Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.4% → 100.0%

Time: 10.2s

Alternatives: 10

Speedup: 1.2×

• 60.8% 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: 74.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot \left({b}^{2} + {a}^{3}\right) + \left({a}^{2} \cdot \left(4 + 2 \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \end{array} \]

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))))))
      5e+305)
   (+
    (+
     (* -12.0 (* a (pow b 2.0)))
     (+
      (* 4.0 (+ (pow b 2.0) (pow a 3.0)))
      (+
       (* (pow a 2.0) (+ 4.0 (* 2.0 (pow b 2.0))))
       (+ (pow a 4.0) (pow b 4.0)))))
    -1.0)
   (+ (pow (cbrt (pow (hypot a b) 4.0)) 3.0) (+ (* 4.0 (pow b 2.0)) -1.0))))

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)))))) <= 5e+305) {
		tmp = ((-12.0 * (a * pow(b, 2.0))) + ((4.0 * (pow(b, 2.0) + pow(a, 3.0))) + ((pow(a, 2.0) * (4.0 + (2.0 * pow(b, 2.0)))) + (pow(a, 4.0) + pow(b, 4.0))))) + -1.0;
	} else {
		tmp = pow(cbrt(pow(hypot(a, b), 4.0)), 3.0) + ((4.0 * pow(b, 2.0)) + -1.0);
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (a + 1.0)) + ((b * b) * (1.0 - (a * 3.0)))))) <= 5e+305) {
		tmp = ((-12.0 * (a * Math.pow(b, 2.0))) + ((4.0 * (Math.pow(b, 2.0) + Math.pow(a, 3.0))) + ((Math.pow(a, 2.0) * (4.0 + (2.0 * Math.pow(b, 2.0)))) + (Math.pow(a, 4.0) + Math.pow(b, 4.0))))) + -1.0;
	} else {
		tmp = Math.pow(Math.cbrt(Math.pow(Math.hypot(a, b), 4.0)), 3.0) + ((4.0 * Math.pow(b, 2.0)) + -1.0);
	}
	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)))))) <= 5e+305)
		tmp = Float64(Float64(Float64(-12.0 * Float64(a * (b ^ 2.0))) + Float64(Float64(4.0 * Float64((b ^ 2.0) + (a ^ 3.0))) + Float64(Float64((a ^ 2.0) * Float64(4.0 + Float64(2.0 * (b ^ 2.0)))) + Float64((a ^ 4.0) + (b ^ 4.0))))) + -1.0);
	else
		tmp = Float64((cbrt((hypot(a, b) ^ 4.0)) ^ 3.0) + Float64(Float64(4.0 * (b ^ 2.0)) + -1.0));
	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], 5e+305], N[(N[(N[(-12.0 * N[(a * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(4.0 + N[(2.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[N[Power[N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. sub-neg 99.6%

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

      5. associate-+l+ 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. cube-mult 99.6%

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

      4. distribute-rgt-in 99.6%

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

      5. add-sqr-sqrt 86.7%

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

      6. pow2 86.7%

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

      7. distribute-rgt-in 86.7%

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

      8. *-un-lft-identity 86.7%

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

      9. cube-mult 86.7%

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

      10. sqr-pow 51.6%

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

      11. hypot-def 51.6%

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

      12. metadata-eval 51.6%

          \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -3, 1\right), {\left(\mathsf{hypot}\left(a, {a}^{\color{blue}{1.5}}\right)\right)}^{2}\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + -1\right) \]

   5. Applied egg-rr 51.6%

      \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -3, 1\right), \color{blue}{{\left(\mathsf{hypot}\left(a, {a}^{1.5}\right)\right)}^{2}}\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + -1\right) \]

   6. Taylor expanded in a around 0 99.9%

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

   if 5.00000000000000009e305 < (+.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 59.5%

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

      1. associate--l+ 59.5%

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

      2. fma-def 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. sqr-neg 59.5%

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

      5. distribute-rgt-in 59.5%

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

   3. Simplified 61.5%

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

      1. fma-def 61.5%

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

      2. unpow2 61.5%

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

      3. add-sqr-sqrt 61.5%

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

      4. associate-*l* 61.5%

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

      5. hypot-def 61.5%

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

      6. hypot-def 61.5%

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

      7. add-sqr-sqrt 61.5%

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

      8. pow2 61.5%

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

      9. hypot-def 61.5%

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

   5. Applied egg-rr 61.5%

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

      1. add-cube-cbrt 61.5%

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

      2. pow3 61.5%

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

      3. associate-*r* 61.5%

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

      4. unpow2 61.5%

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

      5. pow-prod-up 61.5%

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

      6. metadata-eval 61.5%

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in a around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(-12 \cdot \left(a \cdot {b}^{2}\right) + \left(4 \cdot \left({b}^{2} + {a}^{3}\right) + \left({a}^{2} \cdot \left(4 + 2 \cdot {b}^{2}\right) + \left({a}^{4} + {b}^{4}\right)\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left({b}^{4} + \mathsf{fma}\left(2, {\left(a \cdot b\right)}^{2}, {a}^{4}\right)\right) + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \end{array} \]

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))))))
      5e+305)
   (+
    (+ (pow b 4.0) (fma 2.0 (pow (* a b) 2.0) (pow a 4.0)))
    (+ (* 4.0 (fma (* a a) (+ a 1.0) (* (* b b) (+ 1.0 (* a -3.0))))) -1.0))
   (+ (pow (cbrt (pow (hypot a b) 4.0)) 3.0) (+ (* 4.0 (pow b 2.0)) -1.0))))

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)))))) <= 5e+305) {
		tmp = (pow(b, 4.0) + fma(2.0, pow((a * b), 2.0), pow(a, 4.0))) + ((4.0 * fma((a * a), (a + 1.0), ((b * b) * (1.0 + (a * -3.0))))) + -1.0);
	} else {
		tmp = pow(cbrt(pow(hypot(a, b), 4.0)), 3.0) + ((4.0 * pow(b, 2.0)) + -1.0);
	}
	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)))))) <= 5e+305)
		tmp = Float64(Float64((b ^ 4.0) + fma(2.0, (Float64(a * b) ^ 2.0), (a ^ 4.0))) + Float64(Float64(4.0 * fma(Float64(a * a), Float64(a + 1.0), Float64(Float64(b * b) * Float64(1.0 + Float64(a * -3.0))))) + -1.0));
	else
		tmp = Float64((cbrt((hypot(a, b) ^ 4.0)) ^ 3.0) + Float64(Float64(4.0 * (b ^ 2.0)) + -1.0));
	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], 5e+305], N[(N[(N[Power[b, 4.0], $MachinePrecision] + N[(2.0 * N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. fma-def 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. sqr-neg 99.6%

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

      5. distribute-rgt-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around 0 99.9%

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

      1. associate-+r+ 99.9%

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

      2. +-commutative 99.9%

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

      3. fma-def 99.9%

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

      4. unpow2 99.9%

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

      5. unpow2 99.9%

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

      6. swap-sqr 99.9%

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

      7. unpow2 99.9%

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

      8. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   if 5.00000000000000009e305 < (+.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 59.5%

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

      1. associate--l+ 59.5%

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

      2. fma-def 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. sqr-neg 59.5%

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

      5. distribute-rgt-in 59.5%

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

   3. Simplified 61.5%

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

      1. fma-def 61.5%

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

      2. unpow2 61.5%

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

      3. add-sqr-sqrt 61.5%

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

      4. associate-*l* 61.5%

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

      5. hypot-def 61.5%

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

      6. hypot-def 61.5%

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

      7. add-sqr-sqrt 61.5%

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

      8. pow2 61.5%

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

      9. hypot-def 61.5%

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

   5. Applied egg-rr 61.5%

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

      1. add-cube-cbrt 61.5%

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

      2. pow3 61.5%

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

      3. associate-*r* 61.5%

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

      4. unpow2 61.5%

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

      5. pow-prod-up 61.5%

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

      6. metadata-eval 61.5%

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in a around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left({b}^{4} + \mathsf{fma}\left(2, {\left(a \cdot b\right)}^{2}, {a}^{4}\right)\right) + \left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + -1\right) + \mathsf{hypot}\left(a, b\right) \cdot \left(\mathsf{hypot}\left(a, b\right) \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \end{array} \]

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))))))
      5e+305)
   (+
    (+ (* 4.0 (fma (* a a) (+ a 1.0) (* (* b b) (+ 1.0 (* a -3.0))))) -1.0)
    (* (hypot a b) (* (hypot a b) (pow (hypot a b) 2.0))))
   (+ (pow (cbrt (pow (hypot a b) 4.0)) 3.0) (+ (* 4.0 (pow b 2.0)) -1.0))))

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)))))) <= 5e+305) {
		tmp = ((4.0 * fma((a * a), (a + 1.0), ((b * b) * (1.0 + (a * -3.0))))) + -1.0) + (hypot(a, b) * (hypot(a, b) * pow(hypot(a, b), 2.0)));
	} else {
		tmp = pow(cbrt(pow(hypot(a, b), 4.0)), 3.0) + ((4.0 * pow(b, 2.0)) + -1.0);
	}
	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)))))) <= 5e+305)
		tmp = Float64(Float64(Float64(4.0 * fma(Float64(a * a), Float64(a + 1.0), Float64(Float64(b * b) * Float64(1.0 + Float64(a * -3.0))))) + -1.0) + Float64(hypot(a, b) * Float64(hypot(a, b) * (hypot(a, b) ^ 2.0))));
	else
		tmp = Float64((cbrt((hypot(a, b) ^ 4.0)) ^ 3.0) + Float64(Float64(4.0 * (b ^ 2.0)) + -1.0));
	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], 5e+305], N[(N[(N[(4.0 * N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 + N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[(N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision] * N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. fma-def 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. sqr-neg 99.6%

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

      5. distribute-rgt-in 99.6%

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

   3. Simplified 99.6%

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

      1. fma-def 99.6%

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

      2. unpow2 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. associate-*l* 99.7%

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

      5. hypot-def 99.7%

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

      6. hypot-def 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. pow2 99.7%

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

      9. hypot-def 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 5.00000000000000009e305 < (+.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 59.5%

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

      1. associate--l+ 59.5%

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

      2. fma-def 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. sqr-neg 59.5%

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

      5. distribute-rgt-in 59.5%

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

   3. Simplified 61.5%

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

      1. fma-def 61.5%

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

      2. unpow2 61.5%

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

      3. add-sqr-sqrt 61.5%

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

      4. associate-*l* 61.5%

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

      5. hypot-def 61.5%

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

      6. hypot-def 61.5%

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

      7. add-sqr-sqrt 61.5%

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

      8. pow2 61.5%

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

      9. hypot-def 61.5%

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

   5. Applied egg-rr 61.5%

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

      1. add-cube-cbrt 61.5%

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

      2. pow3 61.5%

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

      3. associate-*r* 61.5%

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

      4. unpow2 61.5%

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

      5. pow-prod-up 61.5%

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

      6. metadata-eval 61.5%

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in a around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left(4 \cdot \mathsf{fma}\left(a \cdot a, a + 1, \left(b \cdot b\right) \cdot \left(1 + a \cdot -3\right)\right) + -1\right) + \mathsf{hypot}\left(a, b\right) \cdot \left(\mathsf{hypot}\left(a, b\right) \cdot {\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \end{array} \]

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 5e+305)
     (+ t_0 -1.0)
     (+ (pow (cbrt (pow (hypot a b) 4.0)) 3.0) (+ (* 4.0 (pow b 2.0)) -1.0)))))

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 <= 5e+305) {
		tmp = t_0 + -1.0;
	} else {
		tmp = pow(cbrt(pow(hypot(a, b), 4.0)), 3.0) + ((4.0 * pow(b, 2.0)) + -1.0);
	}
	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 <= 5e+305) {
		tmp = t_0 + -1.0;
	} else {
		tmp = Math.pow(Math.cbrt(Math.pow(Math.hypot(a, b), 4.0)), 3.0) + ((4.0 * Math.pow(b, 2.0)) + -1.0);
	}
	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 <= 5e+305)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64((cbrt((hypot(a, b) ^ 4.0)) ^ 3.0) + Float64(Float64(4.0 * (b ^ 2.0)) + -1.0));
	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, 5e+305], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[Power[N[Power[N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(4.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) 2) (*.f64 4 (+.f64 (*.f64 (*.f64 a a) (+.f64 1 a)) (*.f64 (*.f64 b b) (-.f64 1 (*.f64 3 a)))))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

   if 5.00000000000000009e305 < (+.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 59.5%

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

      1. associate--l+ 59.5%

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

      2. fma-def 59.5%

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

      3. distribute-rgt-in 59.5%

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

      4. sqr-neg 59.5%

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

      5. distribute-rgt-in 59.5%

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

   3. Simplified 61.5%

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

      1. fma-def 61.5%

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

      2. unpow2 61.5%

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

      3. add-sqr-sqrt 61.5%

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

      4. associate-*l* 61.5%

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

      5. hypot-def 61.5%

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

      6. hypot-def 61.5%

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

      7. add-sqr-sqrt 61.5%

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

      8. pow2 61.5%

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

      9. hypot-def 61.5%

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

   5. Applied egg-rr 61.5%

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

      1. add-cube-cbrt 61.5%

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

      2. pow3 61.5%

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

      3. associate-*r* 61.5%

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

      4. unpow2 61.5%

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

      5. pow-prod-up 61.5%

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

      6. metadata-eval 61.5%

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

   7. Applied egg-rr 61.5%

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

   8. Taylor expanded in a around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - a \cdot 3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}}\right)}^{3} + \left(4 \cdot {b}^{2} + -1\right)\\ \end{array} \]

Alternative 5: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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. distribute-rgt-in 0.0%

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

      4. sqr-neg 0.0%

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

      5. distribute-rgt-in 0.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in a around inf 92.1%

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

4. Final simplification 98.0%

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

Alternative 6: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.95 \lor \neg \left(a \leq 1.95\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.95) (not (<= a 1.95)))
   (pow a 4.0)
   (* (+ 1.0 (* a 2.0)) (+ (* a 2.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.95) || !(a <= 1.95)) {
		tmp = pow(a, 4.0);
	} else {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.95d0)) .or. (.not. (a <= 1.95d0))) then
        tmp = a ** 4.0d0
    else
        tmp = (1.0d0 + (a * 2.0d0)) * ((a * 2.0d0) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.95) || !(a <= 1.95)) {
		tmp = Math.pow(a, 4.0);
	} else {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.95) or not (a <= 1.95):
		tmp = math.pow(a, 4.0)
	else:
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.95) || !(a <= 1.95))
		tmp = a ^ 4.0;
	else
		tmp = Float64(Float64(1.0 + Float64(a * 2.0)) * Float64(Float64(a * 2.0) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.95) || ~((a <= 1.95)))
		tmp = a ^ 4.0;
	else
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.95], N[Not[LessEqual[a, 1.95]], $MachinePrecision]], N[Power[a, 4.0], $MachinePrecision], N[(N[(1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.94999999999999996 or 1.94999999999999996 < a

   1. Initial program 54.0%

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

      1. associate--l+ 54.0%

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

      2. fma-def 54.0%

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

      3. distribute-rgt-in 54.0%

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

      4. sqr-neg 54.0%

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

      5. distribute-rgt-in 54.0%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in a around inf 78.2%

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

   if -1.94999999999999996 < a < 1.94999999999999996

   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. associate--l+ 99.8%

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

      2. fma-def 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-rgt-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in b around 0 51.0%

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

   5. Taylor expanded in a around 0 49.4%

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

      1. add-sqr-sqrt 49.4%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 49.4%

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

      3. *-commutative 49.4%

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

      4. sqrt-prod 49.4%

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

      5. unpow2 49.4%

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

      6. sqrt-prod 23.9%

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

      7. add-sqr-sqrt 49.1%

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

      8. metadata-eval 49.1%

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

      9. *-commutative 49.1%

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

      10. sqrt-prod 49.1%

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

      11. unpow2 49.1%

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

      12. sqrt-prod 23.9%

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

      13. add-sqr-sqrt 49.4%

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

      14. metadata-eval 49.4%

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

   7. Applied egg-rr 49.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.95 \lor \neg \left(a \leq 1.95\right):\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \end{array} \]

Alternative 7: 81.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5) (+ (pow a 4.0) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5) {
		tmp = pow(a, 4.0) + -1.0;
	} 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 <= 6.5d0) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.5) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.5:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.5)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.5)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.5], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5

   1. Initial program 77.8%

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

      1. associate--l+ 77.8%

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

      2. fma-def 77.8%

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

      3. distribute-rgt-in 77.8%

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

      4. sqr-neg 77.8%

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

      5. distribute-rgt-in 77.8%

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

   3. Simplified 78.8%

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

      1. fma-def 78.8%

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

      2. unpow2 78.8%

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

      3. add-sqr-sqrt 78.8%

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

      4. associate-*l* 78.9%

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

      5. hypot-def 78.9%

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

      6. hypot-def 78.9%

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

      7. add-sqr-sqrt 78.9%

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

      8. pow2 78.9%

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

      9. hypot-def 78.9%

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

   5. Applied egg-rr 78.9%

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

      1. add-cube-cbrt 78.6%

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

      2. pow3 78.6%

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

      3. associate-*r* 78.6%

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

      4. unpow2 78.6%

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

      5. pow-prod-up 78.7%

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

      6. metadata-eval 78.7%

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

   7. Applied egg-rr 78.7%

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

   8. Taylor expanded in a around 0 95.8%

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

   9. Taylor expanded in b around 0 74.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot {a}^{4} - 1} \]
   10. Step-by-step derivation

       1. sub-neg 74.5%

          \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot {a}^{4} + \left(-1\right)} \]

       2. pow-base-1 74.5%

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

       3. *-lft-identity 74.5%

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

       4. metadata-eval 74.5%

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

   11. Simplified 74.5%

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

   if 6.5 < b

   1. Initial program 71.9%

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

      1. associate--l+ 71.9%

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

      2. fma-def 71.9%

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

      3. distribute-rgt-in 71.9%

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

      4. sqr-neg 71.9%

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

      5. distribute-rgt-in 71.9%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in b around inf 88.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 8: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.8) (* (+ 1.0 (* a 2.0)) (+ (* a 2.0) -1.0)) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.8) {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	} 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 <= 4.8d0) then
        tmp = (1.0d0 + (a * 2.0d0)) * ((a * 2.0d0) + (-1.0d0))
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.8) {
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.8:
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0)
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(1.0 + Float64(a * 2.0)) * Float64(Float64(a * 2.0) + -1.0));
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.8)
		tmp = (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.8], N[(N[(1.0 + N[(a * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(a * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 77.8%

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

      1. associate--l+ 77.8%

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

      2. fma-def 77.8%

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

      3. distribute-rgt-in 77.8%

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

      4. sqr-neg 77.8%

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

      5. distribute-rgt-in 77.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in b around 0 63.6%

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

   5. Taylor expanded in a around 0 56.1%

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

      1. add-sqr-sqrt 56.1%

         \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

      2. difference-of-sqr-1 56.1%

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

      3. *-commutative 56.1%

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

      4. sqrt-prod 56.1%

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

      5. unpow2 56.1%

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

      6. sqrt-prod 28.9%

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

      7. add-sqr-sqrt 45.1%

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

      8. metadata-eval 45.1%

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

      9. *-commutative 45.1%

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

      10. sqrt-prod 45.1%

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

      11. unpow2 45.1%

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

      12. sqrt-prod 28.9%

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

      13. add-sqr-sqrt 56.1%

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

      14. metadata-eval 56.1%

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

   7. Applied egg-rr 56.1%

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

   if 4.79999999999999982 < b

   1. Initial program 71.9%

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

      1. associate--l+ 71.9%

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

      2. fma-def 71.9%

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

      3. distribute-rgt-in 71.9%

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

      4. sqr-neg 71.9%

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

      5. distribute-rgt-in 71.9%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in b around inf 88.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 9: 50.7% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0);
}

Python

def code(a, b):
	return (1.0 + (a * 2.0)) * ((a * 2.0) + -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate--l+ 76.4%

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

   2. fma-def 76.4%

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

   3. distribute-rgt-in 76.4%

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

   4. sqr-neg 76.4%

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

   5. distribute-rgt-in 76.4%

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

3. Simplified 77.6%

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

4. Taylor expanded in b around 0 52.8%

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

5. Taylor expanded in a around 0 46.0%

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

   1. add-sqr-sqrt 46.0%

      \[\leadsto \color{blue}{\sqrt{4 \cdot {a}^{2}} \cdot \sqrt{4 \cdot {a}^{2}}} - 1 \]

   2. difference-of-sqr-1 46.0%

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

   3. *-commutative 46.0%

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

   4. sqrt-prod 46.0%

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

   5. unpow2 46.0%

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

   6. sqrt-prod 23.5%

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

   7. add-sqr-sqrt 35.9%

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

   8. metadata-eval 35.9%

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

   9. *-commutative 35.9%

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

   10. sqrt-prod 35.9%

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

   11. unpow2 35.9%

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

   12. sqrt-prod 23.5%

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

   13. add-sqr-sqrt 46.0%

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

   14. metadata-eval 46.0%

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

7. Applied egg-rr 46.0%

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

8. Final simplification 46.0%

   \[\leadsto \left(1 + a \cdot 2\right) \cdot \left(a \cdot 2 + -1\right) \]

Alternative 10: 24.7% 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 76.4%

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

   1. associate--l+ 76.4%

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

   2. fma-def 76.4%

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

   3. distribute-rgt-in 76.4%

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

   4. sqr-neg 76.4%

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

   5. distribute-rgt-in 76.4%

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

3. Simplified 77.6%

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

4. Taylor expanded in b around 0 52.8%

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

5. Taylor expanded in a around 0 24.1%

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

6. Final simplification 24.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023337 
(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))