Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.6% → 99.9%

Time: 8.4s

Alternatives: 13

Speedup: 5.3×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% 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, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, -12\right), a, b \cdot b\right) \cdot b + 4 \cdot b, b, \left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+
  (fma
   (+ (* (fma (fma a 2.0 -12.0) a (* b b)) b) (* 4.0 b))
   b
   (* (* (fma (+ 4.0 a) a 4.0) a) a))
  -1.0))

C

double code(double a, double b) {
	return fma(((fma(fma(a, 2.0, -12.0), a, (b * b)) * b) + (4.0 * b)), b, ((fma((4.0 + a), a, 4.0) * a) * a)) + -1.0;
}

Julia

function code(a, b)
	return Float64(fma(Float64(Float64(fma(fma(a, 2.0, -12.0), a, Float64(b * b)) * b) + Float64(4.0 * b)), b, Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a)) + -1.0)
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(N[(N[(a * 2.0 + -12.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] + N[(4.0 * b), $MachinePrecision]), $MachinePrecision] * b + N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft1-in N/A

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

   4. distribute-lft-neg-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 90.9%

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

10. Applied rewrites 99.9%

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

12. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a, 2, -12\right), a, b \cdot b\right) \cdot b + 4 \cdot b, b, \left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a\right) + -1 \]
13. Add Preprocessing

Alternative 2: 99.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a\right) + -1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (+
  (fma
   (* (fma b b (fma (fma 2.0 a -12.0) a 4.0)) b)
   b
   (* (* (fma (+ 4.0 a) a 4.0) a) a))
  -1.0))

C

double code(double a, double b) {
	return fma((fma(b, b, fma(fma(2.0, a, -12.0), a, 4.0)) * b), b, ((fma((4.0 + a), a, 4.0) * a) * a)) + -1.0;
}

Julia

function code(a, b)
	return Float64(fma(Float64(fma(b, b, fma(fma(2.0, a, -12.0), a, 4.0)) * b), b, Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a)) + -1.0)
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(b * b + N[(N[(2.0 * a + -12.0), $MachinePrecision] * a + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft1-in N/A

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

   4. distribute-lft-neg-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 90.9%

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

10. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, -12\right), a, 4\right)\right) \cdot b, b, \left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a\right) + \color{blue}{-1} \]
11. Add Preprocessing

Alternative 3: 98.0% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5.0)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (- (* (* (fma b b (fma a (fma 2.0 a -12.0) 4.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5.0) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else {
		tmp = ((fma(b, b, fma(a, fma(2.0, a, -12.0), 4.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5.0)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(a, fma(2.0, a, -12.0), 4.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5.0], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b + N[(a * N[(2.0 * a + -12.0), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5

   1. Initial program 78.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

   if 5 < (*.f64 b b)

   1. Initial program 68.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.7%

      \[\leadsto \left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, \mathsf{fma}\left(2, a, -12\right), 4\right)\right) \cdot b\right) \cdot \color{blue}{b} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.8% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft1-in N/A

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

   4. distribute-lft-neg-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 90.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 99.4%

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

Alternative 5: 93.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-12, a, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e+35)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 2.05e+71)
     (fma (* (fma -12.0 a (fma b b 4.0)) b) b -1.0)
     (fma (* (+ 4.0 a) a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.55e+35) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 2.05e+71) {
		tmp = fma((fma(-12.0, a, fma(b, b, 4.0)) * b), b, -1.0);
	} else {
		tmp = fma(((4.0 + a) * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.55e+35)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 2.05e+71)
		tmp = fma(Float64(fma(-12.0, a, fma(b, b, 4.0)) * b), b, -1.0);
	else
		tmp = fma(Float64(Float64(4.0 + a) * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.55e+35], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 2.05e+71], N[(N[(N[(-12.0 * a + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.54999999999999993e35

   1. Initial program 21.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   if -1.54999999999999993e35 < a < 2.0500000000000001e71

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-in N/A

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

      11. associate-+r- N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-lft-in N/A

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

   8. Applied rewrites 94.8%

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

   if 2.0500000000000001e71 < a

   1. Initial program 63.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

Alternative 6: 98.9% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft1-in N/A

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

   4. distribute-lft-neg-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 90.9%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 99.4%

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

12. Taylor expanded in a around inf

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

14. Applied rewrites 98.8%

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

Alternative 7: 93.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+35} \lor \neg \left(a \leq 2.05 \cdot 10^{+71}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.55e+35) (not (<= a 2.05e+71)))
   (fma (* (+ 4.0 a) a) (* a a) -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.55e+35) || !(a <= 2.05e+71)) {
		tmp = fma(((4.0 + a) * a), (a * a), -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.55e+35) || !(a <= 2.05e+71))
		tmp = fma(Float64(Float64(4.0 + a) * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.55e+35], N[Not[LessEqual[a, 2.05e+71]], $MachinePrecision]], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.54999999999999993e35 or 2.0500000000000001e71 < a

   1. Initial program 40.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

   if -1.54999999999999993e35 < a < 2.0500000000000001e71

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 94.6

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

   8. Applied rewrites 94.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+35} \lor \neg \left(a \leq 2.05 \cdot 10^{+71}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(4 + a\right) \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.55e+35)
   (fma (* (fma (+ 4.0 a) a 4.0) a) a -1.0)
   (if (<= a 2.05e+71)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (* (+ 4.0 a) a) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.55e+35) {
		tmp = fma((fma((4.0 + a), a, 4.0) * a), a, -1.0);
	} else if (a <= 2.05e+71) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma(((4.0 + a) * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.55e+35)
		tmp = fma(Float64(fma(Float64(4.0 + a), a, 4.0) * a), a, -1.0);
	elseif (a <= 2.05e+71)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(Float64(4.0 + a) * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.55e+35], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[a, 2.05e+71], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.54999999999999993e35

   1. Initial program 21.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

   if -1.54999999999999993e35 < a < 2.0500000000000001e71

   1. Initial program 95.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 94.7

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 94.6

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

   8. Applied rewrites 94.6%

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

   if 2.0500000000000001e71 < a

   1. Initial program 63.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 100.0%

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

Alternative 9: 87.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.8e+153)
   (fma 4.0 (* a a) -1.0)
   (if (<= a 1.1e+99)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (fma 4.0 a 4.0) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.8e+153) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if (a <= 1.1e+99) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma(fma(4.0, a, 4.0), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.8e+153)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (a <= 1.1e+99)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(fma(4.0, a, 4.0), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.8e+153], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 1.1e+99], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(4.0 * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.79999999999999985e153

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 97.7%

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

   if -4.79999999999999985e153 < a < 1.09999999999999989e99

   1. Initial program 89.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 84.3

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

   8. Applied rewrites 84.3%

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

   if 1.09999999999999989e99 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 96.0%

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

Alternative 10: 70.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+38)
   (fma 4.0 (* a a) -1.0)
   (if (<= a 1.1e+99)
     (fma (* b b) 4.0 -1.0)
     (fma (fma 4.0 a 4.0) (* a a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+38) {
		tmp = fma(4.0, (a * a), -1.0);
	} else if (a <= 1.1e+99) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = fma(fma(4.0, a, 4.0), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.3e+38)
		tmp = fma(4.0, Float64(a * a), -1.0);
	elseif (a <= 1.1e+99)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = fma(fma(4.0, a, 4.0), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.3e+38], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[a, 1.1e+99], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(4.0 * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.2999999999999999e38

   1. Initial program 21.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 63.4%

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

   if -3.2999999999999999e38 < a < 1.09999999999999989e99

   1. Initial program 94.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 73.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 68.2%

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

   if 1.09999999999999989e99 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 96.0%

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

Alternative 11: 69.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 4.8 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(4, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 4.8e+304) (fma 4.0 (* a a) -1.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 4.8e+304) {
		tmp = fma(4.0, (a * a), -1.0);
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 4.8e+304)
		tmp = fma(4.0, Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 4.8e+304], N[(4.0 * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.79999999999999951e304

   1. Initial program 75.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. distribute-lft-out N/A

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

      10. associate-+r+ N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 60.1%

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

   if 4.79999999999999951e304 < (*.f64 b b)

   1. Initial program 65.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft1-in N/A

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

      4. distribute-lft-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 97.3%

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

Alternative 12: 51.7% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft1-in N/A

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

   4. distribute-lft-neg-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 75.2%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 47.1%

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

Alternative 13: 25.0% accurate, 160.0× 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 73.3%

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

3. Taylor expanded in a around 0

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft1-in N/A

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

   4. distribute-lft-neg-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 89.8

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in b around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. distribute-lft-out N/A

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

   10. associate-+r+ N/A

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

   11. unpow2 N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

8. Applied rewrites 66.1%

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

9. Taylor expanded in a around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 22.6%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(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))