Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.4% → 98.4%

Time: 5.6s

Alternatives: 9

Speedup: 1.2×

• 60.9% 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: 73.4% 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 4.4%

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

   4. Taylor expanded in a around inf 95.8%

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

4. Final simplification 98.9%

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

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 4.4%

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

   4. Taylor expanded in a around inf 95.8%

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

4. Final simplification 98.8%

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

Alternative 3: 93.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+75} \lor \neg \left(a \leq 62\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + b \cdot \left(b \cdot \left(4 + a \cdot -12\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.6e+75) (not (<= a 62.0)))
   (pow a 4.0)
   (+ (+ (pow b 4.0) (* b (* b (+ 4.0 (* a -12.0))))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.6e+75) || !(a <= 62.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (pow(b, 4.0) + (b * (b * (4.0 + (a * -12.0))))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-3.6d+75)) .or. (.not. (a <= 62.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = ((b ** 4.0d0) + (b * (b * (4.0d0 + (a * (-12.0d0)))))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -3.6e+75) || !(a <= 62.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (Math.pow(b, 4.0) + (b * (b * (4.0 + (a * -12.0))))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -3.6e+75) or not (a <= 62.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (math.pow(b, 4.0) + (b * (b * (4.0 + (a * -12.0))))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.6e+75) || !(a <= 62.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64((b ^ 4.0) + Float64(b * Float64(b * Float64(4.0 + Float64(a * -12.0))))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -3.6e+75) || ~((a <= 62.0)))
		tmp = a ^ 4.0;
	else
		tmp = ((b ^ 4.0) + (b * (b * (4.0 + (a * -12.0))))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.6e+75], N[Not[LessEqual[a, 62.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * N[(4.0 + N[(a * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.6e75 or 62 < a

   1. Initial program 36.4%

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

      1. associate--l+ 36.4%

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

      2. sqr-pow 36.4%

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

      3. sqr-pow 36.4%

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

      4. fma-def 36.4%

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

      5. distribute-lft-in 36.4%

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

      6. sqr-neg 36.4%

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

      7. distribute-lft-in 36.4%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in a around inf 96.7%

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

   if -3.6e75 < a < 62

   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. Taylor expanded in a around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+l+ 84.2%

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

      4. associate-*r* 84.2%

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

      5. *-commutative 84.2%

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

      6. metadata-eval 84.2%

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

      7. associate-*l* 84.2%

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

      8. *-commutative 84.2%

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

      9. distribute-rgt-in 97.0%

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

      10. unpow2 97.0%

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

      11. metadata-eval 97.0%

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

      12. distribute-rgt-in 97.0%

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

      13. associate-*l* 97.0%

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

      14. distribute-lft-in 97.0%

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

      15. metadata-eval 97.0%

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

      16. associate-*r* 97.0%

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

      17. metadata-eval 97.0%

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

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\left({b}^{4} + b \cdot \left(b \cdot \left(4 + -12 \cdot a\right)\right)\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+75} \lor \neg \left(a \leq 62\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + b \cdot \left(b \cdot \left(4 + a \cdot -12\right)\right)\right) + -1\\ \end{array} \]

Alternative 4: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{+74} \lor \neg \left(a \leq 4200\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + b \cdot \left(b \cdot 4\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.08e+74) (not (<= a 4200.0)))
   (pow a 4.0)
   (+ (+ (pow b 4.0) (* b (* b 4.0))) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.08e+74) || !(a <= 4200.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (pow(b, 4.0) + (b * (b * 4.0))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.08d+74)) .or. (.not. (a <= 4200.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = ((b ** 4.0d0) + (b * (b * 4.0d0))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.08e+74) || !(a <= 4200.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (Math.pow(b, 4.0) + (b * (b * 4.0))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.08e+74) or not (a <= 4200.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (math.pow(b, 4.0) + (b * (b * 4.0))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.08e+74) || !(a <= 4200.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64((b ^ 4.0) + Float64(b * Float64(b * 4.0))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.08e+74) || ~((a <= 4200.0)))
		tmp = a ^ 4.0;
	else
		tmp = ((b ^ 4.0) + (b * (b * 4.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.08e+74], N[Not[LessEqual[a, 4200.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.08e74 or 4200 < a

   1. Initial program 36.4%

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

      1. associate--l+ 36.4%

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

      2. sqr-pow 36.4%

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

      3. sqr-pow 36.4%

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

      4. fma-def 36.4%

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

      5. distribute-lft-in 36.4%

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

      6. sqr-neg 36.4%

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

      7. distribute-lft-in 36.4%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in a around inf 96.7%

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

   if -1.08e74 < a < 4200

   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. Taylor expanded in a around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+l+ 84.2%

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

      4. associate-*r* 84.2%

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

      5. *-commutative 84.2%

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

      6. metadata-eval 84.2%

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

      7. associate-*l* 84.2%

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

      8. *-commutative 84.2%

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

      9. distribute-rgt-in 97.0%

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

      10. unpow2 97.0%

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

      11. metadata-eval 97.0%

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

      12. distribute-rgt-in 97.0%

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

      13. associate-*l* 97.0%

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

      14. distribute-lft-in 97.0%

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

      15. metadata-eval 97.0%

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

      16. associate-*r* 97.0%

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

      17. metadata-eval 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in a around 0 97.0%

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

      1. *-commutative 97.0%

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

   7. Simplified 97.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{+74} \lor \neg \left(a \leq 4200\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({b}^{4} + b \cdot \left(b \cdot 4\right)\right) + -1\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+74} \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(-12, a, 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.4e+74) (not (<= a 0.41)))
   (pow a 4.0)
   (+ (* (* b b) (fma -12.0 a 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.4e+74) || !(a <= 0.41)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = ((b * b) * fma(-12.0, a, 4.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.4e+74) || !(a <= 0.41))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(Float64(b * b) * fma(-12.0, a, 4.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.4e+74], N[Not[LessEqual[a, 0.41]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3999999999999999e74 or 0.409999999999999976 < a

   1. Initial program 36.4%

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

      1. associate--l+ 36.4%

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

      2. sqr-pow 36.4%

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

      3. sqr-pow 36.4%

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

      4. fma-def 36.4%

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

      5. distribute-lft-in 36.4%

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

      6. sqr-neg 36.4%

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

      7. distribute-lft-in 36.4%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in a around inf 96.7%

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

   if -3.3999999999999999e74 < a < 0.409999999999999976

   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. Taylor expanded in a around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-+l+ 84.2%

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

      4. associate-*r* 84.2%

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

      5. *-commutative 84.2%

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

      6. metadata-eval 84.2%

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

      7. associate-*l* 84.2%

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

      8. *-commutative 84.2%

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

      9. distribute-rgt-in 97.0%

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

      10. unpow2 97.0%

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

      11. metadata-eval 97.0%

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

      12. distribute-rgt-in 97.0%

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

      13. associate-*l* 97.0%

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

      14. distribute-lft-in 97.0%

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

      15. metadata-eval 97.0%

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

      16. associate-*r* 97.0%

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

      17. metadata-eval 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in b around 0 80.1%

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

      1. unpow2 80.1%

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

      2. +-commutative 80.1%

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

      3. fma-udef 80.1%

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

      4. *-commutative 80.1%

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

   7. Simplified 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-12, a, 4\right) \cdot \left(b \cdot b\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+74} \lor \neg \left(a \leq 0.41\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(-12, a, 4\right) + -1\\ \end{array} \]

Alternative 6: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+77}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{+34}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{+26} \lor \neg \left(a \leq 1550\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.05e+77)
   (pow a 4.0)
   (if (<= a -4.8e+34)
     (pow b 4.0)
     (if (or (<= a -3.5e+26) (not (<= a 1550.0)))
       (pow a 4.0)
       (+ (* b (* b 4.0)) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.05e+77) {
		tmp = pow(a, 4.0);
	} else if (a <= -4.8e+34) {
		tmp = pow(b, 4.0);
	} else if ((a <= -3.5e+26) || !(a <= 1550.0)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.05d+77)) then
        tmp = a ** 4.0d0
    else if (a <= (-4.8d+34)) then
        tmp = b ** 4.0d0
    else if ((a <= (-3.5d+26)) .or. (.not. (a <= 1550.0d0))) then
        tmp = a ** 4.0d0
    else
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.05e+77) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= -4.8e+34) {
		tmp = Math.pow(b, 4.0);
	} else if ((a <= -3.5e+26) || !(a <= 1550.0)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.05e+77:
		tmp = math.pow(a, 4.0)
	elif a <= -4.8e+34:
		tmp = math.pow(b, 4.0)
	elif (a <= -3.5e+26) or not (a <= 1550.0):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (b * (b * 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.05e+77)
		tmp = a ^ 4.0;
	elseif (a <= -4.8e+34)
		tmp = b ^ 4.0;
	elseif ((a <= -3.5e+26) || !(a <= 1550.0))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.05e+77)
		tmp = a ^ 4.0;
	elseif (a <= -4.8e+34)
		tmp = b ^ 4.0;
	elseif ((a <= -3.5e+26) || ~((a <= 1550.0)))
		tmp = a ^ 4.0;
	else
		tmp = (b * (b * 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.05e+77], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, -4.8e+34], N[Power[b, 4.0], $MachinePrecision], If[Or[LessEqual[a, -3.5e+26], N[Not[LessEqual[a, 1550.0]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.0499999999999999e77 or -4.79999999999999974e34 < a < -3.4999999999999999e26 or 1550 < a

   1. Initial program 37.6%

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

      1. associate--l+ 37.6%

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

      2. sqr-pow 37.6%

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

      3. sqr-pow 37.6%

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

      4. fma-def 37.6%

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

      5. distribute-lft-in 37.6%

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

      6. sqr-neg 37.6%

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

      7. distribute-lft-in 37.6%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in a around inf 96.8%

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

   if -1.0499999999999999e77 < a < -4.79999999999999974e34

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sqr-pow 99.7%

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

      3. sqr-pow 99.7%

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

      4. fma-def 99.7%

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

      5. distribute-lft-in 99.7%

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

      6. sqr-neg 99.7%

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

      7. distribute-lft-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in b around inf 78.5%

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

   if -3.4999999999999999e26 < a < 1550

   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. Taylor expanded in a around 0 85.8%

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

      1. +-commutative 85.8%

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

      2. +-commutative 85.8%

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

      3. associate-+l+ 85.8%

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

      4. associate-*r* 85.8%

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

      5. *-commutative 85.8%

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

      6. metadata-eval 85.8%

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

      7. associate-*l* 85.8%

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

      8. *-commutative 85.8%

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

      9. distribute-rgt-in 99.5%

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

      10. unpow2 99.5%

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

      11. metadata-eval 99.5%

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

      12. distribute-rgt-in 99.5%

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

      13. associate-*l* 99.5%

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

      14. distribute-lft-in 99.5%

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

      15. metadata-eval 99.5%

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

      16. associate-*r* 99.5%

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

      17. metadata-eval 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in b around 0 82.0%

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

      1. unpow2 82.0%

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

      2. +-commutative 82.0%

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

      3. fma-udef 82.0%

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

      4. *-commutative 82.0%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in a around 0 82.0%

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

      1. unpow2 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

   10. Simplified 82.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+77}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{+34}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{+26} \lor \neg \left(a \leq 1550\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \]

Alternative 7: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+26}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3400:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.35e+26)
   (pow a 4.0)
   (if (<= a 3400.0) (+ (* b (* b 4.0)) -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.35e+26) {
		tmp = pow(a, 4.0);
	} else if (a <= 3400.0) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.35d+26)) then
        tmp = a ** 4.0d0
    else if (a <= 3400.0d0) then
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.35e+26) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 3400.0) {
		tmp = (b * (b * 4.0)) + -1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.35e+26:
		tmp = math.pow(a, 4.0)
	elif a <= 3400.0:
		tmp = (b * (b * 4.0)) + -1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.35e+26)
		tmp = a ^ 4.0;
	elseif (a <= 3400.0)
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.35e+26)
		tmp = a ^ 4.0;
	elseif (a <= 3400.0)
		tmp = (b * (b * 4.0)) + -1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.35e+26], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 3400.0], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35e26 or 3400 < a

   1. Initial program 42.3%

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

      1. associate--l+ 42.3%

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

      2. sqr-pow 42.3%

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

      3. sqr-pow 42.3%

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

      4. fma-def 42.3%

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

      5. distribute-lft-in 42.3%

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

      6. sqr-neg 42.3%

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

      7. distribute-lft-in 42.3%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in a around inf 91.5%

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

   if -1.35e26 < a < 3400

   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. Taylor expanded in a around 0 85.8%

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

      1. +-commutative 85.8%

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

      2. +-commutative 85.8%

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

      3. associate-+l+ 85.8%

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

      4. associate-*r* 85.8%

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

      5. *-commutative 85.8%

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

      6. metadata-eval 85.8%

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

      7. associate-*l* 85.8%

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

      8. *-commutative 85.8%

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

      9. distribute-rgt-in 99.5%

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

      10. unpow2 99.5%

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

      11. metadata-eval 99.5%

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

      12. distribute-rgt-in 99.5%

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

      13. associate-*l* 99.5%

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

      14. distribute-lft-in 99.5%

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

      15. metadata-eval 99.5%

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

      16. associate-*r* 99.5%

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

      17. metadata-eval 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in b around 0 82.0%

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

      1. unpow2 82.0%

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

      2. +-commutative 82.0%

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

      3. fma-udef 82.0%

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

      4. *-commutative 82.0%

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

   7. Simplified 82.0%

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

   8. Taylor expanded in a around 0 82.0%

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

      1. unpow2 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 82.0%

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

   10. Simplified 82.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+26}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3400:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 8: 53.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+26}:\\ \;\;\;\;-12 \cdot \left(a \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e+26) (+ (* -12.0 (* a (* b b))) -1.0) (+ (* b (* b 4.0)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e+26) {
		tmp = (-12.0 * (a * (b * b))) + -1.0;
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-9d+26)) then
        tmp = ((-12.0d0) * (a * (b * b))) + (-1.0d0)
    else
        tmp = (b * (b * 4.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9e+26) {
		tmp = (-12.0 * (a * (b * b))) + -1.0;
	} else {
		tmp = (b * (b * 4.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e+26:
		tmp = (-12.0 * (a * (b * b))) + -1.0
	else:
		tmp = (b * (b * 4.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e+26)
		tmp = Float64(Float64(-12.0 * Float64(a * Float64(b * b))) + -1.0);
	else
		tmp = Float64(Float64(b * Float64(b * 4.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e+26)
		tmp = (-12.0 * (a * (b * b))) + -1.0;
	else
		tmp = (b * (b * 4.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e+26], N[(N[(-12.0 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.99999999999999957e26

   1. Initial program 28.3%

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

   2. Taylor expanded in a around 0 38.2%

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

      1. +-commutative 38.2%

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

      2. +-commutative 38.2%

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

      3. associate-+l+ 38.2%

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

      4. associate-*r* 38.2%

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

      5. *-commutative 38.2%

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

      6. metadata-eval 38.2%

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

      7. associate-*l* 38.2%

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

      8. *-commutative 38.2%

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

      9. distribute-rgt-in 38.2%

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

      10. unpow2 38.2%

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

      11. metadata-eval 38.2%

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

      12. distribute-rgt-in 38.2%

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

      13. associate-*l* 38.2%

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

      14. distribute-lft-in 38.2%

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

      15. metadata-eval 38.2%

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

      16. associate-*r* 38.2%

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

      17. metadata-eval 38.2%

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

   4. Simplified 38.2%

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

   5. Taylor expanded in a around inf 36.7%

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

      1. unpow2 36.7%

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

      2. associate-*r* 36.7%

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

      3. *-commutative 36.7%

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

      4. associate-*r* 36.7%

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

      5. *-commutative 36.7%

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

      6. associate-*r* 36.7%

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

      7. *-commutative 36.7%

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

   7. Simplified 36.7%

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

   8. Taylor expanded in b around 0 36.7%

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

      1. unpow2 36.7%

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

   10. Simplified 36.7%

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

   if -8.99999999999999957e26 < a

   1. Initial program 87.2%

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

   2. Taylor expanded in a around 0 60.5%

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

      1. +-commutative 60.5%

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

      2. +-commutative 60.5%

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

      3. associate-+l+ 60.5%

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

      4. associate-*r* 60.5%

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

      5. *-commutative 60.5%

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

      6. metadata-eval 60.5%

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

      7. associate-*l* 60.5%

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

      8. *-commutative 60.5%

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

      9. distribute-rgt-in 70.2%

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

      10. unpow2 70.2%

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

      11. metadata-eval 70.2%

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

      12. distribute-rgt-in 70.2%

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

      13. associate-*l* 70.2%

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

      14. distribute-lft-in 70.2%

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

      15. metadata-eval 70.2%

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

      16. associate-*r* 70.2%

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

      17. metadata-eval 70.2%

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

   4. Simplified 70.2%

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

   5. Taylor expanded in b around 0 57.8%

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

      1. unpow2 57.8%

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

      2. +-commutative 57.8%

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

      3. fma-udef 57.8%

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

      4. *-commutative 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in a around 0 66.3%

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

      1. unpow2 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-*l* 66.3%

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

   10. Simplified 66.3%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+26}:\\ \;\;\;\;-12 \cdot \left(a \cdot \left(b \cdot b\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 4\right) + -1\\ \end{array} \]

Alternative 9: 50.8% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in a around 0 55.3%

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

   1. +-commutative 55.3%

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

   2. +-commutative 55.3%

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

   3. associate-+l+ 55.3%

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

   4. associate-*r* 55.3%

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

   5. *-commutative 55.3%

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

   6. metadata-eval 55.3%

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

   7. associate-*l* 55.3%

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

   8. *-commutative 55.3%

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

   9. distribute-rgt-in 62.7%

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

   10. unpow2 62.7%

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

   11. metadata-eval 62.7%

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

   12. distribute-rgt-in 62.7%

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

   13. associate-*l* 62.7%

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

   14. distribute-lft-in 62.7%

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

   15. metadata-eval 62.7%

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

   16. associate-*r* 62.7%

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

   17. metadata-eval 62.7%

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

4. Simplified 62.7%

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

5. Taylor expanded in b around 0 52.9%

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

   1. unpow2 52.9%

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

   2. +-commutative 52.9%

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

   3. fma-udef 52.9%

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

   4. *-commutative 52.9%

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

7. Simplified 52.9%

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

8. Taylor expanded in a around 0 55.9%

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

   1. unpow2 55.9%

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

   2. *-commutative 55.9%

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

   3. associate-*l* 55.9%

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

10. Simplified 55.9%

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

11. Final simplification 55.9%

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

Reproduce

herbie shell --seed 2023290 
(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))