Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.2% → 99.1%

Time: 5.7s

Alternatives: 3

Speedup: N/A×

• 61.7% 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: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\left(\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(2, b\_m \cdot b\_m, a \cdot a\right), {b\_m}^{4}\right) + 4 \cdot \left(b\_m \cdot b\_m\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot b\_m, 4, \mathsf{fma}\left(a \cdot \left(4 + 2 \cdot \left(b\_m \cdot b\_m\right)\right), a, {\left(b\_m \cdot b\_m\right)}^{2}\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 2e+70)
   (-
    (+
     (fma (* a a) (fma 2.0 (* b_m b_m) (* a a)) (pow b_m 4.0))
     (* 4.0 (* b_m b_m)))
    1.0)
   (-
    (fma
     (* b_m b_m)
     4.0
     (fma (* a (+ 4.0 (* 2.0 (* b_m b_m)))) a (pow (* b_m b_m) 2.0)))
    1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 2e+70) {
		tmp = (fma((a * a), fma(2.0, (b_m * b_m), (a * a)), pow(b_m, 4.0)) + (4.0 * (b_m * b_m))) - 1.0;
	} else {
		tmp = fma((b_m * b_m), 4.0, fma((a * (4.0 + (2.0 * (b_m * b_m)))), a, pow((b_m * b_m), 2.0))) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 2e+70)
		tmp = Float64(Float64(fma(Float64(a * a), fma(2.0, Float64(b_m * b_m), Float64(a * a)), (b_m ^ 4.0)) + Float64(4.0 * Float64(b_m * b_m))) - 1.0);
	else
		tmp = Float64(fma(Float64(b_m * b_m), 4.0, fma(Float64(a * Float64(4.0 + Float64(2.0 * Float64(b_m * b_m)))), a, (Float64(b_m * b_m) ^ 2.0))) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 2e+70], N[(N[(N[(N[(a * a), $MachinePrecision] * N[(2.0 * N[(b$95$m * b$95$m), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[b$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * 4.0 + N[(N[(a * N[(4.0 + N[(2.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[Power[N[(b$95$m * b$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.00000000000000015e70

   1. Initial program 70.4%

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

   3. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-pow.f64 96.4

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

   8. Applied rewrites 96.4%

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

   if 2.00000000000000015e70 < b

   1. Initial program 63.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 99.0% accurate, N/A× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b_m b_m)) 2.0) (* 4.0 (* b_m b_m))) 1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 3: 86.4% accurate, N/A× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (-
  (fma
   (* b_m b_m)
   4.0
   (fma (* a (+ 4.0 (* 2.0 (* b_m b_m)))) a (pow (* b_m b_m) 2.0)))
  1.0))

C

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

Julia

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * 4.0 + N[(N[(a * N[(4.0 + N[(2.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[Power[N[(b$95$m * b$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 68.7%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 89.3

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

8. Applied rewrites 89.3%

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

Reproduce

herbie shell --seed 2025064 
(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))