Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.2% → 98.3%

Time: 6.2s

Alternatives: 8

Speedup: 1.1×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * Float64(a + 3.0))
	tmp = 0.0
	if (Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + t_0))) <= Inf)
		tmp = 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 + (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[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. 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 2.8%

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

   4. Taylor expanded in a around inf 92.1%

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

4. Final simplification 97.8%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(-1.0 + (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 ^ 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[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. sub-neg 0.0%

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

      2. 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 2.8%

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

   4. Taylor expanded in a around inf 92.1%

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

4. Final simplification 97.7%

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

Alternative 3: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + {a}^{4}\\ \mathbf{if}\;b \cdot b \leq 20:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+44}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + a \cdot 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ -1.0 (pow a 4.0))))
   (if (<= (* b b) 20.0)
     t_0
     (if (<= (* b b) 2e+44)
       (+ -1.0 (+ (pow b 4.0) (* (* b b) (+ 12.0 (* a 4.0)))))
       (if (<= (* b b) 5e+84) t_0 (+ -1.0 (pow b 4.0)))))))

C

double code(double a, double b) {
	double t_0 = -1.0 + pow(a, 4.0);
	double tmp;
	if ((b * b) <= 20.0) {
		tmp = t_0;
	} else if ((b * b) <= 2e+44) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * (12.0 + (a * 4.0))));
	} else if ((b * b) <= 5e+84) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (a ** 4.0d0)
    if ((b * b) <= 20.0d0) then
        tmp = t_0
    else if ((b * b) <= 2d+44) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * (12.0d0 + (a * 4.0d0))))
    else if ((b * b) <= 5d+84) then
        tmp = t_0
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = -1.0 + Math.pow(a, 4.0);
	double tmp;
	if ((b * b) <= 20.0) {
		tmp = t_0;
	} else if ((b * b) <= 2e+44) {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * (12.0 + (a * 4.0))));
	} else if ((b * b) <= 5e+84) {
		tmp = t_0;
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = -1.0 + math.pow(a, 4.0)
	tmp = 0
	if (b * b) <= 20.0:
		tmp = t_0
	elif (b * b) <= 2e+44:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * (12.0 + (a * 4.0))))
	elif (b * b) <= 5e+84:
		tmp = t_0
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(-1.0 + (a ^ 4.0))
	tmp = 0.0
	if (Float64(b * b) <= 20.0)
		tmp = t_0;
	elseif (Float64(b * b) <= 2e+44)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * Float64(12.0 + Float64(a * 4.0)))));
	elseif (Float64(b * b) <= 5e+84)
		tmp = t_0;
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = -1.0 + (a ^ 4.0);
	tmp = 0.0;
	if ((b * b) <= 20.0)
		tmp = t_0;
	elseif ((b * b) <= 2e+44)
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * (12.0 + (a * 4.0))));
	elseif ((b * b) <= 5e+84)
		tmp = t_0;
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 20.0], t$95$0, If[LessEqual[N[(b * b), $MachinePrecision], 2e+44], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(12.0 + N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+84], t$95$0, N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 20 or 2.0000000000000002e44 < (*.f64 b b) < 5.0000000000000001e84

   1. Initial program 80.5%

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

      1. sub-neg 80.5%

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

      2. sqr-pow 80.5%

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

      3. sqr-pow 80.5%

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

      4. sqr-neg 80.5%

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

      5. distribute-rgt-in 80.5%

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

      6. sqr-neg 80.5%

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

      7. distribute-rgt-in 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in a around inf 96.2%

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

   if 20 < (*.f64 b b) < 2.0000000000000002e44

   1. Initial program 78.8%

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

      1. sub-neg 78.8%

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

      2. sqr-pow 78.8%

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

      3. sqr-pow 78.8%

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

      4. sqr-neg 78.8%

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

      5. distribute-rgt-in 78.8%

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

      6. sqr-neg 78.8%

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

      7. distribute-rgt-in 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in a around 0 90.7%

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

      1. +-commutative 90.7%

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

      2. +-commutative 90.7%

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

      3. associate-+l+ 90.7%

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

      4. unpow2 90.7%

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

      5. unpow2 90.7%

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

      6. associate-*r* 90.7%

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

      7. distribute-rgt-in 90.7%

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

      8. metadata-eval 90.7%

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

      9. distribute-lft-in 90.7%

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

      10. *-commutative 90.7%

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

      11. distribute-lft-in 90.7%

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

      12. metadata-eval 90.7%

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

   6. Simplified 90.7%

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

   if 5.0000000000000001e84 < (*.f64 b b)

   1. Initial program 60.7%

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

      1. sub-neg 60.7%

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

      2. sqr-pow 60.7%

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

      3. sqr-pow 60.7%

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

      4. sqr-neg 60.7%

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

      5. distribute-rgt-in 60.7%

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

      6. sqr-neg 60.7%

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

      7. distribute-rgt-in 60.7%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in b around inf 95.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+44}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot \left(12 + a \cdot 4\right)\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 4: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + {a}^{4}\\ \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+44}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ -1.0 (pow a 4.0))))
   (if (<= (* b b) 2e+16)
     t_0
     (if (<= (* b b) 2e+44)
       (+ -1.0 (+ (pow b 4.0) (* (* b b) 12.0)))
       (if (<= (* b b) 5e+84) t_0 (+ -1.0 (pow b 4.0)))))))

C

double code(double a, double b) {
	double t_0 = -1.0 + pow(a, 4.0);
	double tmp;
	if ((b * b) <= 2e+16) {
		tmp = t_0;
	} else if ((b * b) <= 2e+44) {
		tmp = -1.0 + (pow(b, 4.0) + ((b * b) * 12.0));
	} else if ((b * b) <= 5e+84) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) + (a ** 4.0d0)
    if ((b * b) <= 2d+16) then
        tmp = t_0
    else if ((b * b) <= 2d+44) then
        tmp = (-1.0d0) + ((b ** 4.0d0) + ((b * b) * 12.0d0))
    else if ((b * b) <= 5d+84) then
        tmp = t_0
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = -1.0 + Math.pow(a, 4.0);
	double tmp;
	if ((b * b) <= 2e+16) {
		tmp = t_0;
	} else if ((b * b) <= 2e+44) {
		tmp = -1.0 + (Math.pow(b, 4.0) + ((b * b) * 12.0));
	} else if ((b * b) <= 5e+84) {
		tmp = t_0;
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = -1.0 + math.pow(a, 4.0)
	tmp = 0
	if (b * b) <= 2e+16:
		tmp = t_0
	elif (b * b) <= 2e+44:
		tmp = -1.0 + (math.pow(b, 4.0) + ((b * b) * 12.0))
	elif (b * b) <= 5e+84:
		tmp = t_0
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(-1.0 + (a ^ 4.0))
	tmp = 0.0
	if (Float64(b * b) <= 2e+16)
		tmp = t_0;
	elseif (Float64(b * b) <= 2e+44)
		tmp = Float64(-1.0 + Float64((b ^ 4.0) + Float64(Float64(b * b) * 12.0)));
	elseif (Float64(b * b) <= 5e+84)
		tmp = t_0;
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = -1.0 + (a ^ 4.0);
	tmp = 0.0;
	if ((b * b) <= 2e+16)
		tmp = t_0;
	elseif ((b * b) <= 2e+44)
		tmp = -1.0 + ((b ^ 4.0) + ((b * b) * 12.0));
	elseif ((b * b) <= 5e+84)
		tmp = t_0;
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b * b), $MachinePrecision], 2e+16], t$95$0, If[LessEqual[N[(b * b), $MachinePrecision], 2e+44], N[(-1.0 + N[(N[Power[b, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 5e+84], t$95$0, N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 2e16 or 2.0000000000000002e44 < (*.f64 b b) < 5.0000000000000001e84

   1. Initial program 79.5%

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

      1. sub-neg 79.5%

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

      2. sqr-pow 79.5%

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

      3. sqr-pow 79.5%

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

      4. sqr-neg 79.5%

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

      5. distribute-rgt-in 79.5%

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

      6. sqr-neg 79.5%

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

      7. distribute-rgt-in 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in a around inf 95.5%

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

   if 2e16 < (*.f64 b b) < 2.0000000000000002e44

   1. Initial program 98.4%

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

      1. sub-neg 98.4%

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

      2. sqr-pow 98.4%

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

      3. sqr-pow 98.4%

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

      4. sqr-neg 98.4%

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

      5. distribute-rgt-in 98.4%

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

      6. sqr-neg 98.4%

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

      7. distribute-rgt-in 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. unpow2 100.0%

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

      5. unpow2 100.0%

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

      6. associate-*r* 100.0%

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

      7. distribute-rgt-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. distribute-lft-in 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

   if 5.0000000000000001e84 < (*.f64 b b)

   1. Initial program 60.7%

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

      1. sub-neg 60.7%

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

      2. sqr-pow 60.7%

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

      3. sqr-pow 60.7%

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

      4. sqr-neg 60.7%

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

      5. distribute-rgt-in 60.7%

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

      6. sqr-neg 60.7%

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

      7. distribute-rgt-in 60.7%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in b around inf 95.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+16}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{+44}:\\ \;\;\;\;-1 + \left({b}^{4} + \left(b \cdot b\right) \cdot 12\right)\\ \mathbf{elif}\;b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 5: 93.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+16} \lor \neg \left(b \cdot b \leq 2 \cdot 10^{+44}\right) \land b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (* b b) 2e+16) (and (not (<= (* b b) 2e+44)) (<= (* b b) 5e+84)))
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (* (* b b) (+ (* b b) 12.0)))))

C

double code(double a, double b) {
	double tmp;
	if (((b * b) <= 2e+16) || (!((b * b) <= 2e+44) && ((b * b) <= 5e+84))) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((b * b) <= 2d+16) .or. (.not. ((b * b) <= 2d+44)) .and. ((b * b) <= 5d+84)) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + ((b * b) * ((b * b) + 12.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((b * b) <= 2e+16) || (!((b * b) <= 2e+44) && ((b * b) <= 5e+84))) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((b * b) <= 2e+16) or (not ((b * b) <= 2e+44) and ((b * b) <= 5e+84)):
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((Float64(b * b) <= 2e+16) || (!(Float64(b * b) <= 2e+44) && (Float64(b * b) <= 5e+84)))
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * Float64(Float64(b * b) + 12.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((b * b) <= 2e+16) || (~(((b * b) <= 2e+44)) && ((b * b) <= 5e+84)))
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + ((b * b) * ((b * b) + 12.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[(b * b), $MachinePrecision], 2e+16], And[N[Not[LessEqual[N[(b * b), $MachinePrecision], 2e+44]], $MachinePrecision], LessEqual[N[(b * b), $MachinePrecision], 5e+84]]], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e16 or 2.0000000000000002e44 < (*.f64 b b) < 5.0000000000000001e84

   1. Initial program 79.5%

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

      1. sub-neg 79.5%

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

      2. sqr-pow 79.5%

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

      3. sqr-pow 79.5%

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

      4. sqr-neg 79.5%

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

      5. distribute-rgt-in 79.5%

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

      6. sqr-neg 79.5%

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

      7. distribute-rgt-in 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in a around inf 95.5%

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

   if 2e16 < (*.f64 b b) < 2.0000000000000002e44 or 5.0000000000000001e84 < (*.f64 b b)

   1. Initial program 63.0%

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

      1. sub-neg 63.0%

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

      2. sqr-pow 63.0%

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

      3. sqr-pow 63.0%

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

      4. sqr-neg 63.0%

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

      5. distribute-rgt-in 63.0%

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

      6. sqr-neg 63.0%

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

      7. distribute-rgt-in 63.0%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in a around 0 60.7%

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

      1. +-commutative 60.7%

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

      2. +-commutative 60.7%

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

      3. associate-+l+ 60.7%

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

      4. unpow2 60.7%

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

      5. unpow2 60.7%

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

      6. associate-*r* 60.7%

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

      7. distribute-rgt-in 76.5%

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

      8. metadata-eval 76.5%

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

      9. distribute-lft-in 76.5%

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

      10. *-commutative 76.5%

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

      11. distribute-lft-in 76.5%

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

      12. metadata-eval 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in a around 0 95.9%

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

      1. unpow2 95.9%

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

   9. Simplified 95.9%

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

       1. sqr-pow 95.8%

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

       2. metadata-eval 95.8%

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

       3. pow2 95.8%

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

       4. metadata-eval 95.8%

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

       5. pow2 95.8%

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

       6. distribute-rgt-out 95.8%

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

   11. Applied egg-rr 95.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+16} \lor \neg \left(b \cdot b \leq 2 \cdot 10^{+44}\right) \land b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(b \cdot b + 12\right)\\ \end{array} \]

Alternative 6: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+16} \lor \neg \left(b \cdot b \leq 2 \cdot 10^{+44}\right) \land b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (* b b) 2e+16) (and (not (<= (* b b) 2e+44)) (<= (* b b) 5e+84)))
   (+ -1.0 (pow a 4.0))
   (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (((b * b) <= 2e+16) || (!((b * b) <= 2e+44) && ((b * b) <= 5e+84))) {
		tmp = -1.0 + pow(a, 4.0);
	} else {
		tmp = -1.0 + pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((b * b) <= 2d+16) .or. (.not. ((b * b) <= 2d+44)) .and. ((b * b) <= 5d+84)) then
        tmp = (-1.0d0) + (a ** 4.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((b * b) <= 2e+16) || (!((b * b) <= 2e+44) && ((b * b) <= 5e+84))) {
		tmp = -1.0 + Math.pow(a, 4.0);
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((b * b) <= 2e+16) or (not ((b * b) <= 2e+44) and ((b * b) <= 5e+84)):
		tmp = -1.0 + math.pow(a, 4.0)
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((Float64(b * b) <= 2e+16) || (!(Float64(b * b) <= 2e+44) && (Float64(b * b) <= 5e+84)))
		tmp = Float64(-1.0 + (a ^ 4.0));
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((b * b) <= 2e+16) || (~(((b * b) <= 2e+44)) && ((b * b) <= 5e+84)))
		tmp = -1.0 + (a ^ 4.0);
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[(b * b), $MachinePrecision], 2e+16], And[N[Not[LessEqual[N[(b * b), $MachinePrecision], 2e+44]], $MachinePrecision], LessEqual[N[(b * b), $MachinePrecision], 5e+84]]], N[(-1.0 + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e16 or 2.0000000000000002e44 < (*.f64 b b) < 5.0000000000000001e84

   1. Initial program 79.5%

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

      1. sub-neg 79.5%

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

      2. sqr-pow 79.5%

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

      3. sqr-pow 79.5%

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

      4. sqr-neg 79.5%

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

      5. distribute-rgt-in 79.5%

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

      6. sqr-neg 79.5%

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

      7. distribute-rgt-in 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in a around inf 95.5%

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

   if 2e16 < (*.f64 b b) < 2.0000000000000002e44 or 5.0000000000000001e84 < (*.f64 b b)

   1. Initial program 63.0%

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

      1. sub-neg 63.0%

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

      2. sqr-pow 63.0%

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

      3. sqr-pow 63.0%

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

      4. sqr-neg 63.0%

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

      5. distribute-rgt-in 63.0%

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

      6. sqr-neg 63.0%

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

      7. distribute-rgt-in 63.0%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in b around inf 95.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+16} \lor \neg \left(b \cdot b \leq 2 \cdot 10^{+44}\right) \land b \cdot b \leq 5 \cdot 10^{+84}:\\ \;\;\;\;-1 + {a}^{4}\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]

Alternative 7: 70.3% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. sub-neg 72.1%

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

   2. sqr-pow 72.1%

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

   3. sqr-pow 72.1%

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

   4. sqr-neg 72.1%

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

   5. distribute-rgt-in 72.1%

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

   6. sqr-neg 72.1%

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

   7. distribute-rgt-in 72.1%

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

3. Simplified 72.9%

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

4. Taylor expanded in a around 0 52.7%

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

   1. +-commutative 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+l+ 52.7%

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

   4. unpow2 52.7%

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

   5. unpow2 52.7%

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

   6. associate-*r* 52.7%

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

   7. distribute-rgt-in 59.7%

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

   8. metadata-eval 59.7%

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

   9. distribute-lft-in 59.7%

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

   10. *-commutative 59.7%

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

   11. distribute-lft-in 59.7%

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

   12. metadata-eval 59.7%

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

6. Simplified 59.7%

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

7. Taylor expanded in a around 0 67.5%

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

   1. unpow2 67.5%

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

9. Simplified 67.5%

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

    1. sqr-pow 67.5%

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

    2. metadata-eval 67.5%

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

    3. pow2 67.5%

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

    4. metadata-eval 67.5%

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

    5. pow2 67.5%

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

    6. distribute-rgt-out 67.5%

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

11. Applied egg-rr 67.5%

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

12. Final simplification 67.5%

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

Alternative 8: 51.5% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

   1. sub-neg 72.1%

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

   2. sqr-pow 72.1%

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

   3. sqr-pow 72.1%

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

   4. sqr-neg 72.1%

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

   5. distribute-rgt-in 72.1%

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

   6. sqr-neg 72.1%

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

   7. distribute-rgt-in 72.1%

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

3. Simplified 72.9%

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

4. Taylor expanded in a around 0 52.7%

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

   1. +-commutative 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+l+ 52.7%

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

   4. unpow2 52.7%

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

   5. unpow2 52.7%

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

   6. associate-*r* 52.7%

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

   7. distribute-rgt-in 59.7%

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

   8. metadata-eval 59.7%

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

   9. distribute-lft-in 59.7%

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

   10. *-commutative 59.7%

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

   11. distribute-lft-in 59.7%

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

   12. metadata-eval 59.7%

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

6. Simplified 59.7%

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

7. Taylor expanded in a around 0 67.5%

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

   1. unpow2 67.5%

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

9. Simplified 67.5%

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

10. Taylor expanded in b around 0 67.5%

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

    1. unpow2 67.5%

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

    2. metadata-eval 67.5%

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

    3. pow-sqr 67.5%

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

    4. unpow2 67.5%

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

    5. unpow2 67.5%

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

    6. distribute-rgt-in 67.5%

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

    7. associate-*l* 67.5%

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

    8. +-commutative 67.5%

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

    9. fma-def 67.5%

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

12. Simplified 67.5%

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

13. Taylor expanded in b around 0 51.5%

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

    1. unpow2 51.5%

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

15. Simplified 51.5%

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

16. Final simplification 51.5%

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

Reproduce

herbie shell --seed 2023282 
(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))