Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 99.9%

Time: 8.1s

Alternatives: 9

Speedup: 5.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+r+ N/A

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

5. Applied rewrites 85.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 99.9%

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

Alternative 2: 99.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+r+ N/A

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

5. Applied rewrites 85.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 99.9%

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

10. Applied rewrites 94.4%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 99.7%

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

Alternative 3: 94.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+43} \lor \neg \left(a \leq 490000000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.7e+43) (not (<= a 490000000000.0)))
   (- (* (* a a) (* a a)) 1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.7e+43) || !(a <= 490000000000.0)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.7e+43) || !(a <= 490000000000.0))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.7e+43], N[Not[LessEqual[a, 490000000000.0]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.70000000000000006e43 or 4.9e11 < a

   1. Initial program 45.7%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   4. Step-by-step derivation

      1. lower-pow.f64 97.6

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 97.5%

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

   if -1.70000000000000006e43 < a < 4.9e11

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 95.6%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 95.0

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

   8. Applied rewrites 95.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.7 \cdot 10^{+43} \lor \neg \left(a \leq 490000000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right) \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.9e+146)
   (fma (* a a) 4.0 -1.0)
   (if (<= a 1.35e+91)
     (fma (* (fma b b 4.0) b) b -1.0)
     (fma (* (fma a a a) a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.9e+146) {
		tmp = fma((a * a), 4.0, -1.0);
	} else if (a <= 1.35e+91) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = fma((fma(a, a, a) * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.9e+146)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	elseif (a <= 1.35e+91)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = fma(Float64(fma(a, a, a) * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.9e+146], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[a, 1.35e+91], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999e146

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

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

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

      11. distribute-lft-in N/A

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

      12. unpow2 N/A

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

      13. *-lft-identity N/A

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

      14. distribute-rgt-in N/A

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

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

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 94.7%

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

   if -1.8999999999999999e146 < a < 1.35e91

   1. Initial program 93.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 85.2%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 82.8

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

   8. Applied rewrites 82.8%

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

   if 1.35e91 < a

   1. Initial program 63.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

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

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

      11. distribute-lft-in N/A

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

      12. unpow2 N/A

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

      13. *-lft-identity N/A

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

      14. distribute-rgt-in N/A

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

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

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 93.4%

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

Alternative 5: 72.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right) \cdot a, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.9e+146)
   (fma (* a a) 4.0 -1.0)
   (if (<= a 3.4e+85)
     (fma (* 4.0 b) b -1.0)
     (fma (* (fma a a a) a) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.9e+146) {
		tmp = fma((a * a), 4.0, -1.0);
	} else if (a <= 3.4e+85) {
		tmp = fma((4.0 * b), b, -1.0);
	} else {
		tmp = fma((fma(a, a, a) * a), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.9e+146)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	elseif (a <= 3.4e+85)
		tmp = fma(Float64(4.0 * b), b, -1.0);
	else
		tmp = fma(Float64(fma(a, a, a) * a), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.9e+146], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[a, 3.4e+85], N[(N[(4.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999e146

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

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

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

      11. distribute-lft-in N/A

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

      12. unpow2 N/A

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

      13. *-lft-identity N/A

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

      14. distribute-rgt-in N/A

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

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

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 94.7%

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

   if -1.8999999999999999e146 < a < 3.4000000000000003e85

   1. Initial program 93.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 85.1%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 82.7

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

   8. Applied rewrites 82.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 60.7%

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

   if 3.4000000000000003e85 < a

   1. Initial program 61.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

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

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

      11. distribute-lft-in N/A

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

      12. unpow2 N/A

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

      13. *-lft-identity N/A

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

      14. distribute-rgt-in N/A

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

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

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 91.8%

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

Alternative 6: 82.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - -4, a, 4\right) \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e+14)
   (fma (* (fma (- a -4.0) a 4.0) a) a -1.0)
   (fma (* (fma b b 4.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e+14) {
		tmp = fma((fma((a - -4.0), a, 4.0) * a), a, -1.0);
	} else {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e+14)
		tmp = fma(Float64(fma(Float64(a - -4.0), a, 4.0) * a), a, -1.0);
	else
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e+14], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1e14

   1. Initial program 74.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 81.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. distribute-lft-out N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. distribute-rgt-in N/A

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

       8. associate-+r+ N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. lower-fma.f64 N/A

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

   11. Applied rewrites 84.5%

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

   if 1.1e14 < b

   1. Initial program 73.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 98.3%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 93.2

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

   8. Applied rewrites 93.2%

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

Alternative 7: 60.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.4e+150) (fma (* a a) 4.0 -1.0) (fma (* 4.0 b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.4e+150) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = fma((4.0 * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.4e+150)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = fma(Float64(4.0 * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.4e+150], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(4.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.39999999999999991e150

   1. Initial program 76.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-in N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. +-commutative N/A

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

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

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

      11. distribute-lft-in N/A

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

      12. unpow2 N/A

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

      13. *-lft-identity N/A

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

      14. distribute-rgt-in N/A

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

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

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

      16. +-commutative N/A

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

      17. metadata-eval N/A

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

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

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

   8. Applied rewrites 77.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 56.4%

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

   if 8.39999999999999991e150 < b

   1. Initial program 60.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 100.0%

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

   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. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 97.2%

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

Alternative 8: 51.7% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+r+ N/A

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

5. Applied rewrites 85.3%

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

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. metadata-eval N/A

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

   2. pow-sqr N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. unpow2 N/A

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

   15. lower-fma.f64 67.6

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

8. Applied rewrites 67.6%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 48.8%

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

Alternative 9: 25.4% accurate, 160.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 74.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+r+ N/A

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

5. Applied rewrites 85.3%

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

6. Taylor expanded in b around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. unpow2 N/A

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

   6. distribute-lft-in N/A

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

   7. +-commutative N/A

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

   8. associate--l+ N/A

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

   9. +-commutative N/A

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

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

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

   11. distribute-lft-in N/A

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

   12. unpow2 N/A

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

   13. *-lft-identity N/A

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

   14. distribute-rgt-in N/A

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

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

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

   16. +-commutative N/A

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

   17. metadata-eval N/A

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

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

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

8. Applied rewrites 72.8%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 24.6%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))