Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.6% → 99.1%

Time: 4.2s

Alternatives: 7

Speedup: 3.1×

• 39.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 74.6% accurate, 1.0× speedup

Math

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

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: 99.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (let* ((t_0 (fma b b (* a a)))) (fma t_0 t_0 (- (* 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, ((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, Float64(Float64(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[(N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.6%

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

   1. lift-pow.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. sum-square-pow N/A

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

   4. associate-+l+ N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fma.f64 71.6%

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

3. Applied rewrites 71.6%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 93.3%

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

6. Applied rewrites 93.3%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

8. Applied rewrites 99.1%

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

Alternative 2: 93.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;{a}^{4} \cdot \frac{4 + a}{a} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \left|b\right| \cdot \left|b\right|, {\left(\left|b\right|\right)}^{4}\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= (fabs b) 3.6e+24)
  (- (* (pow a 4.0) (/ (+ 4.0 a) a)) 1.0)
  (- (fma 4.0 (* (fabs b) (fabs b)) (pow (fabs b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 3.6e+24) {
		tmp = (pow(a, 4.0) * ((4.0 + a) / a)) - 1.0;
	} else {
		tmp = fma(4.0, (fabs(b) * fabs(b)), pow(fabs(b), 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 3.6e+24)
		tmp = Float64(Float64((a ^ 4.0) * Float64(Float64(4.0 + a) / a)) - 1.0);
	else
		tmp = Float64(fma(4.0, Float64(abs(b) * abs(b)), (abs(b) ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 3.6e+24], N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[(4.0 + a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5999999999999998e24

   1. Initial program 74.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

         \[\leadsto {a}^{4} \cdot \frac{4 + a}{a} - 1 \]

      2. lower-+.f64 69.4%

         \[\leadsto {a}^{4} \cdot \frac{4 + a}{a} - 1 \]

   7. Applied rewrites 69.4%

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

   if 3.5999999999999998e24 < b

   1. Initial program 74.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

Alternative 3: 93.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \left|b\right| \cdot \left|b\right|, {\left(\left|b\right|\right)}^{4}\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= (fabs b) 3.6e+24)
  (- (* (* (* (- a -4.0) a) a) a) 1.0)
  (- (fma 4.0 (* (fabs b) (fabs b)) (pow (fabs b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 3.6e+24) {
		tmp = ((((a - -4.0) * a) * a) * a) - 1.0;
	} else {
		tmp = fma(4.0, (fabs(b) * fabs(b)), pow(fabs(b), 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 3.6e+24)
		tmp = Float64(Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a) - 1.0);
	else
		tmp = Float64(fma(4.0, Float64(abs(b) * abs(b)), (abs(b) ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 3.6e+24], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5999999999999998e24

   1. Initial program 74.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-add N/A

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

      6. unpow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

          \[\leadsto \left(4 \cdot \frac{1}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      17. mult-flip-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      24. lower-*.f64 69.4%

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot a - 1 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot a - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      8. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-unsound-/.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

      15. lower-unsound-*.f64 N/A

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

      16. sum-to-mult N/A

          \[\leadsto \left(\left(a + 4\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      17. +-commutative N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      18. lift-+.f64 N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   8. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 69.4%

         \[\leadsto \left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1 \]

   10. Applied rewrites 69.4%

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

   if 3.5999999999999998e24 < b

   1. Initial program 74.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

Alternative 4: 93.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(\left|b\right| \cdot \mathsf{fma}\left(\left|b\right|, \left|b\right|, 4\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= (fabs b) 3.6e+24)
  (- (* (* (* (- a -4.0) a) a) a) 1.0)
  (- (* (fabs b) (* (fabs b) (fma (fabs b) (fabs b) 4.0))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 3.6e+24) {
		tmp = ((((a - -4.0) * a) * a) * a) - 1.0;
	} else {
		tmp = (fabs(b) * (fabs(b) * fma(fabs(b), fabs(b), 4.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 3.6e+24)
		tmp = Float64(Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(abs(b) * Float64(abs(b) * fma(abs(b), abs(b), 4.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 3.6e+24], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5999999999999998e24

   1. Initial program 74.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-add N/A

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

      6. unpow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

          \[\leadsto \left(4 \cdot \frac{1}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      17. mult-flip-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      24. lower-*.f64 69.4%

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot a - 1 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot a - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      8. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-unsound-/.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

      15. lower-unsound-*.f64 N/A

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

      16. sum-to-mult N/A

          \[\leadsto \left(\left(a + 4\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      17. +-commutative N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      18. lift-+.f64 N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   8. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 69.4%

         \[\leadsto \left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1 \]

   10. Applied rewrites 69.4%

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

   if 3.5999999999999998e24 < b

   1. Initial program 74.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 69.5%

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

      9. lower-fma.f64 69.6%

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

   8. Applied rewrites 69.6%

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

Alternative 5: 79.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{+63}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= a 9e+63)
  (- (* b (* b (fma b b 4.0))) 1.0)
  (- (* (* 4.0 a) (* a a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 9e+63) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 9e+63)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 9e+63], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.0000000000000003e63

   1. Initial program 74.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 69.5%

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

      9. lower-fma.f64 69.6%

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

   8. Applied rewrites 69.6%

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

   if 9.0000000000000003e63 < a

   1. Initial program 74.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-add N/A

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

      6. unpow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

          \[\leadsto \left(4 \cdot \frac{1}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      17. mult-flip-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      24. lower-*.f64 69.4%

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot a - 1 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot a - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      8. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-unsound-/.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

      15. lower-unsound-*.f64 N/A

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

      16. sum-to-mult N/A

          \[\leadsto \left(\left(a + 4\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      17. +-commutative N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      18. lift-+.f64 N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 42.4%

       \[\leadsto \left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 9 \cdot 10^{+63}:\\ \;\;\;\;b \cdot \left(4 \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= a 9e+63)
  (- (* b (* 4.0 b)) 1.0)
  (- (* (* 4.0 a) (* a a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 9e+63) {
		tmp = (b * (4.0 * b)) - 1.0;
	} else {
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= 9d+63) then
        tmp = (b * (4.0d0 * b)) - 1.0d0
    else
        tmp = ((4.0d0 * a) * (a * a)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 9e+63) {
		tmp = (b * (4.0 * b)) - 1.0;
	} else {
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 9e+63:
		tmp = (b * (4.0 * b)) - 1.0
	else:
		tmp = ((4.0 * a) * (a * a)) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 9e+63)
		tmp = Float64(Float64(b * Float64(4.0 * b)) - 1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 9e+63)
		tmp = (b * (4.0 * b)) - 1.0;
	else
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 9e+63], N[(N[(b * N[(4.0 * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.0000000000000003e63

   1. Initial program 74.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 69.5%

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

      9. lower-fma.f64 69.6%

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in b around 0

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

       1. lower-*.f64 51.6%

          \[\leadsto b \cdot \left(4 \cdot b\right) - 1 \]

   11. Applied rewrites 51.6%

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

   if 9.0000000000000003e63 < a

   1. Initial program 74.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-add N/A

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

      6. unpow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

          \[\leadsto \left(4 \cdot \frac{1}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]

      17. mult-flip-rev N/A

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

      18. lower-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

      24. lower-*.f64 69.4%

          \[\leadsto \left(\frac{4}{a} - -1\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1 \]

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot a - 1 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\frac{4}{a} - -1\right) \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot a - 1 \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      8. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\frac{4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-unsound-/.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

      15. lower-unsound-*.f64 N/A

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

      16. sum-to-mult N/A

          \[\leadsto \left(\left(a + 4\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      17. +-commutative N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      18. lift-+.f64 N/A

          \[\leadsto \left(\left(4 + a\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1 \]

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 42.4%

       \[\leadsto \left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.6% accurate, 6.0× speedup

Math

\[b \cdot \left(4 \cdot b\right) - 1 \]

FPCore

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

C

double code(double a, double b) {
	return (b * (4.0 * 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 = (b * (4.0d0 * b)) - 1.0d0
end function

Java

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

Python

def code(a, b):
	return (b * (4.0 * b)) - 1.0

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = (b * (4.0 * b)) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 74.6%

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

2. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-pow.f64 69.6%

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

4. Applied rewrites 69.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow-prod-down N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 69.5%

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

6. Applied rewrites 69.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

      \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 4\right)\right) - 1 \]

   9. lower-fma.f64 69.6%

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

8. Applied rewrites 69.6%

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

9. Taylor expanded in b around 0

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

    1. lower-*.f64 51.6%

       \[\leadsto b \cdot \left(4 \cdot b\right) - 1 \]

11. Applied rewrites 51.6%

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

Reproduce

herbie shell --seed 2025214 
(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))