Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.1% → 99.0%

Time: 4.2s

Alternatives: 8

Speedup: 5.7×

• 60.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a)))) (fma t_0 t_0 (- (* (* b b) 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, (((b * b) * 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(b * b) * 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[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.5%

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

4. Applied rewrites 75.6%

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

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 99.2

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

7. Applied rewrites 99.2%

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

Alternative 2: 96.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5200 \lor \neg \left(a \leq 5 \cdot 10^{-63}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), a \cdot a, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -5200.0) (not (<= a 5e-63)))
   (fma (fma b b (* a a)) (* a a) (- (* (* b b) 4.0) 1.0))
   (- (* (fma b b 4.0) (* b b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -5200.0) || !(a <= 5e-63)) {
		tmp = fma(fma(b, b, (a * a)), (a * a), (((b * b) * 4.0) - 1.0));
	} else {
		tmp = (fma(b, b, 4.0) * (b * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -5200.0) || !(a <= 5e-63))
		tmp = fma(fma(b, b, Float64(a * a)), Float64(a * a), Float64(Float64(Float64(b * b) * 4.0) - 1.0));
	else
		tmp = Float64(Float64(fma(b, b, 4.0) * Float64(b * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -5200.0], N[Not[LessEqual[a, 5e-63]], $MachinePrecision]], 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[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5200 or 5.0000000000000002e-63 < a

   1. Initial program 51.4%

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

   4. Applied rewrites 53.6%

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

   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 98.6

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

   7. Applied rewrites 98.6%

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

   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 97.9

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

   10. Applied rewrites 97.9%

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

   if -5200 < a < 5.0000000000000002e-63

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5200 \lor \neg \left(a \leq 5 \cdot 10^{-63}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), a \cdot a, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;b \leq 2800000000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, b \cdot b, \left(a \cdot a\right) \cdot 4 - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<= b 2800000000.0)
     (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 <= 2800000000.0) {
		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 <= 2800000000.0)
		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, 2800000000.0], 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 < 2.8e9

   1. Initial program 75.3%

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

   4. Applied rewrites 75.8%

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

   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 98.9

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

   7. Applied rewrites 98.9%

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

   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 87.3

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

   10. Applied rewrites 87.3%

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

   if 2.8e9 < b

   1. Initial program 72.0%

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in 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 85.1

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

   7. Applied rewrites 85.1%

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

   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. lower-*.f64 96.9

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

   13. Applied rewrites 96.9%

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

Alternative 4: 98.4% 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 74.5%

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

4. Applied rewrites 75.6%

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

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 82.3

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

7. Applied rewrites 82.3%

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

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 98.9

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

10. Applied rewrites 98.9%

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

Alternative 5: 94.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -340000 \lor \neg \left(a \leq 115000\right):\\ \;\;\;\;\left(4 + a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -340000.0) (not (<= a 115000.0)))
   (* (+ 4.0 a) (* (* a a) a))
   (- (* (fma b b 4.0) (* b b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -340000.0) || !(a <= 115000.0)) {
		tmp = (4.0 + a) * ((a * a) * a);
	} else {
		tmp = (fma(b, b, 4.0) * (b * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -340000.0) || !(a <= 115000.0))
		tmp = Float64(Float64(4.0 + a) * Float64(Float64(a * a) * a));
	else
		tmp = Float64(Float64(fma(b, b, 4.0) * Float64(b * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -340000.0], N[Not[LessEqual[a, 115000.0]], $MachinePrecision]], N[(N[(4.0 + a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4e5 or 115000 < a

   1. Initial program 46.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 95.1

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

   8. Applied rewrites 95.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 95.1

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

   10. Applied rewrites 95.1%

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

   if -3.4e5 < a < 115000

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -340000 \lor \neg \left(a \leq 115000\right):\\ \;\;\;\;\left(4 + a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -270000 \lor \neg \left(a \leq 0.86\right):\\ \;\;\;\;\left(4 + a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -270000.0) (not (<= a 0.86)))
   (* (+ 4.0 a) (* (* a a) a))
   (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -270000.0) || !(a <= 0.86)) {
		tmp = (4.0 + a) * ((a * a) * a);
	} else {
		tmp = ((b * b) * 4.0) - 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 <= (-270000.0d0)) .or. (.not. (a <= 0.86d0))) then
        tmp = (4.0d0 + a) * ((a * a) * a)
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -270000.0) || !(a <= 0.86)) {
		tmp = (4.0 + a) * ((a * a) * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -270000.0) or not (a <= 0.86):
		tmp = (4.0 + a) * ((a * a) * a)
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -270000.0) || !(a <= 0.86))
		tmp = Float64(Float64(4.0 + a) * Float64(Float64(a * a) * a));
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -270000.0) || ~((a <= 0.86)))
		tmp = (4.0 + a) * ((a * a) * a);
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -270000.0], N[Not[LessEqual[a, 0.86]], $MachinePrecision]], N[(N[(4.0 + a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7e5 or 0.859999999999999987 < a

   1. Initial program 46.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 95.1

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

   8. Applied rewrites 95.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 95.1

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

   10. Applied rewrites 95.1%

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

   if -2.7e5 < a < 0.859999999999999987

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 76.2

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

   8. Applied rewrites 76.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -270000 \lor \neg \left(a \leq 0.86\right):\\ \;\;\;\;\left(4 + a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -340000 \lor \neg \left(a \leq 390000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -340000.0) (not (<= a 390000.0)))
   (* (* a a) (* a a))
   (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -340000.0) || !(a <= 390000.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 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 <= (-340000.0d0)) .or. (.not. (a <= 390000.0d0))) then
        tmp = (a * a) * (a * a)
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -340000.0) || !(a <= 390000.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -340000.0) or not (a <= 390000.0):
		tmp = (a * a) * (a * a)
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -340000.0) || !(a <= 390000.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -340000.0) || ~((a <= 390000.0)))
		tmp = (a * a) * (a * a);
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -340000.0], N[Not[LessEqual[a, 390000.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4e5 or 3.9e5 < a

   1. Initial program 46.6%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 94.8

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

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 94.6

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

   7. Applied rewrites 94.6%

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

   if -3.4e5 < a < 3.9e5

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 76.2

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

   8. Applied rewrites 76.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -340000 \lor \neg \left(a \leq 390000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.6% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b) {
	return ((b * b) * 4.0) - 1.0;
}

Python

def code(a, b):
	return ((b * b) * 4.0) - 1.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 74.5%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-pow.f64 66.9

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

5. Applied rewrites 66.9%

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

6. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 51.5

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

8. Applied rewrites 51.5%

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

Reproduce

herbie shell --seed 2025038 
(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))