Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.5% → 98.1%

Time: 7.8s

Alternatives: 6

Speedup: 1.2×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(2 - a\right)\\ \mathbf{if}\;a \leq 5.4 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} - {a}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{t_0}^{4}}, {\left(\sqrt[3]{t_0}\right)}^{2}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (- 2.0 a))))
   (if (<= a 5.4e+50)
     (+
      (pow (fma a a (* b b)) 2.0)
      (+ (* 4.0 (- (pow a 2.0) (pow a 3.0))) -1.0))
     (fma (cbrt (pow t_0 4.0)) (pow (cbrt t_0) 2.0) -1.0))))

C

double code(double a, double b) {
	double t_0 = a * (2.0 - a);
	double tmp;
	if (a <= 5.4e+50) {
		tmp = pow(fma(a, a, (b * b)), 2.0) + ((4.0 * (pow(a, 2.0) - pow(a, 3.0))) + -1.0);
	} else {
		tmp = fma(cbrt(pow(t_0, 4.0)), pow(cbrt(t_0), 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(2.0 - a))
	tmp = 0.0
	if (a <= 5.4e+50)
		tmp = Float64((fma(a, a, Float64(b * b)) ^ 2.0) + Float64(Float64(4.0 * Float64((a ^ 2.0) - (a ^ 3.0))) + -1.0));
	else
		tmp = fma(cbrt((t_0 ^ 4.0)), (cbrt(t_0) ^ 2.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.4e+50], N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[t$95$0, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.4e50

   1. Initial program 86.6%

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

      1. associate--l+ 86.5%

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

      2. fma-def 86.5%

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

      3. sqr-neg 86.5%

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

      4. fma-def 86.5%

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

      5. distribute-rgt-in 86.5%

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

      6. sqr-neg 86.5%

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

      7. distribute-rgt-in 86.5%

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

      8. fma-def 86.5%

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

      9. sqr-neg 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in a around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

   6. Simplified 99.1%

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

   if 5.4e50 < a

   1. Initial program 18.3%

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

      1. associate--l+ 18.3%

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

      2. fma-def 18.3%

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

      3. sqr-neg 18.3%

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

      4. fma-def 18.3%

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

      5. distribute-rgt-in 18.3%

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

      6. sqr-neg 18.3%

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

      7. distribute-rgt-in 18.3%

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

      8. fma-def 18.3%

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

      9. sqr-neg 18.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in b around 0 12.2%

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

      1. add-sqr-sqrt 12.2%

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

      2. pow2 12.2%

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

      3. +-commutative 12.2%

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

      4. metadata-eval 12.2%

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

      5. pow-sqr 12.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. hypot-def 68.3%

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

      8. sqrt-prod 68.3%

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

      9. metadata-eval 68.3%

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

      10. sqrt-prod 68.3%

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

      11. unpow2 68.3%

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

      12. sqrt-prod 68.3%

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

      13. add-sqr-sqrt 68.3%

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

   6. Applied egg-rr 68.3%

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

   7. Taylor expanded in a around 0 93.8%

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

      1. neg-mul-1 93.8%

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

      2. +-commutative 93.8%

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

      3. *-commutative 93.8%

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

      4. unsub-neg 93.8%

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

   9. Simplified 93.8%

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

       1. add-cube-cbrt 93.8%

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

       2. fma-neg 93.8%

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

       3. metadata-eval 93.8%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{+50}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} - {a}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{{\left(a \cdot \left(2 - a\right)\right)}^{4}}, {\left(\sqrt[3]{a \cdot \left(2 - a\right)}\right)}^{2}, -1\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(2 - a\right)\\ \mathbf{if}\;a \leq 2 \cdot 10^{+76}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} - {a}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (- 2.0 a))))
   (if (<= a 2e+76)
     (+
      (pow (fma a a (* b b)) 2.0)
      (+ (* 4.0 (- (pow a 2.0) (pow a 3.0))) -1.0))
     (+ (* t_0 t_0) -1.0))))

C

double code(double a, double b) {
	double t_0 = a * (2.0 - a);
	double tmp;
	if (a <= 2e+76) {
		tmp = pow(fma(a, a, (b * b)), 2.0) + ((4.0 * (pow(a, 2.0) - pow(a, 3.0))) + -1.0);
	} else {
		tmp = (t_0 * t_0) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(a * Float64(2.0 - a))
	tmp = 0.0
	if (a <= 2e+76)
		tmp = Float64((fma(a, a, Float64(b * b)) ^ 2.0) + Float64(Float64(4.0 * Float64((a ^ 2.0) - (a ^ 3.0))) + -1.0));
	else
		tmp = Float64(Float64(t_0 * t_0) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2e+76], N[(N[Power[N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(4.0 * N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.0000000000000001e76

   1. Initial program 86.8%

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

      1. associate--l+ 86.8%

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

      2. fma-def 86.8%

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

      3. sqr-neg 86.8%

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

      4. fma-def 86.8%

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

      5. distribute-rgt-in 86.8%

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

      6. sqr-neg 86.8%

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

      7. distribute-rgt-in 86.8%

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

      8. fma-def 86.8%

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

      9. sqr-neg 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in a around inf 99.1%

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

      1. +-commutative 99.1%

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

      2. mul-1-neg 99.1%

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

      3. unsub-neg 99.1%

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

   6. Simplified 99.1%

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

   if 2.0000000000000001e76 < a

   1. Initial program 12.5%

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

      1. associate--l+ 12.5%

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

      2. fma-def 12.5%

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

      3. sqr-neg 12.5%

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

      4. fma-def 12.5%

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

      5. distribute-rgt-in 12.5%

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

      6. sqr-neg 12.5%

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

      7. distribute-rgt-in 12.5%

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

      8. fma-def 12.5%

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

      9. sqr-neg 12.5%

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

   3. Simplified 17.9%

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

   4. Taylor expanded in b around 0 12.5%

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

      1. add-sqr-sqrt 12.5%

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

      2. pow2 12.5%

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

      3. +-commutative 12.5%

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

      4. metadata-eval 12.5%

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

      5. pow-sqr 12.5%

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

      6. add-sqr-sqrt 0.0%

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

      7. hypot-def 73.2%

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

      8. sqrt-prod 73.2%

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

      9. metadata-eval 73.2%

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

      10. sqrt-prod 73.2%

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

      11. unpow2 73.2%

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

      12. sqrt-prod 73.2%

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

      13. add-sqr-sqrt 73.2%

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

   6. Applied egg-rr 73.2%

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

   7. Taylor expanded in a around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. unsub-neg 100.0%

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

   9. Simplified 100.0%

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

       1. unpow2 100.0%

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

       2. unpow2 100.0%

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

       3. distribute-lft-out-- 100.0%

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

       4. unpow2 100.0%

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

       5. distribute-lft-out-- 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{+76}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2} + \left(4 \cdot \left({a}^{2} - {a}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(2 - a\right)\right) \cdot \left(a \cdot \left(2 - a\right)\right) + -1\\ \end{array} \]

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(b \cdot b + a \cdot a\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\\ t_1 := a \cdot \left(2 - a\right)\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (+
          (pow (+ (* b b) (* a a)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))))
        (t_1 (* a (- 2.0 a))))
   (if (<= t_0 INFINITY) (+ t_0 -1.0) (+ (* t_1 t_1) -1.0))))

C

double code(double a, double b) {
	double t_0 = pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double t_1 = a * (2.0 - a);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 + -1.0;
	} else {
		tmp = (t_1 * t_1) + -1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = Math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	double t_1 = a * (2.0 - a);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + -1.0;
	} else {
		tmp = (t_1 * t_1) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.pow(((b * b) + (a * a)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))))
	t_1 = a * (2.0 - a)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 + -1.0
	else:
		tmp = (t_1 * t_1) + -1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64((Float64(Float64(b * b) + Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(a + 3.0)))))
	t_1 = Float64(a * Float64(2.0 - a))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 + -1.0);
	else
		tmp = Float64(Float64(t_1 * t_1) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (((b * b) + (a * a)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0))));
	t_1 = a * (2.0 - a);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 + -1.0;
	else
		tmp = (t_1 * t_1) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Power[N[(N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(a + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 + -1.0), $MachinePrecision], N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $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 3 a))))) < +inf.0

   1. Initial program 99.8%

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

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

   1. Initial program 0.0%

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

      1. associate--l+ 0.0%

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

      2. fma-def 0.0%

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

      3. sqr-neg 0.0%

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

      4. fma-def 0.0%

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

      5. distribute-rgt-in 0.0%

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

      6. sqr-neg 0.0%

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

      7. distribute-rgt-in 0.0%

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

      8. fma-def 0.0%

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

      9. sqr-neg 0.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in b around 0 27.1%

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

      1. add-sqr-sqrt 27.1%

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

      2. pow2 27.1%

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

      3. +-commutative 27.1%

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

      4. metadata-eval 27.1%

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

      5. pow-sqr 27.1%

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

      6. add-sqr-sqrt 27.1%

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

      7. hypot-def 81.8%

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

      8. sqrt-prod 81.8%

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

      9. metadata-eval 81.8%

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

      10. sqrt-prod 81.8%

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

      11. unpow2 81.8%

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

      12. sqrt-prod 72.0%

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

      13. add-sqr-sqrt 81.8%

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

   6. Applied egg-rr 81.8%

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

   7. Taylor expanded in a around 0 92.5%

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

      1. neg-mul-1 92.5%

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

      2. +-commutative 92.5%

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

      3. *-commutative 92.5%

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

      4. unsub-neg 92.5%

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

   9. Simplified 92.5%

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

       1. unpow2 92.5%

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

       2. unpow2 92.5%

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

       3. distribute-lft-out-- 92.5%

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

       4. unpow2 92.5%

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

       5. distribute-lft-out-- 92.5%

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

   11. Applied egg-rr 92.5%

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

4. Final simplification 97.7%

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

Alternative 4: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(2 - a\right)\\ \mathbf{if}\;b \leq 2350000000:\\ \;\;\;\;t_0 \cdot t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (- 2.0 a))))
   (if (<= b 2350000000.0) (+ (* t_0 t_0) -1.0) (pow b 4.0))))

C

double code(double a, double b) {
	double t_0 = a * (2.0 - a);
	double tmp;
	if (b <= 2350000000.0) {
		tmp = (t_0 * t_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) :: t_0
    real(8) :: tmp
    t_0 = a * (2.0d0 - a)
    if (b <= 2350000000.0d0) then
        tmp = (t_0 * t_0) + (-1.0d0)
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (2.0 - a);
	double tmp;
	if (b <= 2350000000.0) {
		tmp = (t_0 * t_0) + -1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * (2.0 - a)
	tmp = 0
	if b <= 2350000000.0:
		tmp = (t_0 * t_0) + -1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(2.0 - a))
	tmp = 0.0
	if (b <= 2350000000.0)
		tmp = Float64(Float64(t_0 * t_0) + -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * (2.0 - a);
	tmp = 0.0;
	if (b <= 2350000000.0)
		tmp = (t_0 * t_0) + -1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2350000000.0], N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.35e9

   1. Initial program 73.2%

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

      1. associate--l+ 73.2%

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

      2. fma-def 73.2%

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

      3. sqr-neg 73.2%

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

      4. fma-def 73.2%

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

      5. distribute-rgt-in 73.2%

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

      6. sqr-neg 73.2%

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

      7. distribute-rgt-in 73.2%

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

      8. fma-def 73.2%

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

      9. sqr-neg 73.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in b around 0 61.5%

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

      1. add-sqr-sqrt 61.5%

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

      2. pow2 61.5%

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

      3. +-commutative 61.5%

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

      4. metadata-eval 61.5%

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

      5. pow-sqr 61.5%

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

      6. add-sqr-sqrt 54.8%

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

      7. hypot-def 71.7%

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

      8. sqrt-prod 71.7%

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

      9. metadata-eval 71.7%

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

      10. sqrt-prod 71.7%

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

      11. unpow2 71.7%

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

      12. sqrt-prod 45.6%

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

      13. add-sqr-sqrt 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in a around 0 82.0%

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

      1. neg-mul-1 82.0%

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

      2. +-commutative 82.0%

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

      3. *-commutative 82.0%

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

      4. unsub-neg 82.0%

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

   9. Simplified 82.0%

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

       1. unpow2 82.0%

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

       2. unpow2 82.0%

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

       3. distribute-lft-out-- 82.0%

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

       4. unpow2 82.0%

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

       5. distribute-lft-out-- 82.0%

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

   11. Applied egg-rr 82.0%

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

   if 2.35e9 < b

   1. Initial program 62.1%

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

      1. associate--l+ 62.1%

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

      2. fma-def 62.1%

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

      3. sqr-neg 62.1%

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

      4. fma-def 62.1%

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

      5. distribute-rgt-in 62.1%

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

      6. sqr-neg 62.1%

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

      7. distribute-rgt-in 62.1%

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

      8. fma-def 62.1%

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

      9. sqr-neg 62.1%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in b around inf 93.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2350000000:\\ \;\;\;\;\left(a \cdot \left(2 - a\right)\right) \cdot \left(a \cdot \left(2 - a\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \]

Alternative 5: 70.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(2 - a\right)\\ t_0 \cdot t_0 + -1 \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (- 2.0 a)))) (+ (* t_0 t_0) -1.0)))

C

double code(double a, double b) {
	double t_0 = a * (2.0 - a);
	return (t_0 * t_0) + -1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = a * (2.0d0 - a)
    code = (t_0 * t_0) + (-1.0d0)
end function

Java

public static double code(double a, double b) {
	double t_0 = a * (2.0 - a);
	return (t_0 * t_0) + -1.0;
}

Python

def code(a, b):
	t_0 = a * (2.0 - a)
	return (t_0 * t_0) + -1.0

Julia

function code(a, b)
	t_0 = Float64(a * Float64(2.0 - a))
	return Float64(Float64(t_0 * t_0) + -1.0)
end

MATLAB

function tmp = code(a, b)
	t_0 = a * (2.0 - a);
	tmp = (t_0 * t_0) + -1.0;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(2.0 - a), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Initial program 70.6%

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

   1. associate--l+ 70.6%

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

   2. fma-def 70.6%

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

   3. sqr-neg 70.6%

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

   4. fma-def 70.6%

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

   5. distribute-rgt-in 70.6%

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

   6. sqr-neg 70.6%

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

   7. distribute-rgt-in 70.6%

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

   8. fma-def 70.6%

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

   9. sqr-neg 70.6%

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

3. Simplified 71.7%

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

4. Taylor expanded in b around 0 52.6%

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

   1. add-sqr-sqrt 52.6%

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

   2. pow2 52.6%

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

   3. +-commutative 52.6%

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

   4. metadata-eval 52.6%

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

   5. pow-sqr 52.6%

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

   6. add-sqr-sqrt 46.6%

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

   7. hypot-def 62.6%

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

   8. sqrt-prod 62.6%

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

   9. metadata-eval 62.6%

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

   10. sqrt-prod 62.6%

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

   11. unpow2 62.6%

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

   12. sqrt-prod 40.3%

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

   13. add-sqr-sqrt 62.6%

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

6. Applied egg-rr 62.6%

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

7. Taylor expanded in a around 0 71.7%

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

   1. neg-mul-1 71.7%

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

   2. +-commutative 71.7%

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

   3. *-commutative 71.7%

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

   4. unsub-neg 71.7%

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

9. Simplified 71.7%

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

    1. unpow2 71.7%

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

    2. unpow2 71.7%

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

    3. distribute-lft-out-- 71.7%

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

    4. unpow2 71.7%

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

    5. distribute-lft-out-- 71.7%

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

11. Applied egg-rr 71.7%

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

12. Final simplification 71.7%

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

Alternative 6: 25.2% accurate, 128.0× 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 70.6%

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

   1. associate--l+ 70.6%

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

   2. fma-def 70.6%

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

   3. sqr-neg 70.6%

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

   4. fma-def 70.6%

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

   5. distribute-rgt-in 70.6%

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

   6. sqr-neg 70.6%

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

   7. distribute-rgt-in 70.6%

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

   8. fma-def 70.6%

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

   9. sqr-neg 70.6%

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

3. Simplified 71.7%

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

4. Taylor expanded in b around 0 52.6%

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

5. Taylor expanded in a around 0 23.1%

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

6. Final simplification 23.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023314 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))