Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.5% → 98.4%

Time: 13.6s

Alternatives: 10

Speedup: 11.8×

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

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 100.0%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. fma-def 0.0%

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

      5. distribute-lft-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-lft-in 0.0%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in a around inf 93.5%

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

4. Final simplification 98.1%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY) (+ t_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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = 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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = 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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = 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(a + 1.0)) + Float64(Float64(b * b) * Float64(1.0 - Float64(a * 3.0))))))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = 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) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = 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[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[Power[a, 4.0], $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 1 (*.f64 3 a)))))) < +inf.0

   1. Initial program 99.9%

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

   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 1 (*.f64 3 a))))))

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. sqr-pow 0.0%

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

      3. sqr-pow 0.0%

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

      4. fma-def 0.0%

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

      5. distribute-lft-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-lft-in 0.0%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in a around inf 93.5%

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

4. Final simplification 98.1%

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

Alternative 3: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b + 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.8e+18)
   (pow a 4.0)
   (if (<= a 1.15e+68) (fma (* b b) (+ (* b b) 4.0) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.8e+18) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.15e+68) {
		tmp = fma((b * b), ((b * b) + 4.0), -1.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.8e+18)
		tmp = a ^ 4.0;
	elseif (a <= 1.15e+68)
		tmp = fma(Float64(b * b), Float64(Float64(b * b) + 4.0), -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.8e+18], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.15e+68], N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8e18 or 1.15e68 < a

   1. Initial program 43.2%

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

      1. associate--l+ 43.2%

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

      2. sqr-pow 43.2%

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

      3. sqr-pow 43.2%

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

      4. fma-def 43.2%

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

      5. distribute-lft-in 43.2%

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

      6. sqr-neg 43.2%

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

      7. distribute-lft-in 43.2%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in a around inf 98.4%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -2.8e18 < a < 1.15e68

   1. Initial program 96.2%

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

      1. associate--l+ 96.2%

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

      2. sqr-pow 96.2%

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

      3. sqr-pow 96.2%

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

      4. fma-def 96.2%

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

      5. distribute-lft-in 96.2%

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

      6. sqr-neg 96.2%

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

      7. distribute-lft-in 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in a around 0 94.7%

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

      1. metadata-eval 94.7%

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

      2. pow-sqr 94.6%

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

      3. distribute-rgt-out 94.6%

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

      4. fma-neg 94.6%

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

      5. unpow2 94.6%

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

      6. unpow2 94.6%

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

      7. metadata-eval 94.6%

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

   6. Simplified 94.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b + 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 4: 94.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4e+15) (+ -1.0 (* (pow a 3.0) (+ a 4.0))) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+15) {
		tmp = -1.0 + (pow(a, 3.0) * (a + 4.0));
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 4d+15) then
        tmp = (-1.0d0) + ((a ** 3.0d0) * (a + 4.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4e+15) {
		tmp = -1.0 + (Math.pow(a, 3.0) * (a + 4.0));
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 4e+15:
		tmp = -1.0 + (math.pow(a, 3.0) * (a + 4.0))
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4e+15)
		tmp = Float64(-1.0 + Float64((a ^ 3.0) * Float64(a + 4.0)));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 4e+15)
		tmp = -1.0 + ((a ^ 3.0) * (a + 4.0));
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4e+15], N[(-1.0 + N[(N[Power[a, 3.0], $MachinePrecision] * N[(a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4e15

   1. Initial program 81.9%

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

      1. flip-+ 43.1%

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

   3. Applied egg-rr 43.1%

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

   5. Simplified 43.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{4} - 16 \cdot \left(\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -3, 1\right), \left(a \cdot a\right) \cdot \left(1 + a\right)\right) \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -3, 1\right), \left(a \cdot a\right) \cdot \left(1 + a\right)\right)\right)}{{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + -4 \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -3, 1\right), \left(a \cdot a\right) \cdot \left(1 + a\right)\right)}} - 1 \]

   6. Taylor expanded in b around 0 27.3%

      \[\leadsto \color{blue}{\frac{{a}^{8} - 16 \cdot \left({a}^{4} \cdot {\left(1 + a\right)}^{2}\right)}{-4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}}} - 1 \]
   7. Step-by-step derivation

      1. associate-*r* 27.3%

         \[\leadsto \frac{{a}^{8} - \color{blue}{\left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}}{-4 \cdot \left({a}^{2} \cdot \left(1 + a\right)\right) + {a}^{4}} - 1 \]

      2. fma-def 27.3%

         \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{\color{blue}{\mathsf{fma}\left(-4, {a}^{2} \cdot \left(1 + a\right), {a}^{4}\right)}} - 1 \]

      3. unpow2 27.3%

         \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{\mathsf{fma}\left(-4, \color{blue}{\left(a \cdot a\right)} \cdot \left(1 + a\right), {a}^{4}\right)} - 1 \]

   8. Simplified 27.3%

      \[\leadsto \color{blue}{\frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{\mathsf{fma}\left(-4, \left(a \cdot a\right) \cdot \left(1 + a\right), {a}^{4}\right)}} - 1 \]
   9. Step-by-step derivation

      1. add-exp-log 4.0%

         \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{\color{blue}{e^{\log \left(\mathsf{fma}\left(-4, \left(a \cdot a\right) \cdot \left(1 + a\right), {a}^{4}\right)\right)}}} - 1 \]

      2. distribute-rgt-in 4.0%

         \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{e^{\log \left(\mathsf{fma}\left(-4, \color{blue}{1 \cdot \left(a \cdot a\right) + a \cdot \left(a \cdot a\right)}, {a}^{4}\right)\right)}} - 1 \]

      3. *-un-lft-identity 4.0%

         \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{e^{\log \left(\mathsf{fma}\left(-4, \color{blue}{a \cdot a} + a \cdot \left(a \cdot a\right), {a}^{4}\right)\right)}} - 1 \]

      4. fma-def 4.0%

         \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{e^{\log \left(\mathsf{fma}\left(-4, \color{blue}{\mathsf{fma}\left(a, a, a \cdot \left(a \cdot a\right)\right)}, {a}^{4}\right)\right)}} - 1 \]

      5. cube-unmult 4.0%

         \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{e^{\log \left(\mathsf{fma}\left(-4, \mathsf{fma}\left(a, a, \color{blue}{{a}^{3}}\right), {a}^{4}\right)\right)}} - 1 \]

   10. Applied egg-rr 4.0%

       \[\leadsto \frac{{a}^{8} - \left(16 \cdot {a}^{4}\right) \cdot {\left(1 + a\right)}^{2}}{\color{blue}{e^{\log \left(\mathsf{fma}\left(-4, \mathsf{fma}\left(a, a, {a}^{3}\right), {a}^{4}\right)\right)}}} - 1 \]

   11. Taylor expanded in a around inf 78.7%

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

       1. metadata-eval 78.7%

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

       2. pow-sqr 78.6%

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

       3. unpow2 78.6%

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

       4. associate-*l* 78.6%

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

       5. unpow2 78.6%

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

       6. cube-mult 78.6%

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

       7. distribute-rgt-out 96.6%

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

   13. Simplified 96.6%

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

   if 4e15 < (*.f64 b b)

   1. Initial program 59.7%

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

      1. associate--l+ 59.7%

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

      2. sqr-pow 59.7%

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

      3. sqr-pow 59.7%

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

      4. fma-def 59.7%

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

      5. distribute-lft-in 59.7%

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

      6. sqr-neg 59.7%

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

      7. distribute-lft-in 59.7%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in b around inf 92.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4 \cdot 10^{+15}:\\ \;\;\;\;-1 + {a}^{3} \cdot \left(a + 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.2e+18)
   (pow a 4.0)
   (if (<= a 1.1e+24) (fma 4.0 (* b b) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.2e+18) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.1e+24) {
		tmp = fma(4.0, (b * b), -1.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.2e+18)
		tmp = a ^ 4.0;
	elseif (a <= 1.1e+24)
		tmp = fma(4.0, Float64(b * b), -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.2e+18], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.1e+24], N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.2e18 or 1.10000000000000001e24 < a

   1. Initial program 45.3%

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

      1. associate--l+ 45.3%

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

      2. sqr-pow 45.3%

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

      3. sqr-pow 45.3%

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

      4. fma-def 45.3%

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

      5. distribute-lft-in 45.3%

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

      6. sqr-neg 45.3%

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

      7. distribute-lft-in 45.3%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in a around inf 93.5%

      \[\leadsto \color{blue}{{a}^{4}} \]

   if -3.2e18 < a < 1.10000000000000001e24

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. sqr-pow 99.0%

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

      3. sqr-pow 99.0%

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

      4. fma-def 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. sqr-neg 99.0%

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

      7. distribute-lft-in 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in a around 0 98.2%

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

      1. metadata-eval 98.2%

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

      2. pow-sqr 98.0%

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

      3. distribute-rgt-out 98.0%

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

      4. fma-neg 98.0%

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

      5. unpow2 98.0%

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

      6. unpow2 98.0%

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

      7. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in b around 0 76.9%

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

      1. fma-neg 76.9%

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

      2. unpow2 76.9%

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

      3. metadata-eval 76.9%

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

   9. Simplified 76.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 6: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 20000.0) (+ -1.0 (* (* a a) 4.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000.0) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 20000.0d0) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000.0) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 20000.0:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 20000.0)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 20000.0)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 20000.0], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e4

   1. Initial program 81.4%

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

   2. Taylor expanded in b around 0 80.4%

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

      1. fma-def 80.4%

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

      2. unpow2 80.4%

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

   4. Simplified 80.4%

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

   5. Taylor expanded in a around 0 76.9%

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

      1. unpow2 76.9%

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

   7. Simplified 76.9%

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

   if 2e4 < (*.f64 b b)

   1. Initial program 61.0%

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

      1. associate--l+ 61.0%

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

      2. sqr-pow 61.0%

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

      3. sqr-pow 61.0%

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

      4. fma-def 61.0%

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

      5. distribute-lft-in 61.0%

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

      6. sqr-neg 61.0%

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

      7. distribute-lft-in 61.0%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in b around inf 90.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 7: 68.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-11) -1.0 (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-11) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d-11) then
        tmp = -1.0d0
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-11) {
		tmp = -1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-11:
		tmp = -1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-11)
		tmp = -1.0;
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-11)
		tmp = -1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-11], -1.0, N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.99999999999999988e-11

   1. Initial program 81.4%

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

      1. associate--l+ 81.4%

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

      2. sqr-pow 81.4%

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

      3. sqr-pow 81.4%

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

      4. fma-def 81.4%

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

      5. distribute-lft-in 81.4%

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

      6. sqr-neg 81.4%

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

      7. distribute-lft-in 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in a around 0 48.6%

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

      1. metadata-eval 48.6%

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

      2. pow-sqr 48.6%

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

      3. distribute-rgt-out 48.6%

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

      4. fma-neg 48.6%

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

      5. unpow2 48.6%

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

      6. unpow2 48.6%

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

      7. metadata-eval 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in b around 0 48.3%

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

   if 1.99999999999999988e-11 < (*.f64 b b)

   1. Initial program 61.8%

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

      1. associate--l+ 61.8%

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

      2. sqr-pow 61.8%

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

      3. sqr-pow 61.8%

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

      4. fma-def 61.8%

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

      5. distribute-lft-in 61.8%

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

      6. sqr-neg 61.8%

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

      7. distribute-lft-in 61.8%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in b around inf 86.9%

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

      1. metadata-eval 86.9%

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

      2. pow-prod-up 86.8%

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. pow-prod-down 86.8%

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

      4. pow2 86.8%

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

   6. Applied egg-rr 86.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 8: 82.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 20000.0) (+ -1.0 (* (* a a) 4.0)) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000.0) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 20000.0d0) then
        tmp = (-1.0d0) + ((a * a) * 4.0d0)
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000.0) {
		tmp = -1.0 + ((a * a) * 4.0);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 20000.0:
		tmp = -1.0 + ((a * a) * 4.0)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 20000.0)
		tmp = Float64(-1.0 + Float64(Float64(a * a) * 4.0));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 20000.0)
		tmp = -1.0 + ((a * a) * 4.0);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 20000.0], N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e4

   1. Initial program 81.4%

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

   2. Taylor expanded in b around 0 80.4%

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

      1. fma-def 80.4%

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

      2. unpow2 80.4%

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

   4. Simplified 80.4%

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

   5. Taylor expanded in a around 0 76.9%

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

      1. unpow2 76.9%

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

   7. Simplified 76.9%

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

   if 2e4 < (*.f64 b b)

   1. Initial program 61.0%

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

      1. associate--l+ 61.0%

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

      2. sqr-pow 61.0%

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

      3. sqr-pow 61.0%

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

      4. fma-def 61.0%

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

      5. distribute-lft-in 61.0%

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

      6. sqr-neg 61.0%

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

      7. distribute-lft-in 61.0%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in b around inf 90.2%

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

      1. metadata-eval 90.2%

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

      2. pow-prod-up 90.0%

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} \]

      3. pow-prod-down 90.0%

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

      4. pow2 90.0%

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

   6. Applied egg-rr 90.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20000:\\ \;\;\;\;-1 + \left(a \cdot a\right) \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

Alternative 9: 38.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 2.2e-5) -1.0 (* b (* b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (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 <= 2.2d-5) then
        tmp = -1.0d0
    else
        tmp = b * (b * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-5) {
		tmp = -1.0;
	} else {
		tmp = b * (b * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.2e-5:
		tmp = -1.0
	else:
		tmp = b * (b * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.2e-5)
		tmp = -1.0;
	else
		tmp = Float64(b * Float64(b * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.2e-5)
		tmp = -1.0;
	else
		tmp = b * (b * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.2e-5], -1.0, N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1999999999999999e-5

   1. Initial program 73.9%

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

      1. associate--l+ 73.9%

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

      2. sqr-pow 73.9%

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

      3. sqr-pow 73.9%

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

      4. fma-def 73.9%

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

      5. distribute-lft-in 73.9%

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

      6. sqr-neg 73.9%

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

      7. distribute-lft-in 73.9%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in a around 0 62.3%

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

      1. metadata-eval 62.3%

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

      2. pow-sqr 62.3%

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

      3. distribute-rgt-out 62.3%

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

      4. fma-neg 62.3%

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

      5. unpow2 62.3%

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

      6. unpow2 62.3%

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

      7. metadata-eval 62.3%

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

   6. Simplified 62.3%

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

   7. Taylor expanded in b around 0 31.6%

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

   if 2.1999999999999999e-5 < b

   1. Initial program 64.2%

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

      1. associate--l+ 64.2%

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

      2. sqr-pow 64.2%

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

      3. sqr-pow 64.2%

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

      4. fma-def 64.2%

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

      5. distribute-lft-in 64.2%

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

      6. sqr-neg 64.2%

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

      7. distribute-lft-in 64.2%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in a around 0 87.9%

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

      1. metadata-eval 87.9%

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

      2. pow-sqr 87.8%

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

      3. distribute-rgt-out 87.8%

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

      4. fma-neg 87.8%

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

      5. unpow2 87.8%

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

      6. unpow2 87.8%

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

      7. metadata-eval 87.8%

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

   6. Simplified 87.8%

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

   7. Taylor expanded in b around inf 86.8%

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

      1. +-commutative 86.8%

         \[\leadsto \color{blue}{{b}^{4} + 4 \cdot {b}^{2}} \]

      2. metadata-eval 86.8%

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

      3. pow-sqr 86.7%

         \[\leadsto \color{blue}{{b}^{2} \cdot {b}^{2}} + 4 \cdot {b}^{2} \]

      4. distribute-rgt-in 86.7%

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

      5. unpow2 86.7%

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

      6. fma-udef 86.7%

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

      7. unpow2 86.7%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \mathsf{fma}\left(b, b, 4\right) \]

      8. associate-*l* 86.8%

         \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} \]

   9. Simplified 86.8%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right)} \]

   10. Taylor expanded in b around 0 61.3%

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

       1. unpow2 61.3%

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

       2. *-commutative 61.3%

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

       3. associate-*l* 61.3%

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

       4. *-commutative 61.3%

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

   12. Simplified 61.3%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right)\\ \end{array} \]

Alternative 10: 25.3% accurate, 130.0× 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 71.8%

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

   1. associate--l+ 71.8%

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

   2. sqr-pow 71.8%

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

   3. sqr-pow 71.8%

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

   4. fma-def 71.8%

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

   5. distribute-lft-in 71.8%

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

   6. sqr-neg 71.8%

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

   7. distribute-lft-in 71.8%

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

3. Simplified 74.9%

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

4. Taylor expanded in a around 0 67.9%

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

   1. metadata-eval 67.9%

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

   2. pow-sqr 67.9%

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

   3. distribute-rgt-out 67.9%

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

   4. fma-neg 67.9%

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

   5. unpow2 67.9%

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

   6. unpow2 67.9%

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

   7. metadata-eval 67.9%

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

6. Simplified 67.9%

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

7. Taylor expanded in b around 0 24.9%

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

8. Final simplification 24.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))