Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.3% → 81.9%

Time: 9.4s

Alternatives: 7

Speedup: 1.2×

• 61.3% 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: 73.3% 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: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_1 := {a}^{\left(\frac{3}{2}\right)}\\ t_2 := 1 - 3 \cdot a\\ \mathbf{if}\;a \leq 3.1 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a + a, 4 \cdot a, \mathsf{fma}\left(4 \cdot \left(t\_2 \cdot b\right), b, t\_0 \cdot t\_0 - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_1, t\_1, \left(b \cdot b\right) \cdot a\right), a, \mathsf{fma}\left(b, b \cdot t\_0, 4 \cdot \mathsf{fma}\left(t\_2, b \cdot b, \mathsf{fma}\left(a, a, a\right) \cdot a\right) - 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b)))
        (t_1 (pow a (/ 3.0 2.0)))
        (t_2 (- 1.0 (* 3.0 a))))
   (if (<= a 3.1e+48)
     (fma
      (+ (* a a) a)
      (* 4.0 a)
      (fma (* 4.0 (* t_2 b)) b (- (* t_0 t_0) 1.0)))
     (fma
      (fma t_1 t_1 (* (* b b) a))
      a
      (fma b (* b t_0) (- (* 4.0 (fma t_2 (* b b) (* (fma a a a) a))) 1.0))))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double t_1 = pow(a, (3.0 / 2.0));
	double t_2 = 1.0 - (3.0 * a);
	double tmp;
	if (a <= 3.1e+48) {
		tmp = fma(((a * a) + a), (4.0 * a), fma((4.0 * (t_2 * b)), b, ((t_0 * t_0) - 1.0)));
	} else {
		tmp = fma(fma(t_1, t_1, ((b * b) * a)), a, fma(b, (b * t_0), ((4.0 * fma(t_2, (b * b), (fma(a, a, a) * a))) - 1.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	t_1 = a ^ Float64(3.0 / 2.0)
	t_2 = Float64(1.0 - Float64(3.0 * a))
	tmp = 0.0
	if (a <= 3.1e+48)
		tmp = fma(Float64(Float64(a * a) + a), Float64(4.0 * a), fma(Float64(4.0 * Float64(t_2 * b)), b, Float64(Float64(t_0 * t_0) - 1.0)));
	else
		tmp = fma(fma(t_1, t_1, Float64(Float64(b * b) * a)), a, fma(b, Float64(b * t_0), Float64(Float64(4.0 * fma(t_2, Float64(b * b), Float64(fma(a, a, a) * a))) - 1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[a, N[(3.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3.1e+48], N[(N[(N[(a * a), $MachinePrecision] + a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision] + N[(N[(4.0 * N[(t$95$2 * b), $MachinePrecision]), $MachinePrecision] * b + N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$1 + N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a + N[(b * N[(b * t$95$0), $MachinePrecision] + N[(N[(4.0 * N[(t$95$2 * N[(b * b), $MachinePrecision] + N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.10000000000000005e48

   1. Initial program 76.9%

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

   3. Applied rewrites 76.9%

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

   5. Applied rewrites 83.8%

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

   7. Applied rewrites 83.8%

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

   if 3.10000000000000005e48 < a

   1. Initial program 59.6%

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

   3. Applied rewrites 74.6%

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 74.6%

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

Alternative 2: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(t\_0, t\_0, -1\right)\\ t_2 := 1 - 3 \cdot a\\ \mathbf{if}\;b \leq 1.3 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a + a, 4 \cdot a, \mathsf{fma}\left(t\_2, \left(b \cdot b\right) \cdot 4, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \mathsf{fma}\left(t\_2 \cdot \left(b \cdot 4\right), b, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b)))
        (t_1 (fma t_0 t_0 -1.0))
        (t_2 (- 1.0 (* 3.0 a))))
   (if (<= b 1.3e+153)
     (fma (+ (* a a) a) (* 4.0 a) (fma t_2 (* (* b b) 4.0) t_1))
     (fma (fma a a a) (* 4.0 a) (fma (* t_2 (* b 4.0)) b t_1)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double t_1 = fma(t_0, t_0, -1.0);
	double t_2 = 1.0 - (3.0 * a);
	double tmp;
	if (b <= 1.3e+153) {
		tmp = fma(((a * a) + a), (4.0 * a), fma(t_2, ((b * b) * 4.0), t_1));
	} else {
		tmp = fma(fma(a, a, a), (4.0 * a), fma((t_2 * (b * 4.0)), b, t_1));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	t_1 = fma(t_0, t_0, -1.0)
	t_2 = Float64(1.0 - Float64(3.0 * a))
	tmp = 0.0
	if (b <= 1.3e+153)
		tmp = fma(Float64(Float64(a * a) + a), Float64(4.0 * a), fma(t_2, Float64(Float64(b * b) * 4.0), t_1));
	else
		tmp = fma(fma(a, a, a), Float64(4.0 * a), fma(Float64(t_2 * Float64(b * 4.0)), b, t_1));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.3e+153], N[(N[(N[(a * a), $MachinePrecision] + a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision] + N[(t$95$2 * N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(a * a + a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision] + N[(N[(t$95$2 * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] * b + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2999999999999999e153

   1. Initial program 75.1%

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

   3. Applied rewrites 78.7%

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 82.5%

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

   9. Applied rewrites 83.8%

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

   if 1.2999999999999999e153 < b

   1. Initial program 59.5%

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

   3. Applied rewrites 59.6%

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 68.3%

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

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_1 := 1 - 3 \cdot a\\ \mathbf{if}\;a \leq 3 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a + a, 4 \cdot a, \mathsf{fma}\left(4 \cdot \left(t\_1 \cdot b\right), b, t\_0 \cdot t\_0 - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_1, b \cdot b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, \mathsf{fma}\left(t\_0, t\_0, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))) (t_1 (- 1.0 (* 3.0 a))))
   (if (<= a 3e+23)
     (fma
      (+ (* a a) a)
      (* 4.0 a)
      (fma (* 4.0 (* t_1 b)) b (- (* t_0 t_0) 1.0)))
     (fma (fma t_1 (* b b) (* (fma a a a) a)) 4.0 (fma t_0 t_0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double t_1 = 1.0 - (3.0 * a);
	double tmp;
	if (a <= 3e+23) {
		tmp = fma(((a * a) + a), (4.0 * a), fma((4.0 * (t_1 * b)), b, ((t_0 * t_0) - 1.0)));
	} else {
		tmp = fma(fma(t_1, (b * b), (fma(a, a, a) * a)), 4.0, fma(t_0, t_0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	t_1 = Float64(1.0 - Float64(3.0 * a))
	tmp = 0.0
	if (a <= 3e+23)
		tmp = fma(Float64(Float64(a * a) + a), Float64(4.0 * a), fma(Float64(4.0 * Float64(t_1 * b)), b, Float64(Float64(t_0 * t_0) - 1.0)));
	else
		tmp = fma(fma(t_1, Float64(b * b), Float64(fma(a, a, a) * a)), 4.0, fma(t_0, t_0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3e+23], N[(N[(N[(a * a), $MachinePrecision] + a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision] + N[(N[(4.0 * N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision] * b + N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(b * b), $MachinePrecision] + N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.0000000000000001e23

   1. Initial program 77.2%

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

   3. Applied rewrites 77.2%

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 83.5%

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

   if 3.0000000000000001e23 < a

   1. Initial program 60.5%

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

   3. Applied rewrites 74.1%

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 74.1%

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

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(t\_0, t\_0, -1\right)\\ t_2 := 1 - 3 \cdot a\\ \mathbf{if}\;a \leq 3 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), 4 \cdot a, \mathsf{fma}\left(t\_2 \cdot \left(b \cdot 4\right), b, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_2, b \cdot b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b)))
        (t_1 (fma t_0 t_0 -1.0))
        (t_2 (- 1.0 (* 3.0 a))))
   (if (<= a 3e+23)
     (fma (fma a a a) (* 4.0 a) (fma (* t_2 (* b 4.0)) b t_1))
     (fma (fma t_2 (* b b) (* (fma a a a) a)) 4.0 t_1))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double t_1 = fma(t_0, t_0, -1.0);
	double t_2 = 1.0 - (3.0 * a);
	double tmp;
	if (a <= 3e+23) {
		tmp = fma(fma(a, a, a), (4.0 * a), fma((t_2 * (b * 4.0)), b, t_1));
	} else {
		tmp = fma(fma(t_2, (b * b), (fma(a, a, a) * a)), 4.0, t_1);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	t_1 = fma(t_0, t_0, -1.0)
	t_2 = Float64(1.0 - Float64(3.0 * a))
	tmp = 0.0
	if (a <= 3e+23)
		tmp = fma(fma(a, a, a), Float64(4.0 * a), fma(Float64(t_2 * Float64(b * 4.0)), b, t_1));
	else
		tmp = fma(fma(t_2, Float64(b * b), Float64(fma(a, a, a) * a)), 4.0, t_1);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3e+23], N[(N[(a * a + a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision] + N[(N[(t$95$2 * N[(b * 4.0), $MachinePrecision]), $MachinePrecision] * b + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(b * b), $MachinePrecision] + N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 4.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.0000000000000001e23

   1. Initial program 77.2%

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

   3. Applied rewrites 77.2%

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 83.5%

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

   if 3.0000000000000001e23 < a

   1. Initial program 60.5%

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

   3. Applied rewrites 74.1%

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 74.1%

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

Alternative 5: 77.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 3 \cdot a\\ t_1 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, 4 \cdot \mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), a, \left(b \cdot b\right) \cdot t\_0\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_0, b \cdot b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, t\_1 \cdot t\_1\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 3.0 a))) (t_1 (fma a a (* b b))))
   (if (<= a -1e+103)
     (fma t_1 t_1 (- (* 4.0 (fma (fma a a a) a (* (* b b) t_0))) 1.0))
     (- (fma (fma t_0 (* b b) (* (fma a a a) a)) 4.0 (* t_1 t_1)) 1.0))))

C

double code(double a, double b) {
	double t_0 = 1.0 - (3.0 * a);
	double t_1 = fma(a, a, (b * b));
	double tmp;
	if (a <= -1e+103) {
		tmp = fma(t_1, t_1, ((4.0 * fma(fma(a, a, a), a, ((b * b) * t_0))) - 1.0));
	} else {
		tmp = fma(fma(t_0, (b * b), (fma(a, a, a) * a)), 4.0, (t_1 * t_1)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(1.0 - Float64(3.0 * a))
	t_1 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (a <= -1e+103)
		tmp = fma(t_1, t_1, Float64(Float64(4.0 * fma(fma(a, a, a), a, Float64(Float64(b * b) * t_0))) - 1.0));
	else
		tmp = Float64(fma(fma(t_0, Float64(b * b), Float64(fma(a, a, a) * a)), 4.0, Float64(t_1 * t_1)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1e+103], N[(t$95$1 * t$95$1 + N[(N[(4.0 * N[(N[(a * a + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(b * b), $MachinePrecision] + N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1e103

   1. Initial program 0.0%

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

   3. Applied rewrites 0.0%

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

   5. Applied rewrites 8.1%

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

   if -1e103 < a

   1. Initial program 88.2%

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

   3. Applied rewrites 92.0%

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

Alternative 6: 77.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ t_1 := 1 - 3 \cdot a\\ \mathbf{if}\;a \leq -9.4 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right), a, t\_1 \cdot \left(b \cdot b\right)\right), 4, \mathsf{fma}\left(t\_0, t\_0, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_1, b \cdot b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, t\_0 \cdot t\_0\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))) (t_1 (- 1.0 (* 3.0 a))))
   (if (<= a -9.4e+132)
     (fma (fma (fma a a a) a (* t_1 (* b b))) 4.0 (fma t_0 t_0 -1.0))
     (- (fma (fma t_1 (* b b) (* (fma a a a) a)) 4.0 (* t_0 t_0)) 1.0))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double t_1 = 1.0 - (3.0 * a);
	double tmp;
	if (a <= -9.4e+132) {
		tmp = fma(fma(fma(a, a, a), a, (t_1 * (b * b))), 4.0, fma(t_0, t_0, -1.0));
	} else {
		tmp = fma(fma(t_1, (b * b), (fma(a, a, a) * a)), 4.0, (t_0 * t_0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	t_1 = Float64(1.0 - Float64(3.0 * a))
	tmp = 0.0
	if (a <= -9.4e+132)
		tmp = fma(fma(fma(a, a, a), a, Float64(t_1 * Float64(b * b))), 4.0, fma(t_0, t_0, -1.0));
	else
		tmp = Float64(fma(fma(t_1, Float64(b * b), Float64(fma(a, a, a) * a)), 4.0, Float64(t_0 * t_0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9.4e+132], N[(N[(N[(a * a + a), $MachinePrecision] * a + N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * N[(b * b), $MachinePrecision] + N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.400000000000001e132

   1. Initial program 0.0%

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

   3. Applied rewrites 0.0%

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

   5. Applied rewrites 11.3%

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

   7. Applied rewrites 0.0%

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

   9. Applied rewrites 3.6%

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

   if -9.400000000000001e132 < a

   1. Initial program 85.3%

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

   3. Applied rewrites 89.0%

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

Alternative 7: 76.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(1 - 3 \cdot a, b \cdot b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, \mathsf{fma}\left(t\_0, t\_0, -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (fma
    (fma (- 1.0 (* 3.0 a)) (* b b) (* (fma a a a) a))
    4.0
    (fma t_0 t_0 -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	return fma(fma((1.0 - (3.0 * a)), (b * b), (fma(a, a, a) * a)), 4.0, fma(t_0, t_0, -1.0));
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	return fma(fma(Float64(1.0 - Float64(3.0 * a)), Float64(b * b), Float64(fma(a, a, a) * a)), 4.0, fma(t_0, t_0, -1.0))
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.3%

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

3. Applied rewrites 76.4%

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

5. Applied rewrites 80.8%

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

7. Applied rewrites 76.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1 - 3 \cdot a, b \cdot b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025121 
(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))