Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.1% → 99.9%

Time: 8.2s

Alternatives: 11

Speedup: 4.2×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 95.2%

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

10. Applied rewrites 99.9%

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

Alternative 2: 99.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

8. Applied rewrites 95.2%

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

10. Applied rewrites 99.9%

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

11. Taylor expanded in a around inf

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

13. Applied rewrites 99.5%

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

Alternative 3: 97.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\left(\mathsf{fma}\left(1 - a, 4, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+41)
   (- (* (* (fma (- 1.0 a) 4.0 (* a a)) a) a) 1.0)
   (fma (fma (fma 2.0 a 4.0) a (fma b b 12.0)) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+41) {
		tmp = ((fma((1.0 - a), 4.0, (a * a)) * a) * a) - 1.0;
	} else {
		tmp = fma(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+41)
		tmp = Float64(Float64(Float64(fma(Float64(1.0 - a), 4.0, Float64(a * a)) * a) * a) - 1.0);
	else
		tmp = fma(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+41], N[(N[(N[(N[(N[(1.0 - a), $MachinePrecision] * 4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.00000000000000001e41

   1. Initial program 84.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. distribute-rgt-in N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. lower--.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 99.8

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

   11. Applied rewrites 99.8%

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

   if 2.00000000000000001e41 < (*.f64 b b)

   1. Initial program 56.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.8%

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

Alternative 4: 94.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\left(\mathsf{fma}\left(1 - a, 4, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4 \cdot a\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+48)
   (- (* (* (fma (- 1.0 a) 4.0 (* a a)) a) a) 1.0)
   (fma (* (fma b b (* 4.0 a)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+48) {
		tmp = ((fma((1.0 - a), 4.0, (a * a)) * a) * a) - 1.0;
	} else {
		tmp = fma((fma(b, b, (4.0 * a)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+48)
		tmp = Float64(Float64(Float64(fma(Float64(1.0 - a), 4.0, Float64(a * a)) * a) * a) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, Float64(4.0 * a)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+48], N[(N[(N[(N[(N[(1.0 - a), $MachinePrecision] * 4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + N[(4.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999973e48

   1. Initial program 85.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-pow.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. metadata-eval N/A

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

       4. pow-sqr N/A

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

       5. distribute-rgt-in N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 N/A

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

       13. lower--.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 99.8

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

   11. Applied rewrites 99.8%

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

   if 4.99999999999999973e48 < (*.f64 b b)

   1. Initial program 56.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 93.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\left(\mathsf{fma}\left(1 - a, 4, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4 \cdot a\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \left(1 - a\right) \cdot 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4 \cdot a\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+48)
   (fma (* a a) (fma a a (* (- 1.0 a) 4.0)) -1.0)
   (fma (* (fma b b (* 4.0 a)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+48) {
		tmp = fma((a * a), fma(a, a, ((1.0 - a) * 4.0)), -1.0);
	} else {
		tmp = fma((fma(b, b, (4.0 * a)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+48)
		tmp = fma(Float64(a * a), fma(a, a, Float64(Float64(1.0 - a) * 4.0)), -1.0);
	else
		tmp = fma(Float64(fma(b, b, Float64(4.0 * a)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+48], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(N[(1.0 - a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(4.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.99999999999999973e48

   1. Initial program 85.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.7

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

   8. Applied rewrites 99.7%

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

   if 4.99999999999999973e48 < (*.f64 b b)

   1. Initial program 56.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 93.6%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 93.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a, \left(1 - a\right) \cdot 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4 \cdot a\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

8. Applied rewrites 70.9%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 70.9%

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

12. Final simplification 70.9%

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

Alternative 7: 54.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.4:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.4) (- (* (* b b) 12.0) 1.0) (* (* (* b b) a) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.4) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = ((b * b) * a) * 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.4d0) then
        tmp = ((b * b) * 12.0d0) - 1.0d0
    else
        tmp = ((b * b) * a) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 2.4) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = ((b * b) * a) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.4:
		tmp = ((b * b) * 12.0) - 1.0
	else:
		tmp = ((b * b) * a) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.4)
		tmp = Float64(Float64(Float64(b * b) * 12.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * a) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2.4)
		tmp = ((b * b) * 12.0) - 1.0;
	else
		tmp = ((b * b) * a) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.4], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.39999999999999991

   1. Initial program 81.7%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 58.4%

      \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} - 1 \]

   if 2.39999999999999991 < a

   1. Initial program 37.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

   8. Applied rewrites 51.3%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 44.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot a\right) \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-rgt-in N/A

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

   4. sub-neg N/A

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(12 + {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(12 + {b}^{2}\right) \cdot {b}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(12 + {b}^{2}\right) \cdot \color{blue}{\left(b \cdot b\right)} + \left(\mathsf{neg}\left(1\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(12 + {b}^{2}\right) \cdot b\right) \cdot b} + \left(\mathsf{neg}\left(1\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(b \cdot \left(12 + {b}^{2}\right)\right)} \cdot b + \left(\mathsf{neg}\left(1\right)\right) \]

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 70.5

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

8. Applied rewrites 70.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)} \]
9. Add Preprocessing

Alternative 9: 68.5% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

8. Applied rewrites 70.9%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 70.5%

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

Alternative 10: 50.7% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot b\right) \cdot 12 - 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* b b) 12.0) 1.0))

C

double code(double a, double b) {
	return ((b * b) * 12.0) - 1.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((b * b) * 12.0d0) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((b * b) * 12.0) - 1.0;
}

Python

def code(a, b):
	return ((b * b) * 12.0) - 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(b * b) * 12.0) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((b * b) * 12.0) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 76.7%

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

6. Taylor expanded in a around 0

   \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
7. Step-by-step derivation

8. Applied rewrites 51.4%

   \[\leadsto 12 \cdot \color{blue}{\left(b \cdot b\right)} - 1 \]

9. Final simplification 51.4%

   \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
10. Add Preprocessing

Alternative 11: 24.7% accurate, 155.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

8. Applied rewrites 70.9%

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

9. Taylor expanded in b around 0

   \[\leadsto -1 \]
10. Step-by-step derivation

11. Applied rewrites 24.0%

    \[\leadsto -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024282 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))