Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.7% → 98.1%

Time: 8.0s

Alternatives: 8

Speedup: 5.7×

• 61.2% 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: 73.7% 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: 98.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 0.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b\_m, b\_m, a \cdot a\right), b\_m \cdot b\_m, \left(b\_m \cdot b\_m\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 0.0005)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (- (fma (fma b_m b_m (* a a)) (* b_m b_m) (* (* b_m b_m) 4.0)) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 0.0005) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma(fma(b_m, b_m, (a * a)), (b_m * b_m), ((b_m * b_m) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 0.0005)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(fma(fma(b_m, b_m, Float64(a * a)), Float64(b_m * b_m), Float64(Float64(b_m * b_m) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 0.0005], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0000000000000001e-4

   1. Initial program 81.7%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-pow.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 84.1

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

   8. Applied rewrites 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-+.f64 84.1

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

   10. Applied rewrites 84.1%

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

   if 5.0000000000000001e-4 < b

   1. Initial program 73.8%

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 96.6

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

   10. Applied rewrites 96.6%

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

Alternative 2: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b\_m \cdot b\_m\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b\_m \cdot b\_m\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b_m b_m)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b_m b_m) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.5)
   (- (* (* a a) 4.0) 1.0)
   (* (* a a) (* a a))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (((pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((((((a * a) + (b_m * b_m)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b_m * b_m) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.5d0)) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (a * a) * (a * a)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (((Math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if ((math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (a * a) * (a * a)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b_m * b_m)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b_m * b_m) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if ((((((a * a) + (b_m * b_m)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

      2. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      4. pow2 N/A

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

      5. lift-*.f64 96.3

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

   8. Applied rewrites 96.3%

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

   if -0.5 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 71.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 60.9

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

   5. Applied rewrites 60.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

Alternative 3: 50.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b\_m \cdot b\_m\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b\_m \cdot b\_m\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.01:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b_m b_m)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b_m b_m) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.01)
   -1.0
   (* (* a a) 4.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (((pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.01) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((((((a * a) + (b_m * b_m)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b_m * b_m) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.01d0)) then
        tmp = -1.0d0
    else
        tmp = (a * a) * 4.0d0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (((Math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.01) {
		tmp = -1.0;
	} else {
		tmp = (a * a) * 4.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if ((math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.01:
		tmp = -1.0
	else:
		tmp = (a * a) * 4.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b_m * b_m)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b_m * b_m) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.01)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * a) * 4.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if ((((((a * a) + (b_m * b_m)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.01)
		tmp = -1.0;
	else
		tmp = (a * a) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[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[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.01], -1.0, N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.0100000000000000002

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

      \[\leadsto -1 \]

   if -0.0100000000000000002 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 71.4%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-pow.f64 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

      2. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      4. pow2 N/A

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

      5. lift-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

   9. Taylor expanded in a around inf

      \[\leadsto 4 \cdot {a}^{\color{blue}{2}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {a}^{2} \cdot 4 \]

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 36.8

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

   11. Applied rewrites 36.8%

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

Alternative 4: 99.1% accurate, 3.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (let* ((t_0 (fma b_m b_m (* a a))))
   (- (fma t_0 t_0 (* (* b_m b_m) 4.0)) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

4. Applied rewrites 80.3%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 5: 93.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.6 \cdot 10^{+39}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 2.6e+39)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (* (* b_m b_m) (* b_m b_m))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 2.6e+39) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = (b_m * b_m) * (b_m * b_m);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 2.6e+39)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(b_m * b_m) * Float64(b_m * b_m));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 2.6e+39], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e39

   1. Initial program 82.0%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-pow.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 82.1

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

   8. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-+.f64 82.1

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

   10. Applied rewrites 82.1%

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

   if 2.6e39 < b

   1. Initial program 70.8%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.3

         \[\leadsto {b}^{\color{blue}{4}} \]

   5. Applied rewrites 98.3%

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

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{4}} \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 6: 79.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;b\_m \leq 2.6 \cdot 10^{+39}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 3.1e-47)
   (- (* (* a a) 4.0) 1.0)
   (if (<= b_m 2.6e+39) (* (* a a) (* a a)) (* (* b_m b_m) (* b_m b_m)))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 3.1e-47) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b_m <= 2.6e+39) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b_m * b_m) * (b_m * b_m);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b_m <= 3.1d-47) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else if (b_m <= 2.6d+39) then
        tmp = (a * a) * (a * a)
    else
        tmp = (b_m * b_m) * (b_m * b_m)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (b_m <= 3.1e-47) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b_m <= 2.6e+39) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b_m * b_m) * (b_m * b_m);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if b_m <= 3.1e-47:
		tmp = ((a * a) * 4.0) - 1.0
	elif b_m <= 2.6e+39:
		tmp = (a * a) * (a * a)
	else:
		tmp = (b_m * b_m) * (b_m * b_m)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 3.1e-47)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (b_m <= 2.6e+39)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b_m * b_m) * Float64(b_m * b_m));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if (b_m <= 3.1e-47)
		tmp = ((a * a) * 4.0) - 1.0;
	elseif (b_m <= 2.6e+39)
		tmp = (a * a) * (a * a);
	else
		tmp = (b_m * b_m) * (b_m * b_m);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 3.1e-47], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[b$95$m, 2.6e+39], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.0999999999999998e-47

   1. Initial program 81.3%

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

   3. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-pow.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto 4 \cdot {a}^{2} - 1 \]

      2. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      4. pow2 N/A

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

      5. lift-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

   if 3.0999999999999998e-47 < b < 2.6e39

   1. Initial program 86.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if 2.6e39 < b

   1. Initial program 70.8%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.3

         \[\leadsto {b}^{\color{blue}{4}} \]

   5. Applied rewrites 98.3%

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

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{4}} \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 7: 50.7% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in b around 0

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

   1. associate--l+ N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-pow.f64 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

      \[\leadsto 4 \cdot {a}^{2} - 1 \]

   2. *-commutative N/A

      \[\leadsto {a}^{2} \cdot 4 - 1 \]

   3. lower-*.f64 N/A

      \[\leadsto {a}^{2} \cdot 4 - 1 \]

   4. pow2 N/A

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

   5. lift-*.f64 53.0

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

8. Applied rewrites 53.0%

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

Alternative 8: 24.3% accurate, 160.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ -1 \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m) :precision binary64 -1.0)

C

b_m = fabs(b);
double code(double a, double b_m) {
	return -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 = -1.0d0
end function

Java

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

Python

b_m = math.fabs(b)
def code(a, b_m):
	return -1.0

Julia

b_m = abs(b)
function code(a, b_m)
	return -1.0
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m)
	tmp = -1.0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := -1.0

Derivation

1. Initial program 79.6%

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

3. Taylor expanded in b around 0

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

   1. associate--l+ N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-pow.f64 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 27.4%

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

Reproduce

herbie shell --seed 2025045 
(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))