Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 7.7s

Alternatives: 12

Speedup: 2.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

   3. pow-sqr N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-+r+ N/A

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

   12. metadata-eval N/A

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

   13. pow-sqr N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

   3. pow-sqr N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-+r+ N/A

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

   12. metadata-eval N/A

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

   13. pow-sqr N/A

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 99.9%

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

Alternative 3: 89.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 700000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \left(1 \cdot a\right) \cdot a, \mathsf{fma}\left(4, b \cdot b, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a \cdot a, 2, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 700000000000.0)
   (fma (fma b b (* a a)) (* (* 1.0 a) a) (fma 4.0 (* b b) -1.0))
   (* (* (fma (* a a) 2.0 (fma b b 4.0)) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 700000000000.0) {
		tmp = fma(fma(b, b, (a * a)), ((1.0 * a) * a), fma(4.0, (b * b), -1.0));
	} else {
		tmp = (fma((a * a), 2.0, fma(b, b, 4.0)) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 700000000000.0)
		tmp = fma(fma(b, b, Float64(a * a)), Float64(Float64(1.0 * a) * a), fma(4.0, Float64(b * b), -1.0));
	else
		tmp = Float64(Float64(fma(Float64(a * a), 2.0, fma(b, b, 4.0)) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 700000000000.0], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 * a), $MachinePrecision] * a), $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0 + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7e11

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. fp-cancel-sub-sign-inv N/A

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

      17. lift-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 89.1%

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

   if 7e11 < b

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. fp-cancel-sub-sign-inv N/A

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

      17. lift-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 86.4

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 700000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \left(1 \cdot a\right) \cdot a, \mathsf{fma}\left(4, b \cdot b, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a \cdot a, 2, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(4, b \cdot b, -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a)))) (fma t_0 t_0 (fma 4.0 (* b b) -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	return fma(t_0, t_0, fma(4.0, (b * b), -1.0));
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(t_0, t_0, fma(4.0, Float64(b * b), -1.0))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval N/A

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

   16. fp-cancel-sub-sign-inv N/A

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

   17. lift-*.f64 N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 N/A

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

   22. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

Alternative 5: 84.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 17:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 17.0)
   (fma (* (fma b b 4.0) b) b -1.0)
   (* (* (fma (* b b) 2.0 (* a a)) a) a)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 17.0) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = (fma((b * b), 2.0, (a * a)) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 17.0)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 17.0], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 17

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt-in N/A

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

   5. Applied rewrites 100.0%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied rewrites 80.5%

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

   if 17 < a

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. fp-cancel-sub-sign-inv N/A

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

      17. lift-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot {b}^{2}}{{a}^{2}}} \cdot {a}^{4} + 1 \cdot {a}^{4} \]

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r/ N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. distribute-rgt-in N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 96.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 17:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.6 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 8.6e+60)
   (fma (* (fma b b 4.0) b) b -1.0)
   (fma (* (* (* a a) 2.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 8.6e+60) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((((a * a) * 2.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 8.6e+60)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 8.6e+60], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.59999999999999942e60

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt-in N/A

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

   5. Applied rewrites 100.0%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied rewrites 78.1%

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

   if 8.59999999999999942e60 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. distribute-lft-in N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 75.4%

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

Alternative 7: 75.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.6 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 2 \cdot \left(a \cdot a\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 8.6e+60)
   (fma (* (fma b b 4.0) b) b -1.0)
   (fma (* b b) (* 2.0 (* a a)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 8.6e+60) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((b * b), (2.0 * (a * a)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 8.6e+60)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(b * b), Float64(2.0 * Float64(a * a)), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 8.6e+60], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.59999999999999942e60

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. distribute-rgt-in N/A

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

      3. pow-sqr N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

   5. Applied rewrites 100.0%

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

   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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied rewrites 78.1%

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

   if 8.59999999999999942e60 < a

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. distribute-lft-in N/A

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

      11. associate-+r+ N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 75.4%

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

   10. Applied rewrites 63.5%

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

Alternative 8: 70.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

   3. pow-sqr N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. +-commutative N/A

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

   11. associate-+r+ N/A

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

   12. metadata-eval N/A

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

   13. pow-sqr N/A

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

5. Applied rewrites 100.0%

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

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. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. distribute-rgt-in N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

8. Applied rewrites 67.8%

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

Alternative 9: 70.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 67.8

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

5. Applied rewrites 67.8%

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

Alternative 10: 69.7% accurate, 7.7× 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 99.9%

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

3. Taylor expanded in a around 0

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

   1. associate-+r- N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. +-commutative N/A

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

   7. associate--l+ N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. distribute-lft-in N/A

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

   11. associate-+r+ N/A

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

   12. metadata-eval N/A

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

   13. fp-cancel-sub-sign-inv N/A

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

5. Applied rewrites 85.1%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 63.3%

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

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 67.6%

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

Alternative 11: 51.8% accurate, 10.9× 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 99.9%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 67.8

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

5. Applied rewrites 67.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 49.1%

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

Alternative 12: 25.5% accurate, 131.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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 99.9%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. distribute-rgt-out N/A

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

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

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval 67.8

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

5. Applied rewrites 67.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 24.9%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))