Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.5% → 99.1%

Time: 12.7s

Alternatives: 15

Speedup: 5.9×

• 61.7% 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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{\mathsf{fma}\left(1, \frac{b \cdot \left(a \cdot b\right)}{a}, a \cdot a\right)}} + 4 \cdot \left(b \cdot b\right)\right) + -1 \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. clear-num N/A

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

   12. flip-+ N/A

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

   13. lift-+.f64 N/A

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

   14. lower-/.f64 74.5

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

   15. lift-+.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 74.5

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

4. Applied rewrites 74.5%

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

5. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. *-rgt-identity N/A

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

   9. associate-*r/ N/A

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

   10. rgt-mult-inverse N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

12. Applied rewrites 99.4%

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

13. Final simplification 99.4%

    \[\leadsto \left(\frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{\mathsf{fma}\left(1, \frac{b \cdot \left(a \cdot b\right)}{a}, a \cdot a\right)}} + 4 \cdot \left(b \cdot b\right)\right) + -1 \]
14. Add Preprocessing

Alternative 2: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(4 \cdot \left(b \cdot b\right) + \frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{\mathsf{fma}\left(1, \frac{a \cdot \left(b \cdot b\right)}{a}, a \cdot a\right)}}\right) + -1 \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. clear-num N/A

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

   12. flip-+ N/A

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

   13. lift-+.f64 N/A

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

   14. lower-/.f64 73.5

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

   15. lift-+.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lower-fma.f64 73.5

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

4. Applied rewrites 73.5%

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

5. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r/ N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. *-rgt-identity N/A

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

   9. associate-*r/ N/A

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

   10. rgt-mult-inverse N/A

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

   11. *-rgt-identity N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

    \[\leadsto \left(4 \cdot \left(b \cdot b\right) + \frac{\mathsf{fma}\left(a, a, b \cdot b\right)}{\frac{1}{\mathsf{fma}\left(1, \frac{a \cdot \left(b \cdot b\right)}{a}, a \cdot a\right)}}\right) + -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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))