Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.5% → 100.0%

Time: 7.6s

Alternatives: 11

Speedup: 11.7×

• 59.1% 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: 100.0% accurate, 0.2× speedup

Math

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

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)
   (+ (+ (* 2.0 (pow (* a b) 2.0)) (pow a 4.0)) (+ -1.0 (* (* b b) 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 = ((2.0 * pow((a * b), 2.0)) + pow(a, 4.0)) + (-1.0 + ((b * b) * 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 = Float64(fma(4.0, fma(a, fma(a, a, a), Float64(b * Float64(b * fma(a, -3.0, 1.0)))), (hypot(a, b) ^ 4.0)) + -1.0);
	else
		tmp = Float64(Float64(Float64(2.0 * (Float64(a * b) ^ 2.0)) + (a ^ 4.0)) + Float64(-1.0 + Float64(Float64(b * b) * 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[(N[(4.0 * N[(a * N[(a * a + a), $MachinePrecision] + N[(b * N[(b * N[(a * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(2.0 * N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

   3. Simplified 100.0%

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

   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. fma-def 0.0%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in a around inf 6.1%

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

      1. fma-def 6.1%

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

      2. *-commutative 6.1%

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

      3. unpow2 6.1%

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

      4. unpow2 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in a around 0 87.9%

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

      1. unpow2 87.9%

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

   9. Simplified 87.9%

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

       1. fma-udef 87.9%

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

       2. unswap-sqr 100.0%

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

       3. pow2 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (+ a 1.0)) (* (* b b) (- 1.0 (* a 3.0))))))))
   (if (<= t_0 INFINITY)
     (+ t_0 -1.0)
     (+ (+ (* 2.0 (pow (* a b) 2.0)) (pow a 4.0)) (+ -1.0 (* (* b b) 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 = ((2.0 * pow((a * b), 2.0)) + pow(a, 4.0)) + (-1.0 + ((b * b) * 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 = ((2.0 * Math.pow((a * b), 2.0)) + Math.pow(a, 4.0)) + (-1.0 + ((b * b) * 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 = ((2.0 * math.pow((a * b), 2.0)) + math.pow(a, 4.0)) + (-1.0 + ((b * b) * 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 = Float64(Float64(Float64(2.0 * (Float64(a * b) ^ 2.0)) + (a ^ 4.0)) + Float64(-1.0 + Float64(Float64(b * b) * 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 = ((2.0 * ((a * b) ^ 2.0)) + (a ^ 4.0)) + (-1.0 + ((b * b) * 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[(N[(N[(2.0 * N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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. fma-def 0.0%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in a around inf 6.1%

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

      1. fma-def 6.1%

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

      2. *-commutative 6.1%

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

      3. unpow2 6.1%

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

      4. unpow2 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in a around 0 87.9%

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

      1. unpow2 87.9%

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

   9. Simplified 87.9%

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

       1. fma-udef 87.9%

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

       2. unswap-sqr 100.0%

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

       3. pow2 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 97.0% accurate, 0.5× speedup

Math

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

FPCore

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

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 = (-1.0 + ((b * b) * 4.0)) + fma(2.0, ((a * a) * (b * b)), ((a * a) * (a * a)));
	}
	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 = Float64(Float64(-1.0 + Float64(Float64(b * b) * 4.0)) + fma(2.0, Float64(Float64(a * a) * Float64(b * b)), Float64(Float64(a * a) * Float64(a * a))));
	end
	return 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[(N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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. fma-def 0.0%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in a around inf 6.1%

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

      1. fma-def 6.1%

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

      2. *-commutative 6.1%

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

      3. unpow2 6.1%

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

      4. unpow2 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in a around 0 87.9%

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

      1. unpow2 87.9%

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

   9. Simplified 87.9%

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

       1. metadata-eval 87.1%

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

       2. pow-prod-up 87.1%

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

       3. pow-prod-down 87.1%

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

       4. pow2 87.1%

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

   11. Applied egg-rr 87.9%

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

4. Final simplification 96.7%

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

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot 4\\ \mathbf{if}\;a \leq -88:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 510000:\\ \;\;\;\;t_0 + \left(-1 + {b}^{4}\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\left(-1 + t_0\right) + \left({a}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) 4.0)))
   (if (<= a -88.0)
     (pow a 4.0)
     (if (<= a 510000.0)
       (+ t_0 (+ -1.0 (pow b 4.0)))
       (if (<= a 2e+74)
         (+ (+ -1.0 t_0) (+ (pow a 4.0) (* 2.0 (* (* a a) (* b b)))))
         (* (* a a) (* a a)))))))

C

double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if (a <= -88.0) {
		tmp = pow(a, 4.0);
	} else if (a <= 510000.0) {
		tmp = t_0 + (-1.0 + pow(b, 4.0));
	} else if (a <= 2e+74) {
		tmp = (-1.0 + t_0) + (pow(a, 4.0) + (2.0 * ((a * a) * (b * b))));
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) * 4.0d0
    if (a <= (-88.0d0)) then
        tmp = a ** 4.0d0
    else if (a <= 510000.0d0) then
        tmp = t_0 + ((-1.0d0) + (b ** 4.0d0))
    else if (a <= 2d+74) then
        tmp = ((-1.0d0) + t_0) + ((a ** 4.0d0) + (2.0d0 * ((a * a) * (b * b))))
    else
        tmp = (a * a) * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (b * b) * 4.0;
	double tmp;
	if (a <= -88.0) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 510000.0) {
		tmp = t_0 + (-1.0 + Math.pow(b, 4.0));
	} else if (a <= 2e+74) {
		tmp = (-1.0 + t_0) + (Math.pow(a, 4.0) + (2.0 * ((a * a) * (b * b))));
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (b * b) * 4.0
	tmp = 0
	if a <= -88.0:
		tmp = math.pow(a, 4.0)
	elif a <= 510000.0:
		tmp = t_0 + (-1.0 + math.pow(b, 4.0))
	elif a <= 2e+74:
		tmp = (-1.0 + t_0) + (math.pow(a, 4.0) + (2.0 * ((a * a) * (b * b))))
	else:
		tmp = (a * a) * (a * a)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * 4.0)
	tmp = 0.0
	if (a <= -88.0)
		tmp = a ^ 4.0;
	elseif (a <= 510000.0)
		tmp = Float64(t_0 + Float64(-1.0 + (b ^ 4.0)));
	elseif (a <= 2e+74)
		tmp = Float64(Float64(-1.0 + t_0) + Float64((a ^ 4.0) + Float64(2.0 * Float64(Float64(a * a) * Float64(b * b)))));
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (b * b) * 4.0;
	tmp = 0.0;
	if (a <= -88.0)
		tmp = a ^ 4.0;
	elseif (a <= 510000.0)
		tmp = t_0 + (-1.0 + (b ^ 4.0));
	elseif (a <= 2e+74)
		tmp = (-1.0 + t_0) + ((a ^ 4.0) + (2.0 * ((a * a) * (b * b))));
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[a, -88.0], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 510000.0], N[(t$95$0 + N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2e+74], N[(N[(-1.0 + t$95$0), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -88

   1. Initial program 36.6%

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

      1. associate--l+ 36.6%

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

      2. fma-def 36.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in a around inf 90.2%

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

   if -88 < a < 5.1e5

   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. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-def 99.9%

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

      2. add-cbrt-cube 90.5%

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

      3. pow3 90.5%

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

      4. pow-pow 90.5%

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

      5. add-sqr-sqrt 90.5%

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

      6. hypot-udef 90.5%

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

      7. hypot-udef 90.5%

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

      8. pow2 90.5%

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

      9. metadata-eval 90.5%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 87.7%

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

      1. associate--l+ 87.7%

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

      2. +-commutative 87.7%

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

      3. associate--l+ 87.7%

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

      4. associate-+r+ 87.7%

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

      5. +-commutative 87.7%

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

      6. associate-*r* 87.7%

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

      7. distribute-rgt-out 98.5%

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

      8. unpow2 98.5%

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

      9. *-commutative 98.5%

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

      10. sub-neg 98.5%

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

      11. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in a around 0 98.5%

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

   if 5.1e5 < a < 1.9999999999999999e74

   1. Initial program 58.8%

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

      1. associate--l+ 58.8%

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

      2. fma-def 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in a around inf 54.8%

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

      1. fma-def 54.8%

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

      2. *-commutative 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

   6. Simplified 54.8%

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

   7. Taylor expanded in a around 0 95.7%

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

      1. unpow2 95.7%

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

   9. Simplified 95.7%

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

       1. fma-udef 95.7%

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

       2. unswap-sqr 95.7%

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

       3. pow2 95.7%

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

   11. Applied egg-rr 95.7%

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

       1. unpow-prod-down 95.7%

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

       2. pow2 95.7%

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

       3. pow2 95.7%

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

   13. Applied egg-rr 95.7%

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

   if 1.9999999999999999e74 < a

   1. Initial program 55.3%

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

      1. associate--l+ 55.3%

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

      2. fma-def 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in a around inf 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-prod-up 100.0%

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

      3. pow-prod-down 100.0%

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

      4. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -88:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 510000:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + \left(-1 + {b}^{4}\right)\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+74}:\\ \;\;\;\;\left(-1 + \left(b \cdot b\right) \cdot 4\right) + \left({a}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 5: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -88:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+17}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + \left(-1 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -88.0)
   (pow a 4.0)
   (if (<= a 3.5e+17)
     (+ (* (* b b) 4.0) (+ -1.0 (pow b 4.0)))
     (+ (pow a 4.0) (+ -1.0 (* (+ a 1.0) (* (* a a) 4.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -88.0) {
		tmp = pow(a, 4.0);
	} else if (a <= 3.5e+17) {
		tmp = ((b * b) * 4.0) + (-1.0 + pow(b, 4.0));
	} else {
		tmp = pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-88.0d0)) then
        tmp = a ** 4.0d0
    else if (a <= 3.5d+17) then
        tmp = ((b * b) * 4.0d0) + ((-1.0d0) + (b ** 4.0d0))
    else
        tmp = (a ** 4.0d0) + ((-1.0d0) + ((a + 1.0d0) * ((a * a) * 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -88.0) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 3.5e+17) {
		tmp = ((b * b) * 4.0) + (-1.0 + Math.pow(b, 4.0));
	} else {
		tmp = Math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -88.0:
		tmp = math.pow(a, 4.0)
	elif a <= 3.5e+17:
		tmp = ((b * b) * 4.0) + (-1.0 + math.pow(b, 4.0))
	else:
		tmp = math.pow(a, 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -88.0)
		tmp = a ^ 4.0;
	elseif (a <= 3.5e+17)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + Float64(-1.0 + (b ^ 4.0)));
	else
		tmp = Float64((a ^ 4.0) + Float64(-1.0 + Float64(Float64(a + 1.0) * Float64(Float64(a * a) * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -88.0)
		tmp = a ^ 4.0;
	elseif (a <= 3.5e+17)
		tmp = ((b * b) * 4.0) + (-1.0 + (b ^ 4.0));
	else
		tmp = (a ^ 4.0) + (-1.0 + ((a + 1.0) * ((a * a) * 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -88.0], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 3.5e+17], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] + N[(-1.0 + N[(N[(a + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -88

   1. Initial program 36.6%

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

      1. associate--l+ 36.6%

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

      2. fma-def 36.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in a around inf 90.2%

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

   if -88 < a < 3.5e17

   1. Initial program 97.7%

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

      1. associate--l+ 97.7%

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

      2. fma-def 97.7%

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

   3. Simplified 97.7%

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

      1. fma-def 97.7%

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

      2. add-cbrt-cube 88.5%

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

      3. pow3 88.5%

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

      4. pow-pow 88.5%

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

      5. add-sqr-sqrt 88.5%

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

      6. hypot-udef 88.5%

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

      7. hypot-udef 88.5%

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

      8. pow2 88.5%

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

      9. metadata-eval 88.5%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 85.7%

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

      1. associate--l+ 85.7%

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

      2. +-commutative 85.7%

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

      3. associate--l+ 85.7%

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

      4. associate-+r+ 85.7%

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

      5. +-commutative 85.7%

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

      6. associate-*r* 85.7%

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

      7. distribute-rgt-out 96.3%

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

      8. unpow2 96.3%

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

      9. *-commutative 96.3%

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

      10. sub-neg 96.3%

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

      11. metadata-eval 96.3%

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

   8. Simplified 96.3%

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

   9. Taylor expanded in a around 0 98.5%

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

   if 3.5e17 < a

   1. Initial program 59.0%

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

      1. associate--l+ 59.0%

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

      2. fma-def 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in b around 0 90.0%

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

      1. associate--l+ 90.0%

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

      2. associate-*r* 90.0%

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

      3. unpow2 90.0%

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

   6. Simplified 90.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -88:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+17}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + \left(-1 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} + \left(-1 + \left(a + 1\right) \cdot \left(\left(a \cdot a\right) \cdot 4\right)\right)\\ \end{array} \]

Alternative 6: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -88:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + \left(-1 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -88.0)
   (pow a 4.0)
   (if (<= a 4.5e+17) (+ (* (* b b) 4.0) (+ -1.0 (pow b 4.0))) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -88.0) {
		tmp = pow(a, 4.0);
	} else if (a <= 4.5e+17) {
		tmp = ((b * b) * 4.0) + (-1.0 + pow(b, 4.0));
	} 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 <= (-88.0d0)) then
        tmp = a ** 4.0d0
    else if (a <= 4.5d+17) then
        tmp = ((b * b) * 4.0d0) + ((-1.0d0) + (b ** 4.0d0))
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -88.0) {
		tmp = Math.pow(a, 4.0);
	} else if (a <= 4.5e+17) {
		tmp = ((b * b) * 4.0) + (-1.0 + Math.pow(b, 4.0));
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -88.0:
		tmp = math.pow(a, 4.0)
	elif a <= 4.5e+17:
		tmp = ((b * b) * 4.0) + (-1.0 + math.pow(b, 4.0))
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -88.0)
		tmp = a ^ 4.0;
	elseif (a <= 4.5e+17)
		tmp = Float64(Float64(Float64(b * b) * 4.0) + Float64(-1.0 + (b ^ 4.0)));
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -88.0)
		tmp = a ^ 4.0;
	elseif (a <= 4.5e+17)
		tmp = ((b * b) * 4.0) + (-1.0 + (b ^ 4.0));
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -88.0], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 4.5e+17], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -88 or 4.5e17 < a

   1. Initial program 48.6%

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

      1. associate--l+ 48.6%

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

      2. fma-def 48.6%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in a around inf 90.1%

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

   if -88 < a < 4.5e17

   1. Initial program 97.7%

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

      1. associate--l+ 97.7%

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

      2. fma-def 97.7%

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

   3. Simplified 97.7%

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

      1. fma-def 97.7%

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

      2. add-cbrt-cube 88.5%

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

      3. pow3 88.5%

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

      4. pow-pow 88.5%

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

      5. add-sqr-sqrt 88.5%

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

      6. hypot-udef 88.5%

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

      7. hypot-udef 88.5%

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

      8. pow2 88.5%

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

      9. metadata-eval 88.5%

         \[\leadsto \sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{\color{blue}{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   5. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{6}}} + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + -3 \cdot a\right)\right) - 1\right) \]

   6. Taylor expanded in a around 0 85.7%

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

      1. associate--l+ 85.7%

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

      2. +-commutative 85.7%

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

      3. associate--l+ 85.7%

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

      4. associate-+r+ 85.7%

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

      5. +-commutative 85.7%

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

      6. associate-*r* 85.7%

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

      7. distribute-rgt-out 96.3%

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

      8. unpow2 96.3%

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

      9. *-commutative 96.3%

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

      10. sub-neg 96.3%

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

      11. metadata-eval 96.3%

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

   8. Simplified 96.3%

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

   9. Taylor expanded in a around 0 98.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -88:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 + \left(-1 + {b}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \]

Alternative 7: 93.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+26)
   (+ (* (* a a) (* a a)) (+ -1.0 (* (* a a) 4.0)))
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+26) {
		tmp = ((a * a) * (a * a)) + (-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) <= 2d+26) then
        tmp = ((a * a) * (a * a)) + ((-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) <= 2e+26) {
		tmp = ((a * a) * (a * a)) + (-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) <= 2e+26:
		tmp = ((a * a) * (a * a)) + (-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) <= 2e+26)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) + 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) <= 2e+26)
		tmp = ((a * a) * (a * a)) + (-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], 2e+26], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e26

   1. Initial program 85.6%

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

      1. associate--l+ 85.6%

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

      2. fma-def 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in b around 0 83.1%

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

      1. associate--l+ 83.1%

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

      2. associate-*r* 83.1%

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

      3. unpow2 83.1%

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

   6. Simplified 83.1%

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

      1. metadata-eval 49.2%

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

      2. pow-prod-up 49.1%

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

      3. pow-prod-down 49.1%

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

      4. pow2 49.1%

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

   8. Applied egg-rr 83.0%

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

   9. Taylor expanded in a around 0 95.0%

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

       1. unpow2 95.0%

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

   11. Simplified 95.0%

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

   if 2.0000000000000001e26 < (*.f64 b b)

   1. Initial program 60.2%

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

      1. associate--l+ 60.2%

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

      2. fma-def 60.2%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in b around inf 89.5%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 8: 84.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.3 \cdot 10^{+307}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 3.3e+307)
   (+ (* (* a a) (* a a)) (+ -1.0 (* (* a a) 4.0)))
   (+ -1.0 (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.3e+307) {
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = -1.0 + ((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 * b) <= 3.3d+307) then
        tmp = ((a * a) * (a * a)) + ((-1.0d0) + ((a * a) * 4.0d0))
    else
        tmp = (-1.0d0) + ((b * b) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 3.3e+307) {
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	} else {
		tmp = -1.0 + ((b * b) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 3.3e+307:
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0))
	else:
		tmp = -1.0 + ((b * b) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 3.3e+307)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) + Float64(-1.0 + Float64(Float64(a * a) * 4.0)));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 3.3e+307)
		tmp = ((a * a) * (a * a)) + (-1.0 + ((a * a) * 4.0));
	else
		tmp = -1.0 + ((b * b) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 3.3e+307], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 3.2999999999999999e307

   1. Initial program 81.6%

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

      1. associate--l+ 81.6%

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

      2. fma-def 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in b around 0 67.3%

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

      1. associate--l+ 67.3%

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

      2. associate-*r* 67.3%

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

      3. unpow2 67.3%

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

   6. Simplified 67.3%

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

      1. metadata-eval 47.0%

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

      2. pow-prod-up 46.9%

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

      3. pow-prod-down 46.9%

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

      4. pow2 46.9%

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

   8. Applied egg-rr 67.2%

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

   9. Taylor expanded in a around 0 79.4%

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

       1. unpow2 79.4%

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

   11. Simplified 79.4%

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

   if 3.2999999999999999e307 < (*.f64 b b)

   1. Initial program 49.2%

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

      1. associate--l+ 49.2%

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

      2. fma-def 49.2%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in a around inf 33.9%

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

      1. fma-def 33.9%

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

      2. *-commutative 33.9%

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

      3. unpow2 33.9%

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

      4. unpow2 33.9%

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

   6. Simplified 33.9%

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

   7. Taylor expanded in a around 0 40.7%

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

      1. sub-neg 40.7%

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

      2. metadata-eval 40.7%

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

      3. +-commutative 40.7%

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

      4. +-commutative 40.7%

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

      5. associate-*r* 40.7%

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

      6. distribute-rgt-out 64.4%

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

      7. unpow2 64.4%

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

      8. *-commutative 64.4%

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

   9. Simplified 64.4%

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

   10. Taylor expanded in a around 0 100.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 3.3 \cdot 10^{+307}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) + \left(-1 + \left(a \cdot a\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \end{array} \]

Alternative 9: 67.6% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-14} \lor \neg \left(a \leq 3.3 \cdot 10^{-15}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -4.7e-14) (not (<= a 3.3e-15))) (* (* a a) (* a a)) -1.0))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -4.7e-14) || !(a <= 3.3e-15)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = -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 <= (-4.7d-14)) .or. (.not. (a <= 3.3d-15))) then
        tmp = (a * a) * (a * a)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -4.7e-14) || !(a <= 3.3e-15)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -4.7e-14) or not (a <= 3.3e-15):
		tmp = (a * a) * (a * a)
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -4.7e-14) || !(a <= 3.3e-15))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -4.7e-14) || ~((a <= 3.3e-15)))
		tmp = (a * a) * (a * a);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -4.7e-14], N[Not[LessEqual[a, 3.3e-15]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if a < -4.7000000000000002e-14 or 3.3e-15 < a

   1. Initial program 49.9%

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

      1. associate--l+ 49.9%

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

      2. fma-def 49.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in a around inf 84.5%

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

      1. metadata-eval 84.5%

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

      2. pow-prod-up 84.4%

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

      3. pow-prod-down 84.4%

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

      4. pow2 84.4%

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

   6. Applied egg-rr 84.4%

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

   if -4.7000000000000002e-14 < a < 3.3e-15

   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. associate--l+ 99.9%

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

      2. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 53.8%

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

      1. associate--l+ 53.8%

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

      2. associate-*r* 53.8%

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

      3. unpow2 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in a around 0 53.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-14} \lor \neg \left(a \leq 3.3 \cdot 10^{-15}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 82.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -88 \lor \neg \left(a \leq 3.4 \cdot 10^{+17}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -88.0) (not (<= a 3.4e+17)))
   (* (* a a) (* a a))
   (+ -1.0 (* (* b b) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -88.0) || !(a <= 3.4e+17)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = -1.0 + ((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 ((a <= (-88.0d0)) .or. (.not. (a <= 3.4d+17))) then
        tmp = (a * a) * (a * a)
    else
        tmp = (-1.0d0) + ((b * b) * 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -88.0) || !(a <= 3.4e+17)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = -1.0 + ((b * b) * 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -88.0) or not (a <= 3.4e+17):
		tmp = (a * a) * (a * a)
	else:
		tmp = -1.0 + ((b * b) * 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -88.0) || !(a <= 3.4e+17))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(-1.0 + Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -88.0) || ~((a <= 3.4e+17)))
		tmp = (a * a) * (a * a);
	else
		tmp = -1.0 + ((b * b) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -88.0], N[Not[LessEqual[a, 3.4e+17]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -88 or 3.4e17 < a

   1. Initial program 48.6%

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

      1. associate--l+ 48.6%

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

      2. fma-def 48.6%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in a around inf 90.1%

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

      1. metadata-eval 90.1%

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

      2. pow-prod-up 90.0%

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

      3. pow-prod-down 90.0%

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

      4. pow2 90.0%

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

   6. Applied egg-rr 90.0%

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

   if -88 < a < 3.4e17

   1. Initial program 97.7%

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

      1. associate--l+ 97.7%

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

      2. fma-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in a around inf 63.8%

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

      1. fma-def 63.8%

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

      2. *-commutative 63.8%

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

      3. unpow2 63.8%

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

      4. unpow2 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in a around 0 61.4%

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

      1. sub-neg 61.4%

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

      2. metadata-eval 61.4%

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

      3. +-commutative 61.4%

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

      4. +-commutative 61.4%

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

      5. associate-*r* 61.4%

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

      6. distribute-rgt-out 72.0%

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

      7. unpow2 72.0%

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

      8. *-commutative 72.0%

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

   9. Simplified 72.0%

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

   10. Taylor expanded in a around 0 74.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -88 \lor \neg \left(a \leq 3.4 \cdot 10^{+17}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot 4\\ \end{array} \]

Alternative 11: 26.1% 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 74.1%

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

   1. associate--l+ 74.1%

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

   2. fma-def 74.1%

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

3. Simplified 75.7%

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

4. Taylor expanded in b around 0 56.8%

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

   1. associate--l+ 56.8%

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

   2. associate-*r* 56.8%

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

   3. unpow2 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in a around 0 26.4%

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

8. Final simplification 26.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023234 
(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))