Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 99.9%

Time: 8.9s

Alternatives: 4

Speedup: 1.2×

• 61.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (+ (pow a 4.0) -1.0)
   (+
    (+ (pow (hypot b a) 4.0) (* 4.0 (+ (* (* a a) (+ a 1.0)) (* b b))))
    -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = (pow(hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + (b * b)))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = (Math.pow(Math.hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + (b * b)))) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+103:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = (math.pow(math.hypot(b, a), 4.0) + (4.0 * (((a * a) * (a + 1.0)) + (b * b)))) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64((hypot(b, a) ^ 4.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(b * b)))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+103)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = ((hypot(b, a) ^ 4.0) + (4.0 * (((a * a) * (a + 1.0)) + (b * b)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[Power[N[Sqrt[b ^ 2 + a ^ 2], $MachinePrecision], 4.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1e103

   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. sub-neg 0.0%

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

   3. Simplified 7.7%

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

   5. Taylor expanded in a around inf 100.0%

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

   if -1e103 < a

   1. Initial program 88.1%

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

      1. associate-*r* 88.1%

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

      2. cancel-sign-sub-inv 88.1%

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

      3. metadata-eval 88.1%

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

      4. add-exp-log 69.0%

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

      5. metadata-eval 69.0%

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

      6. cancel-sign-sub-inv 69.0%

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

      7. associate-*r* 73.9%

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

      8. cancel-sign-sub-inv 73.9%

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

      9. metadata-eval 73.9%

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

      10. +-commutative 73.9%

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

      11. pow2 73.9%

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

      12. *-commutative 73.9%

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

      13. fma-undefine 73.9%

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in a around 0 99.9%

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

      1. log-pow 49.4%

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

   7. Simplified 49.4%

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

      1. log-pow 99.9%

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

      2. pow2 99.9%

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

      3. add-exp-log 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. unpow2 99.9%

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

       2. add-exp-log 99.4%

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

       3. pow2 99.4%

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

       4. log-pow 49.1%

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

       5. distribute-lft-in 45.7%

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

       6. pow1 45.7%

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

       7. metadata-eval 45.7%

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

       8. sqrt-pow2 45.7%

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

       9. hypot-define 45.7%

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

       10. pow2 45.7%

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

       11. pow1 45.7%

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

       12. metadata-eval 45.7%

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

       13. sqrt-pow2 45.7%

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

       14. hypot-define 45.7%

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

       15. log-pow 88.1%

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

       16. add-exp-log 88.6%

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

   11. Applied egg-rr 88.6%

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

       1. distribute-lft-out 99.9%

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

       2. rem-square-sqrt 99.9%

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

       3. unpow2 99.9%

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

       4. unpow2 99.9%

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

       5. hypot-undefine 99.9%

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

       6. unpow2 99.9%

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

       7. unpow2 99.9%

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

       8. hypot-undefine 99.9%

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

       9. unpow2 99.9%

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

       10. pow-sqr 100.0%

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

       11. metadata-eval 100.0%

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

       12. hypot-undefine 100.0%

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

       13. unpow2 100.0%

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

       14. unpow2 100.0%

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

       15. +-commutative 100.0%

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

       16. unpow2 100.0%

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

       17. unpow2 100.0%

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

       18. hypot-define 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{hypot}\left(b, a\right)\right)}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+68}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.5e+68)
   (+ (pow a 4.0) -1.0)
   (+
    (+ (* 4.0 (+ (* (* a a) (+ a 1.0)) (* b b))) (pow (+ (* a a) (* b b)) 2.0))
    -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+68) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (((a * a) * (a + 1.0)) + (b * b))) + pow(((a * a) + (b * b)), 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 <= (-3.5d+68)) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = ((4.0d0 * (((a * a) * (a + 1.0d0)) + (b * b))) + (((a * a) + (b * b)) ** 2.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+68) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = ((4.0 * (((a * a) * (a + 1.0)) + (b * b))) + Math.pow(((a * a) + (b * b)), 2.0)) + -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.5e+68:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = ((4.0 * (((a * a) * (a + 1.0)) + (b * b))) + math.pow(((a * a) + (b * b)), 2.0)) + -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.5e+68)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(a + 1.0)) + Float64(b * b))) + (Float64(Float64(a * a) + Float64(b * b)) ^ 2.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.5e+68)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = ((4.0 * (((a * a) * (a + 1.0)) + (b * b))) + (((a * a) + (b * b)) ^ 2.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.5e+68], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(a + 1.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.49999999999999977e68

   1. Initial program 11.8%

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

      1. sub-neg 11.8%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in a around inf 100.0%

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

   if -3.49999999999999977e68 < a

   1. Initial program 87.7%

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

      1. associate-*r* 87.7%

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

      2. cancel-sign-sub-inv 87.7%

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

      3. metadata-eval 87.7%

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

      4. add-exp-log 67.9%

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

      5. metadata-eval 67.9%

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

      6. cancel-sign-sub-inv 67.9%

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

      7. associate-*r* 73.0%

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

      8. cancel-sign-sub-inv 73.0%

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

      9. metadata-eval 73.0%

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

      10. +-commutative 73.0%

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

      11. pow2 73.0%

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

      12. *-commutative 73.0%

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

      13. fma-undefine 73.0%

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

   4. Applied egg-rr 73.0%

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

   5. Taylor expanded in a around 0 99.9%

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

      1. log-pow 49.2%

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

   7. Simplified 49.2%

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

      1. log-pow 99.9%

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

      2. pow2 99.9%

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

      3. add-exp-log 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+68}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + b \cdot b\right) + {\left(a \cdot a + b \cdot b\right)}^{2}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e+30) (+ (pow a 4.0) -1.0) (+ -1.0 (pow b 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.5e+30) {
		tmp = pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + 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 <= 2.5d+30) then
        tmp = (a ** 4.0d0) + (-1.0d0)
    else
        tmp = (-1.0d0) + (b ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.5e+30) {
		tmp = Math.pow(a, 4.0) + -1.0;
	} else {
		tmp = -1.0 + Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.5e+30:
		tmp = math.pow(a, 4.0) + -1.0
	else:
		tmp = -1.0 + math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.5e+30)
		tmp = Float64((a ^ 4.0) + -1.0);
	else
		tmp = Float64(-1.0 + (b ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.5e+30)
		tmp = (a ^ 4.0) + -1.0;
	else
		tmp = -1.0 + (b ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.5e+30], N[(N[Power[a, 4.0], $MachinePrecision] + -1.0), $MachinePrecision], N[(-1.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4999999999999999e30

   1. Initial program 73.0%

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

      1. sub-neg 73.0%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in a around inf 77.7%

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

   if 2.4999999999999999e30 < b

   1. Initial program 58.2%

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

      1. sub-neg 58.2%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in b around inf 98.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;{a}^{4} + -1\\ \mathbf{else}:\\ \;\;\;\;-1 + {b}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {a}^{4} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. sub-neg 70.2%

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

3. Simplified 71.8%

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

5. Taylor expanded in a around inf 71.8%

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

6. Final simplification 71.8%

   \[\leadsto {a}^{4} + -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))