Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.1% → 99.1%

Time: 2.8s

Alternatives: 6

Speedup: 5.5×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot 12 - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a)))) (fma t_0 t_0 (- (* (* b b) 12.0) 1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	return fma(t_0, t_0, (((b * b) * 12.0) - 1.0));
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(t_0, t_0, Float64(Float64(Float64(b * b) * 12.0) - 1.0))
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.0%

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

4. Applied rewrites 76.8%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 99.1

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

7. Applied rewrites 99.1%

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

Alternative 2: 87.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.4 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), a \cdot a, \left(\left(a \cdot a\right) \cdot 1\right) \cdot 4 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.4e+17)
   (fma (fma b b (* a a)) (* a a) (- (* (* (* a a) 1.0) 4.0) 1.0))
   (- (* (* (fma b b (fma (fma 2.0 a 4.0) a 12.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.4e+17) {
		tmp = fma(fma(b, b, (a * a)), (a * a), ((((a * a) * 1.0) * 4.0) - 1.0));
	} else {
		tmp = ((fma(b, b, fma(fma(2.0, a, 4.0), a, 12.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.4e+17)
		tmp = fma(fma(b, b, Float64(a * a)), Float64(a * a), Float64(Float64(Float64(Float64(a * a) * 1.0) * 4.0) - 1.0));
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(fma(2.0, a, 4.0), a, 12.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.4e+17], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * 1.0), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4e17

   1. Initial program 80.0%

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in b around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in a around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 67.5

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

   10. Applied rewrites 67.5%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 83.3%

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

   if 7.4e17 < b

   1. Initial program 65.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 77.4

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 99.8%

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

   13. Applied rewrites 99.9%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right)\right) \cdot b\right) \cdot b - 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+35} \lor \neg \left(a \leq 2.65 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1e+35) (not (<= a 2.65e+14)))
   (* (* a a) (* a a))
   (- (* (* (fma b b (fma (fma 2.0 a 4.0) a 12.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1e+35) || !(a <= 2.65e+14)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((fma(b, b, fma(fma(2.0, a, 4.0), a, 12.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1e+35) || !(a <= 2.65e+14))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(fma(2.0, a, 4.0), a, 12.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1e+35], N[Not[LessEqual[a, 2.65e+14]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999997e34 or 2.65e14 < a

   1. Initial program 49.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.8

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

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 93.7

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

   7. Applied rewrites 93.7%

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

   if -9.9999999999999997e34 < a < 2.65e14

   1. Initial program 99.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 70.2

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 97.9%

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

   13. Applied rewrites 98.0%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right)\right) \cdot b\right) \cdot b - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+35} \lor \neg \left(a \leq 2.65 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, 12\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+35} \lor \neg \left(a \leq 2.65 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1e+35) (not (<= a 2.65e+14)))
   (* (* a a) (* a a))
   (- (* (fma b b 12.0) (* b b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1e+35) || !(a <= 2.65e+14)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (fma(b, b, 12.0) * (b * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1e+35) || !(a <= 2.65e+14))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(fma(b, b, 12.0) * Float64(b * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1e+35], N[Not[LessEqual[a, 2.65e+14]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999997e34 or 2.65e14 < a

   1. Initial program 49.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.8

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

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 93.7

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

   7. Applied rewrites 93.7%

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

   if -9.9999999999999997e34 < a < 2.65e14

   1. Initial program 99.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 70.2

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 97.9%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. pow2 N/A

          \[\leadsto \left(b \cdot b + 12\right) \cdot \left(b \cdot b\right) - 1 \]

       3. lower-fma.f64 97.9

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

   14. Applied rewrites 97.9%

       \[\leadsto \mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+35} \lor \neg \left(a \leq 2.65 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+34} \lor \neg \left(a \leq 68000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -9.5e+34) (not (<= a 68000000.0)))
   (* (* a a) (* a a))
   (- (* b (* b 12.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -9.5e+34) || !(a <= 68000000.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * (b * 12.0)) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-9.5d+34)) .or. (.not. (a <= 68000000.0d0))) then
        tmp = (a * a) * (a * a)
    else
        tmp = (b * (b * 12.0d0)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -9.5e+34) || !(a <= 68000000.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * (b * 12.0)) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -9.5e+34) or not (a <= 68000000.0):
		tmp = (a * a) * (a * a)
	else:
		tmp = (b * (b * 12.0)) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -9.5e+34) || !(a <= 68000000.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b * Float64(b * 12.0)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -9.5e+34) || ~((a <= 68000000.0)))
		tmp = (a * a) * (a * a);
	else
		tmp = (b * (b * 12.0)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -9.5e+34], N[Not[LessEqual[a, 68000000.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.4999999999999999e34 or 6.8e7 < a

   1. Initial program 49.5%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.8

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

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 93.7

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

   7. Applied rewrites 93.7%

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

   if -9.4999999999999999e34 < a < 6.8e7

   1. Initial program 99.1%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 70.2

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

   8. Applied rewrites 70.2%

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

   9. Taylor expanded in a around 0

      \[\leadsto 12 \cdot {b}^{\color{blue}{2}} - 1 \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {b}^{2} \cdot 12 - 1 \]

       2. pow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

       3. lift-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

       4. lift-*.f64 69.5

          \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

   11. Applied rewrites 69.5%

       \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

       2. lift-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

       3. associate-*l* N/A

          \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

       4. lower-*.f64 N/A

          \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

       5. lower-*.f64 69.5

          \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

   13. Applied rewrites 69.5%

       \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+34} \lor \neg \left(a \leq 68000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.3% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(b \cdot 12\right) - 1 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * (b * 12.0d0)) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (b * (b * 12.0)) - 1.0;
}

Python

def code(a, b):
	return (b * (b * 12.0)) - 1.0

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = (b * (b * 12.0)) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 58.6%

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

6. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. pow2 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-in N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 63.4

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

8. Applied rewrites 63.4%

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

9. Taylor expanded in a around 0

   \[\leadsto 12 \cdot {b}^{\color{blue}{2}} - 1 \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto {b}^{2} \cdot 12 - 1 \]

    2. pow2 N/A

       \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

    3. lift-*.f64 N/A

       \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

    4. lift-*.f64 48.6

       \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

11. Applied rewrites 48.6%

    \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

    2. lift-*.f64 N/A

       \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

    3. associate-*l* N/A

       \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

    4. lower-*.f64 N/A

       \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

    5. lower-*.f64 48.6

       \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

13. Applied rewrites 48.6%

    \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025083 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))