8%

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

      3. fma-def 53.8%

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

      4. +-commutative 53.8%

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

      5. metadata-eval 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 97.6%

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

      1. unpow2 97.6%

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

      2. *-lft-identity 97.6%

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

      3. cube-mult 97.6%

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

      4. unpow2 97.6%

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

      5. associate-*r* 97.6%

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

      6. metadata-eval 97.6%

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

      7. distribute-lft-neg-in 97.6%

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

      8. *-commutative 97.6%

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

      9. distribute-lft-neg-in 97.6%

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

      10. associate-*r* 97.6%

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

      11. unpow2 97.6%

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

      12. distribute-rgt-in 97.6%

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

      13. sub-neg 97.6%

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

      14. unpow2 97.6%

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

      15. associate-*r* 97.6%

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

      16. *-commutative 97.6%

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

      17. associate-*l* 97.6%

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

      18. unpow2 97.6%

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

   9. Simplified 97.6%

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

   if -1.0499999999999999e90 < a < 8.00000000000000014e141

   1. Initial program 93.6%

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

      1. sub-neg 93.6%

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

      2. fma-def 93.6%

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

      3. fma-def 94.6%

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

      4. +-commutative 94.6%

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

      5. metadata-eval 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in a around 0 73.0%

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

      1. associate-+r+ 73.0%

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

      2. associate-*r* 73.0%

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

      3. distribute-rgt-out 80.9%

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

      4. metadata-eval 80.9%

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

      5. distribute-lft-in 80.9%

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

      6. unpow2 80.9%

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

      7. distribute-rgt-in 80.9%

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

      8. metadata-eval 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in a around 0 83.5%

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

      1. unpow2 83.5%

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

   9. Simplified 83.5%

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

       1. sqr-pow 83.4%

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

       2. metadata-eval 83.4%

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

       3. pow2 83.4%

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

       4. metadata-eval 83.4%

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

       5. pow2 83.4%

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

       6. distribute-rgt-out 83.4%

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

   11. Applied egg-rr 83.4%

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

   if 8.00000000000000014e141 < a

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. fma-def 0.0%

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

      3. fma-def 0.0%

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

      4. +-commutative 0.0%

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

      5. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in b around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in a around 0 93.0%

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

      1. unpow2 93.0%

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

   9. Simplified 93.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+90}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 \cdot \left(1 - a\right)\right)\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+141}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \end{array} \]

Alternative 7: 71.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+90}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 \cdot \left(1 - a\right)\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.1e+90)
   (+ -1.0 (* (* a a) (* 4.0 (- 1.0 a))))
   (if (<= a 1.4e+138) (+ -1.0 (* (* b b) 12.0)) (+ -1.0 (* (* a a) 4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.1e+90) {
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)));
	} else if (a <= 1.4e+138) {
		tmp = -1.0 + ((b * b) * 12.0);
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.1d+90)) then
        tmp = (-1.0d0) + ((a * a) * (4.0d0 * (1.0d0 - a)))
    else if (a <= 1.4d+138) then
        tmp = (-1.0d0) + ((b * b) * 12.0d0)
    else
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.1e+90) {
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)));
	} else if (a <= 1.4e+138) {
		tmp = -1.0 + ((b * b) * 12.0);
	} else {
		tmp = -1.0 + ((a * a) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.1e+90:
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)))
	elif a <= 1.4e+138:
		tmp = -1.0 + ((b * b) * 12.0)
	else:
		tmp = -1.0 + ((a * a) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.1e+90)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(4.0 * Float64(1.0 - a))));
	elseif (a <= 1.4e+138)
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 12.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.1e+90)
		tmp = -1.0 + ((a * a) * (4.0 * (1.0 - a)));
	elseif (a <= 1.4e+138)
		tmp = -1.0 + ((b * b) * 12.0);
	else
		tmp = -1.0 + ((a * a) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.1e+90], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(4.0 * N[(1.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.4e+138], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.09999999999999995e90

   1. Initial program 53.8%

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

      1. sub-neg 53.8%

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

      2. fma-def 53.8%

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

      3. fma-def 53.8%

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

      4. +-commutative 53.8%

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

      5. metadata-eval 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 97.6%

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

      1. unpow2 97.6%

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

      2. *-lft-identity 97.6%

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

      3. cube-mult 97.6%

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

      4. unpow2 97.6%

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

      5. associate-*r* 97.6%

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

      6. metadata-eval 97.6%

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

      7. distribute-lft-neg-in 97.6%

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

      8. *-commutative 97.6%

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

      9. distribute-lft-neg-in 97.6%

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

      10. associate-*r* 97.6%

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

      11. unpow2 97.6%

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

      12. distribute-rgt-in 97.6%

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

      13. sub-neg 97.6%

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

      14. unpow2 97.6%

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

      15. associate-*r* 97.6%

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

      16. *-commutative 97.6%

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

      17. associate-*l* 97.6%

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

      18. unpow2 97.6%

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

   9. Simplified 97.6%

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

   if -1.09999999999999995e90 < a < 1.4e138

   1. Initial program 95.1%

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

      1. sub-neg 95.1%

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

      2. fma-def 95.1%

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

      3. fma-def 95.6%

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

      4. +-commutative 95.6%

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

      5. metadata-eval 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in a around 0 73.6%

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

      1. associate-+r+ 73.6%

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

      2. associate-*r* 73.6%

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

      3. distribute-rgt-out 81.6%

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

      4. metadata-eval 81.6%

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

      5. distribute-lft-in 81.6%

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

      6. unpow2 81.6%

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

      7. distribute-rgt-in 81.6%

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

      8. metadata-eval 81.6%

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

   6. Simplified 81.6%

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

   7. Taylor expanded in a around 0 84.3%

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

      1. unpow2 84.3%

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

   9. Simplified 84.3%

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

       1. sqr-pow 84.2%

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

       2. metadata-eval 84.2%

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

       3. pow2 84.2%

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

       4. metadata-eval 84.2%

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

       5. pow2 84.2%

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

       6. distribute-rgt-out 84.2%

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

   11. Applied egg-rr 84.2%

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

   12. Taylor expanded in b around 0 66.9%

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

       1. unpow2 66.9%

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

   14. Simplified 66.9%

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

   if 1.4e138 < 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 3.4%

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

      4. +-commutative 3.4%

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

      5. metadata-eval 3.4%

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

   3. Simplified 3.4%

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

   4. Taylor expanded in b around 0 0.0%

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

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in a around 0 84.2%

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

      1. unpow2 84.2%

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

   9. Simplified 84.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+90}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(4 \cdot \left(1 - a\right)\right)\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+138}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \end{array} \]

Alternative 8: 69.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+306}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+306) (+ -1.0 (* (* a a) 4.0)) (+ -1.0 (* (* b b) 12.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+306) {
		tmp = -1.0 + ((a * a) * 4.0);
	} 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 ((b * b) <= 2d+306) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    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 ((b * b) <= 2e+306) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = -1.0 + ((b * b) * 12.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+306:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = -1.0 + ((b * b) * 12.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+306)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	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 ((b * b) <= 2e+306)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = -1.0 + ((b * b) * 12.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+306], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000003e306

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. fma-def 81.3%

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

      3. fma-def 81.8%

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

      4. +-commutative 81.8%

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

      5. metadata-eval 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in b around 0 70.3%

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

      1. associate-*r* 70.3%

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

      2. unpow2 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in a around 0 62.6%

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

      1. unpow2 62.6%

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

   9. Simplified 62.6%

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

   if 2.00000000000000003e306 < (*.f64 b b)

   1. Initial program 67.7%

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

      1. sub-neg 67.7%

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

      2. fma-def 67.7%

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

      3. fma-def 69.4%

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

      4. +-commutative 69.4%

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

      5. metadata-eval 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in a around 0 56.5%

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

      1. associate-+r+ 56.5%

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

      2. associate-*r* 56.5%

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

      3. distribute-rgt-out 80.6%

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

      4. metadata-eval 80.6%

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

      5. distribute-lft-in 80.6%

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

      6. unpow2 80.6%

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

      7. distribute-rgt-in 80.6%

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

      8. metadata-eval 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

       1. sqr-pow 100.0%

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

       2. metadata-eval 100.0%

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

       3. pow2 100.0%

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

       4. metadata-eval 100.0%

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

       5. pow2 100.0%

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

       6. distribute-rgt-out 100.0%

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

   11. Applied egg-rr 100.0%

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

   12. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+306}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 12\\ \end{array} \]

Alternative 9: 50.9% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

   1. sub-neg 78.0%

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

   2. fma-def 78.0%

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

   3. fma-def 78.8%

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

   4. +-commutative 78.8%

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

   5. metadata-eval 78.8%

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

3. Simplified 78.8%

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

4. Taylor expanded in b around 0 57.8%

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

   1. associate-*r* 57.8%

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

   2. unpow2 57.8%

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

6. Simplified 57.8%

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

7. Taylor expanded in a around 0 53.2%

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

   1. unpow2 53.2%

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

9. Simplified 53.2%

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

10. Final simplification 53.2%

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

Reproduce

herbie shell --seed 2023221 
(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))