Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.7% → 98.4%

Time: 6.4s

Alternatives: 9

Speedup: 8.5×

• 60.6% 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: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(4.0 * N[(a * N[(a - N[(a * a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. fma-def 99.8%

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

      5. associate-*l* 99.8%

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

      6. fma-def 99.8%

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

      7. distribute-lft-out-- 99.8%

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

      8. *-rgt-identity 99.8%

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

      9. +-commutative 99.8%

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

   3. Simplified 100.0%

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

   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. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-rgt-in 0.0%

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

   3. Simplified 1.4%

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

   4. Taylor expanded in a around inf 36.7%

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

      1. unpow3 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. metadata-eval 36.7%

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

      2. pow-prod-up 36.7%

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

      3. pow1 36.7%

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

      4. pow3 36.7%

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

      5. distribute-rgt-in 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-*l* 90.3%

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

      8. *-commutative 90.3%

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

   8. Applied egg-rr 90.3%

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

4. Final simplification 97.4%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-rgt-in 0.0%

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

   3. Simplified 1.4%

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

   4. Taylor expanded in a around inf 36.7%

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

      1. unpow3 36.7%

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

   6. Applied egg-rr 36.7%

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

      1. metadata-eval 36.7%

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

      2. pow-prod-up 36.7%

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

      3. pow1 36.7%

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

      4. pow3 36.7%

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

      5. distribute-rgt-in 90.3%

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

      6. +-commutative 90.3%

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

      7. associate-*l* 90.3%

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

      8. *-commutative 90.3%

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

   8. Applied egg-rr 90.3%

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

4. Final simplification 97.3%

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

Alternative 3: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+30}:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+30)
   (+ -1.0 (* (+ a -4.0) (pow a 3.0)))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+30) {
		tmp = -1.0 + ((a + -4.0) * pow(a, 3.0));
	} 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+30) then
        tmp = (-1.0d0) + ((a + (-4.0d0)) * (a ** 3.0d0))
    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+30) {
		tmp = -1.0 + ((a + -4.0) * Math.pow(a, 3.0));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+30:
		tmp = -1.0 + ((a + -4.0) * math.pow(a, 3.0))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+30)
		tmp = Float64(-1.0 + Float64(Float64(a + -4.0) * (a ^ 3.0)));
	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+30)
		tmp = -1.0 + ((a + -4.0) * (a ^ 3.0));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+30], N[(-1.0 + N[(N[(a + -4.0), $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e30

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. sqr-pow 83.8%

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

      3. sqr-pow 83.8%

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

      4. sqr-neg 83.8%

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

      5. distribute-rgt-in 83.8%

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

      6. sqr-neg 83.8%

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

      7. distribute-rgt-in 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in a around inf 82.5%

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

      1. expm1-log1p-u 81.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot {a}^{3} + {a}^{4}\right)\right)} + -1 \]

      2. expm1-udef 81.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-4 \cdot {a}^{3} + {a}^{4}\right)} - 1\right)} + -1 \]

      3. fma-def 81.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)}\right)} - 1\right) + -1 \]

   6. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)\right)} - 1\right)} + -1 \]
   7. Step-by-step derivation

      1. expm1-def 81.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)\right)\right)} + -1 \]

      2. expm1-log1p 82.5%

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

      3. fma-udef 82.5%

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

      4. +-commutative 82.5%

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

      5. metadata-eval 82.5%

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

      6. pow-plus 82.5%

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

      7. *-commutative 82.5%

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

      8. distribute-lft-out 98.5%

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

   8. Simplified 98.5%

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

   if 2e30 < (*.f64 b b)

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. sqr-pow 61.5%

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

      3. sqr-pow 61.5%

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

      4. sqr-neg 61.5%

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

      5. distribute-rgt-in 61.5%

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

      6. sqr-neg 61.5%

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

      7. distribute-rgt-in 61.5%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in b around inf 93.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+30}:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 4: 93.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+30}:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+30)
   (+ -1.0 (* (+ a -4.0) (* a (* a a))))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+30) {
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)));
	} 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+30) then
        tmp = (-1.0d0) + ((a + (-4.0d0)) * (a * (a * a)))
    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+30) {
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)));
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+30:
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)))
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+30)
		tmp = Float64(-1.0 + Float64(Float64(a + -4.0) * Float64(a * Float64(a * a))));
	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+30)
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)));
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+30], N[(-1.0 + N[(N[(a + -4.0), $MachinePrecision] * N[(a * N[(a * a), $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) < 2e30

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. sqr-pow 83.8%

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

      3. sqr-pow 83.8%

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

      4. sqr-neg 83.8%

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

      5. distribute-rgt-in 83.8%

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

      6. sqr-neg 83.8%

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

      7. distribute-rgt-in 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in a around inf 82.5%

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

      1. expm1-log1p-u 81.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot {a}^{3} + {a}^{4}\right)\right)} + -1 \]

      2. expm1-udef 81.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-4 \cdot {a}^{3} + {a}^{4}\right)} - 1\right)} + -1 \]

      3. fma-def 81.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)}\right)} - 1\right) + -1 \]

   6. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)\right)} - 1\right)} + -1 \]
   7. Step-by-step derivation

      1. expm1-def 81.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)\right)\right)} + -1 \]

      2. expm1-log1p 82.5%

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

      3. fma-udef 82.5%

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

      4. +-commutative 82.5%

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

      5. metadata-eval 82.5%

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

      6. pow-plus 82.5%

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

      7. *-commutative 82.5%

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

      8. distribute-lft-out 98.5%

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

   8. Simplified 98.5%

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

      1. unpow3 82.5%

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

   10. Applied egg-rr 98.5%

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

   if 2e30 < (*.f64 b b)

   1. Initial program 61.5%

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

      1. sub-neg 61.5%

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

      2. sqr-pow 61.5%

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

      3. sqr-pow 61.5%

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

      4. sqr-neg 61.5%

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

      5. distribute-rgt-in 61.5%

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

      6. sqr-neg 61.5%

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

      7. distribute-rgt-in 61.5%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in b around inf 93.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+30}:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 5: 84.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+288}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot \left(a + -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+288)
   (+ -1.0 (* (* a a) (* a (+ a -4.0))))
   (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+288) {
		tmp = -1.0 + ((a * a) * (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+288) then
        tmp = (-1.0d0) + ((a * a) * (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+288) {
		tmp = -1.0 + ((a * a) * (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+288:
		tmp = -1.0 + ((a * a) * (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+288)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * Float64(a * Float64(a + -4.0))));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+288)
		tmp = -1.0 + ((a * a) * (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+288], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * N[(a * N[(a + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e288

   1. Initial program 79.4%

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

      1. sub-neg 79.4%

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

      2. sqr-pow 79.4%

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

      3. sqr-pow 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-in 79.4%

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

      6. sqr-neg 79.4%

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

      7. distribute-rgt-in 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in a around inf 65.5%

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

      1. unpow3 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. metadata-eval 65.5%

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

      2. pow-prod-up 65.5%

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

      3. pow1 65.5%

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

      4. pow3 65.4%

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

      5. distribute-rgt-in 80.9%

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

      6. +-commutative 80.9%

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

      7. associate-*l* 80.9%

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

      8. *-commutative 80.9%

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

   8. Applied egg-rr 80.9%

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

   if 2e288 < (*.f64 b b)

   1. Initial program 57.3%

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

      1. sub-neg 57.3%

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

      2. sqr-pow 57.3%

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

      3. sqr-pow 57.3%

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

      4. sqr-neg 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. sqr-neg 57.3%

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

      7. distribute-rgt-in 57.3%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in a around 0 46.7%

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

      1. +-commutative 46.7%

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

      2. +-commutative 46.7%

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

      3. associate-+l+ 46.7%

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

      4. unpow2 46.7%

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

      5. unpow2 46.7%

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

      6. associate-*r* 46.7%

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

      7. distribute-rgt-in 69.3%

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

      8. metadata-eval 69.3%

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

      9. distribute-lft-in 69.3%

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

      10. *-commutative 69.3%

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

      11. distribute-lft-in 69.3%

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

      12. metadata-eval 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in b around 0 65.8%

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

      1. unpow2 65.8%

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

      2. +-commutative 65.8%

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

      3. *-commutative 65.8%

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

      4. fma-udef 65.8%

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

   9. Simplified 65.8%

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

       1. fma-udef 65.8%

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

       2. distribute-rgt-in 43.1%

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

       3. *-commutative 43.1%

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

       4. associate-*l* 52.5%

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

   11. Applied egg-rr 52.5%

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

   12. Taylor expanded in a around 0 94.1%

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

       1. unpow2 94.1%

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

       2. *-commutative 94.1%

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

       3. associate-*r* 94.1%

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

   14. Simplified 94.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+288}:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot \left(a \cdot \left(a + -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 6: 84.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+288}:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+288)
   (+ -1.0 (* (+ a -4.0) (* a (* a a))))
   (+ -1.0 (* b (* b 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+288) {
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)));
	} 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+288) then
        tmp = (-1.0d0) + ((a + (-4.0d0)) * (a * (a * a)))
    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+288) {
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)));
	} else {
		tmp = -1.0 + (b * (b * 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+288:
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)))
	else:
		tmp = -1.0 + (b * (b * 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+288)
		tmp = Float64(-1.0 + Float64(Float64(a + -4.0) * Float64(a * Float64(a * a))));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+288)
		tmp = -1.0 + ((a + -4.0) * (a * (a * a)));
	else
		tmp = -1.0 + (b * (b * 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+288], N[(-1.0 + N[(N[(a + -4.0), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e288

   1. Initial program 79.4%

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

      1. sub-neg 79.4%

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

      2. sqr-pow 79.4%

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

      3. sqr-pow 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-in 79.4%

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

      6. sqr-neg 79.4%

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

      7. distribute-rgt-in 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in a around inf 65.5%

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

      1. expm1-log1p-u 64.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-4 \cdot {a}^{3} + {a}^{4}\right)\right)} + -1 \]

      2. expm1-udef 64.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-4 \cdot {a}^{3} + {a}^{4}\right)} - 1\right)} + -1 \]

      3. fma-def 64.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)}\right)} - 1\right) + -1 \]

   6. Applied egg-rr 64.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)\right)} - 1\right)} + -1 \]
   7. Step-by-step derivation

      1. expm1-def 64.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-4, {a}^{3}, {a}^{4}\right)\right)\right)} + -1 \]

      2. expm1-log1p 65.5%

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

      3. fma-udef 65.5%

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

      4. +-commutative 65.5%

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

      5. metadata-eval 65.5%

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

      6. pow-plus 65.5%

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

      7. *-commutative 65.5%

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

      8. distribute-lft-out 80.9%

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

   8. Simplified 80.9%

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

      1. unpow3 65.5%

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

   10. Applied egg-rr 80.9%

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

   if 2e288 < (*.f64 b b)

   1. Initial program 57.3%

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

      1. sub-neg 57.3%

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

      2. sqr-pow 57.3%

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

      3. sqr-pow 57.3%

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

      4. sqr-neg 57.3%

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

      5. distribute-rgt-in 57.3%

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

      6. sqr-neg 57.3%

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

      7. distribute-rgt-in 57.3%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in a around 0 46.7%

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

      1. +-commutative 46.7%

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

      2. +-commutative 46.7%

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

      3. associate-+l+ 46.7%

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

      4. unpow2 46.7%

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

      5. unpow2 46.7%

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

      6. associate-*r* 46.7%

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

      7. distribute-rgt-in 69.3%

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

      8. metadata-eval 69.3%

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

      9. distribute-lft-in 69.3%

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

      10. *-commutative 69.3%

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

      11. distribute-lft-in 69.3%

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

      12. metadata-eval 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in b around 0 65.8%

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

      1. unpow2 65.8%

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

      2. +-commutative 65.8%

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

      3. *-commutative 65.8%

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

      4. fma-udef 65.8%

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

   9. Simplified 65.8%

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

       1. fma-udef 65.8%

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

       2. distribute-rgt-in 43.1%

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

       3. *-commutative 43.1%

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

       4. associate-*l* 52.5%

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

   11. Applied egg-rr 52.5%

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

   12. Taylor expanded in a around 0 94.1%

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

       1. unpow2 94.1%

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

       2. *-commutative 94.1%

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

       3. associate-*r* 94.1%

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

   14. Simplified 94.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+288}:\\ \;\;\;\;-1 + \left(a + -4\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \end{array} \]

Alternative 7: 53.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-62}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 9e-62) (+ -1.0 (* b (* b 12.0))) (+ -1.0 (* b (* b (* a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 9e-62) {
		tmp = -1.0 + (b * (b * 12.0));
	} else {
		tmp = -1.0 + (b * (b * (a * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 9d-62) then
        tmp = (-1.0d0) + (b * (b * 12.0d0))
    else
        tmp = (-1.0d0) + (b * (b * (a * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 9e-62) {
		tmp = -1.0 + (b * (b * 12.0));
	} else {
		tmp = -1.0 + (b * (b * (a * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 9e-62:
		tmp = -1.0 + (b * (b * 12.0))
	else:
		tmp = -1.0 + (b * (b * (a * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 9e-62)
		tmp = Float64(-1.0 + Float64(b * Float64(b * 12.0)));
	else
		tmp = Float64(-1.0 + Float64(b * Float64(b * Float64(a * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 9e-62)
		tmp = -1.0 + (b * (b * 12.0));
	else
		tmp = -1.0 + (b * (b * (a * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 9e-62], N[(-1.0 + N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(b * N[(b * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.00000000000000036e-62

   1. Initial program 82.5%

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

      1. sub-neg 82.5%

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

      2. sqr-pow 82.5%

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

      3. sqr-pow 82.5%

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

      4. sqr-neg 82.5%

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

      5. distribute-rgt-in 82.5%

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

      6. sqr-neg 82.5%

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

      7. distribute-rgt-in 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in a around 0 51.2%

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

      1. +-commutative 51.2%

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

      2. +-commutative 51.2%

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

      3. associate-+l+ 51.2%

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

      4. unpow2 51.2%

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

      5. unpow2 51.2%

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

      6. associate-*r* 51.2%

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

      7. distribute-rgt-in 60.5%

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

      8. metadata-eval 60.5%

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

      9. distribute-lft-in 60.5%

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

      10. *-commutative 60.5%

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

      11. distribute-lft-in 60.5%

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

      12. metadata-eval 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in b around 0 44.3%

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

      1. unpow2 44.3%

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

      2. +-commutative 44.3%

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

      3. *-commutative 44.3%

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

      4. fma-udef 44.3%

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

   9. Simplified 44.3%

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

       1. fma-udef 44.3%

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

       2. distribute-rgt-in 35.1%

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

       3. *-commutative 35.1%

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

       4. associate-*l* 38.9%

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

   11. Applied egg-rr 38.9%

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

   12. Taylor expanded in a around 0 56.8%

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

       1. unpow2 56.8%

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

       2. *-commutative 56.8%

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

       3. associate-*r* 56.8%

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

   14. Simplified 56.8%

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

   if 9.00000000000000036e-62 < a

   1. Initial program 48.5%

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

      1. sub-neg 48.5%

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

      2. sqr-pow 48.5%

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

      3. sqr-pow 48.5%

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

      4. sqr-neg 48.5%

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

      5. distribute-rgt-in 48.5%

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

      6. sqr-neg 48.5%

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

      7. distribute-rgt-in 48.5%

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

   3. Simplified 49.9%

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

   4. Taylor expanded in a around 0 53.5%

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

      1. +-commutative 53.5%

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

      2. +-commutative 53.5%

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

      3. associate-+l+ 53.5%

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

      4. unpow2 53.5%

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

      5. unpow2 53.5%

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

      6. associate-*r* 53.5%

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

      7. distribute-rgt-in 53.5%

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

      8. metadata-eval 53.5%

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

      9. distribute-lft-in 53.5%

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

      10. *-commutative 53.5%

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

      11. distribute-lft-in 53.5%

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

      12. metadata-eval 53.5%

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

   6. Simplified 53.5%

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

   7. Taylor expanded in a around inf 49.7%

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

      1. *-commutative 49.7%

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

      2. unpow2 49.7%

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

      3. *-commutative 49.7%

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

      4. associate-*r* 49.7%

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

   9. Simplified 49.7%

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

   10. Taylor expanded in b around 0 49.7%

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

       1. *-commutative 49.7%

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

       2. unpow2 49.7%

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

       3. *-commutative 49.7%

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

       4. associate-*r* 49.7%

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

       5. associate-*r* 49.7%

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

       6. *-commutative 49.7%

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

   12. Simplified 49.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{-62}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot 12\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + b \cdot \left(b \cdot \left(a \cdot 4\right)\right)\\ \end{array} \]

Alternative 8: 50.9% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

   2. sqr-pow 72.9%

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

   3. sqr-pow 72.9%

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

   4. sqr-neg 72.9%

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

   5. distribute-rgt-in 72.9%

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

   6. sqr-neg 72.9%

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

   7. distribute-rgt-in 72.9%

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

3. Simplified 73.3%

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

4. Taylor expanded in a around 0 51.9%

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

   1. +-commutative 51.9%

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

   2. +-commutative 51.9%

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

   3. associate-+l+ 51.9%

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

   4. unpow2 51.9%

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

   5. unpow2 51.9%

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

   6. associate-*r* 51.9%

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

   7. distribute-rgt-in 58.5%

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

   8. metadata-eval 58.5%

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

   9. distribute-lft-in 58.5%

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

   10. *-commutative 58.5%

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

   11. distribute-lft-in 58.5%

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

   12. metadata-eval 58.5%

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

6. Simplified 58.5%

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

7. Taylor expanded in b around 0 45.8%

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

   1. unpow2 45.8%

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

   2. +-commutative 45.8%

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

   3. *-commutative 45.8%

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

   4. fma-udef 45.8%

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

9. Simplified 45.8%

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

    1. fma-udef 45.8%

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

    2. distribute-rgt-in 39.2%

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

    3. *-commutative 39.2%

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

    4. associate-*l* 41.9%

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

11. Applied egg-rr 41.9%

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

12. Taylor expanded in a around 0 52.4%

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

    1. unpow2 52.4%

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

    2. *-commutative 52.4%

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

    3. associate-*r* 52.4%

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

14. Simplified 52.4%

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

15. Final simplification 52.4%

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

Alternative 9: 25.4% accurate, 128.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

   2. sqr-pow 72.9%

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

   3. sqr-pow 72.9%

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

   4. sqr-neg 72.9%

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

   5. distribute-rgt-in 72.9%

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

   6. sqr-neg 72.9%

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

   7. distribute-rgt-in 72.9%

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

3. Simplified 73.3%

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

4. Taylor expanded in a around inf 68.1%

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

5. Taylor expanded in a around 0 23.9%

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

6. Final simplification 23.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023272 
(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))