Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.0% → 98.5%

Time: 5.3s

Alternatives: 10

Speedup: 3.8×

• 61.1% 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: 74.0% 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: 98.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(t_0, t_0, Float64(Float64(Float64(a * a) * 4.0) - 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[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 69.0%

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

4. Applied rewrites 72.2%

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

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 79.6

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

7. Applied rewrites 79.6%

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

8. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 98.5

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

10. Applied rewrites 98.5%

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

Alternative 2: 74.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \lor \neg \left(a \leq 0.0152\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.4) (not (<= a 0.0152)))
   (* (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) a) a)
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.4) || !(a <= 0.0152)) {
		tmp = (fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.4) || !(a <= 0.0152))
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.4], N[Not[LessEqual[a, 0.0152]], $MachinePrecision]], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.39999999999999991 or 0.0152 < a

   1. Initial program 40.9%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 96.3

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

   8. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(4 + a\right) \cdot a + \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot \left(a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(4 + a\right) \cdot a + \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot \left(a \cdot a\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(4 + a\right) \cdot a + \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot \left(a \cdot a\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(4 + a\right) \cdot a + \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot \left(a \cdot a\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(4 + a\right) \cdot a + \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot a\right) \cdot a \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(4 + a\right) \cdot a + \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot a\right) \cdot a \]

   10. Applied rewrites 96.3%

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

   if -3.39999999999999991 < a < 0.0152

   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 inf

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

      1. lower-pow.f64 49.9

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

   5. Applied rewrites 49.9%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 49.8

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

   7. Applied rewrites 49.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \lor \neg \left(a \leq 0.0152\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;b \leq 800000000000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, b \cdot b, \left(a \cdot a\right) \cdot 4 - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<= b 800000000000.0)
     (fma t_0 (* a a) (- (* (* b b) 4.0) 1.0))
     (fma t_0 (* b b) (- (* (* a a) 4.0) 1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (b <= 800000000000.0) {
		tmp = fma(t_0, (a * a), (((b * b) * 4.0) - 1.0));
	} else {
		tmp = fma(t_0, (b * b), (((a * a) * 4.0) - 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (b <= 800000000000.0)
		tmp = fma(t_0, Float64(a * a), Float64(Float64(Float64(b * b) * 4.0) - 1.0));
	else
		tmp = fma(t_0, Float64(b * b), Float64(Float64(Float64(a * a) * 4.0) - 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 800000000000.0], N[(t$95$0 * N[(a * a), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(b * b), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8e11

   1. Initial program 71.2%

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in a around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 85.7

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

   10. Applied rewrites 85.7%

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

   if 8e11 < b

   1. Initial program 62.2%

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

   4. Applied rewrites 65.4%

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

   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 86.8

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

   7. Applied rewrites 86.8%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in a around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 98.3

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

   13. Applied rewrites 98.3%

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

Alternative 4: 86.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 850000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), a \cdot a, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 850000000000.0)
   (fma (fma b b (* a a)) (* a a) (- (* (* b b) 4.0) 1.0))
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 850000000000.0) {
		tmp = fma(fma(b, b, (a * a)), (a * a), (((b * b) * 4.0) - 1.0));
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 850000000000.0)
		tmp = fma(fma(b, b, Float64(a * a)), Float64(a * a), Float64(Float64(Float64(b * b) * 4.0) - 1.0));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 850000000000.0], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5e11

   1. Initial program 71.2%

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in a around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 85.7

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

   10. Applied rewrites 85.7%

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

   if 8.5e11 < b

   1. Initial program 62.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 92.3

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

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 92.2

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

   7. Applied rewrites 92.2%

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

Alternative 5: 72.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -47000:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a\\ \mathbf{elif}\;a \leq 420:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, 4\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -47000.0)
   (* (* (fma a a (fma (* b b) 2.0 4.0)) a) a)
   (if (<= a 420.0) (* (* b b) (* b b)) (* (fma (+ 4.0 a) a 4.0) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -47000.0) {
		tmp = (fma(a, a, fma((b * b), 2.0, 4.0)) * a) * a;
	} else if (a <= 420.0) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = fma((4.0 + a), a, 4.0) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -47000.0)
		tmp = Float64(Float64(fma(a, a, fma(Float64(b * b), 2.0, 4.0)) * a) * a);
	elseif (a <= 420.0)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	else
		tmp = Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -47000.0], N[(N[(N[(a * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 420.0], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -47000

   1. Initial program 32.3%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 95.1

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 93.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

   13. Applied rewrites 93.9%

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

   if -47000 < a < 420

   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 inf

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

      1. lower-pow.f64 50.3

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

   5. Applied rewrites 50.3%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 50.2

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

   7. Applied rewrites 50.2%

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

   if 420 < a

   1. Initial program 51.8%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 97.9

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

   8. Applied rewrites 97.9%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift-+.f64 97.8

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

   11. Applied rewrites 97.8%

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

Alternative 6: 70.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+48} \lor \neg \left(a \leq 480\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.7e+48) (not (<= a 480.0)))
   (* (* a a) (* a a))
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.7e+48) || !(a <= 480.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	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 <= (-2.7d+48)) .or. (.not. (a <= 480.0d0))) then
        tmp = (a * a) * (a * a)
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -2.7e+48) || !(a <= 480.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.7e+48) or not (a <= 480.0):
		tmp = (a * a) * (a * a)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.7e+48) || !(a <= 480.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -2.7e+48) || ~((a <= 480.0)))
		tmp = (a * a) * (a * a);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.7e+48], N[Not[LessEqual[a, 480.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.70000000000000004e48 or 480 < a

   1. Initial program 34.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 97.2

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

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 97.1

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

   7. Applied rewrites 97.1%

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

   if -2.70000000000000004e48 < a < 480

   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 inf

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

      1. lower-pow.f64 50.8

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

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 50.7

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

   7. Applied rewrites 50.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+48} \lor \neg \left(a \leq 480\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 820000000000:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, 4\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 820000000000.0)
   (* (fma (+ 4.0 a) a 4.0) (* a a))
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 820000000000.0) {
		tmp = fma((4.0 + a), a, 4.0) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 820000000000.0)
		tmp = Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 820000000000.0], N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.2e11

   1. Initial program 71.2%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 55.9

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

   8. Applied rewrites 55.9%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift-+.f64 51.3

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

   11. Applied rewrites 51.3%

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

   if 8.2e11 < b

   1. Initial program 62.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 92.3

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

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 92.2

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

   7. Applied rewrites 92.2%

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

Alternative 8: 57.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 820000000000:\\ \;\;\;\;\left(\left(4 + a\right) \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 820000000000.0) (* (* (+ 4.0 a) a) (* a a)) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 820000000000.0) {
		tmp = ((4.0 + a) * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	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 (b <= 820000000000.0d0) then
        tmp = ((4.0d0 + a) * a) * (a * a)
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 820000000000.0) {
		tmp = ((4.0 + a) * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 820000000000.0:
		tmp = ((4.0 + a) * a) * (a * a)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 820000000000.0)
		tmp = Float64(Float64(Float64(4.0 + a) * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 820000000000.0)
		tmp = ((4.0 + a) * a) * (a * a);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 820000000000.0], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.2e11

   1. Initial program 71.2%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 55.9

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

   8. Applied rewrites 55.9%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 51.2

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

   11. Applied rewrites 51.2%

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

   12. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 51.2

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

   14. Applied rewrites 51.2%

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

   if 8.2e11 < b

   1. Initial program 62.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 92.3

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

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 92.2

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

   7. Applied rewrites 92.2%

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

Alternative 9: 45.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot a\right) \cdot \left(a \cdot a\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (* a a) (* a a)))

C

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

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) * (a * a)
end function

Java

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

Python

def code(a, b):
	return (a * a) * (a * a)

Julia

function code(a, b)
	return Float64(Float64(a * a) * Float64(a * a))
end

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Taylor expanded in a around inf

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

   1. lower-pow.f64 48.8

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

5. Applied rewrites 48.8%

   \[\leadsto \color{blue}{{a}^{4}} \]
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 48.7

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

7. Applied rewrites 48.7%

   \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
8. Add Preprocessing

Alternative 10: 19.2% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \left(4 \cdot a\right) \cdot \left(a \cdot a\right) \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return (4.0 * a) * (a * a);
}

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 = (4.0d0 * a) * (a * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Taylor expanded in a around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 54.1%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 56.9

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

8. Applied rewrites 56.9%

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

9. Taylor expanded in a around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-+.f64 N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 49.5

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

11. Applied rewrites 49.5%

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

12. Taylor expanded in a around 0

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

    1. lower-*.f64 19.0

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

14. Applied rewrites 19.0%

    \[\leadsto \left(4 \cdot a\right) \cdot \left(a \cdot a\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025027 
(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))