Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.3% → 99.0%

Time: 6.4s

Alternatives: 14

Speedup: 5.7×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot 12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<=
        (-
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
         1.0)
        50000000.0)
     (fma (* b b) (fma b b 12.0) -1.0)
     (fma t_0 t_0 (* (* b b) 12.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0) <= 50000000.0) {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	} else {
		tmp = fma(t_0, t_0, ((b * b) * 12.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0) <= 50000000.0)
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	else
		tmp = fma(t_0, t_0, Float64(Float64(b * b) * 12.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], 50000000.0], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64)) < 5e7

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + 12 \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot {b}^{2}} + 12 \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5e7 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 69.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 99.3%

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

Alternative 2: 69.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out-- N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.9%

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

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

   1. Initial program 69.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 58.4

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

   5. Applied rewrites 58.4%

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

   7. Applied rewrites 58.4%

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

Alternative 3: 83.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a + -4, 4\right), a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), b \cdot b, \mathsf{fma}\left(b \cdot b, 12, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.6e-6)
   (fma (fma a (+ a -4.0) 4.0) (* a a) -1.0)
   (fma (fma b b (* a a)) (* b b) (fma (* b b) 12.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.6e-6) {
		tmp = fma(fma(a, (a + -4.0), 4.0), (a * a), -1.0);
	} else {
		tmp = fma(fma(b, b, (a * a)), (b * b), fma((b * b), 12.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.6e-6)
		tmp = fma(fma(a, Float64(a + -4.0), 4.0), Float64(a * a), -1.0);
	else
		tmp = fma(fma(b, b, Float64(a * a)), Float64(b * b), fma(Float64(b * b), 12.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.6e-6], N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0 + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e-6

   1. Initial program 78.1%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out-- N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 80.5%

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

   if 7.6000000000000001e-6 < b

   1. Initial program 74.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.9

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

   10. Applied rewrites 96.9%

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

Alternative 4: 99.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval N/A

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 5: 94.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7800000000 \lor \neg \left(a \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7800000000.0) (not (<= a 1.2e+14)))
   (* (* (* a a) a) a)
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7800000000.0) || !(a <= 1.2e+14)) {
		tmp = ((a * a) * a) * a;
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7800000000.0) || !(a <= 1.2e+14))
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7800000000.0], N[Not[LessEqual[a, 1.2e+14]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.8e9 or 1.2e14 < a

   1. Initial program 52.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.4%

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

   if -7.8e9 < a < 1.2e14

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.3%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7800000000 \lor \neg \left(a \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7800000000 \lor \neg \left(a \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7800000000.0) (not (<= a 1.2e+14)))
   (* (* a a) (* a a))
   (fma (* (fma b b 12.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7800000000.0) || !(a <= 1.2e+14)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = fma((fma(b, b, 12.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7800000000.0) || !(a <= 1.2e+14))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = fma(Float64(fma(b, b, 12.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7800000000.0], N[Not[LessEqual[a, 1.2e+14]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.8e9 or 1.2e14 < a

   1. Initial program 52.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.4%

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

   if -7.8e9 < a < 1.2e14

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7800000000 \lor \neg \left(a \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7800000000 \lor \neg \left(a \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7800000000.0) (not (<= a 1.2e+14)))
   (* (* a a) (* a a))
   (fma (* b b) (fma b b 12.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7800000000.0) || !(a <= 1.2e+14)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = fma((b * b), fma(b, b, 12.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7800000000.0) || !(a <= 1.2e+14))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = fma(Float64(b * b), fma(b, b, 12.0), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7800000000.0], N[Not[LessEqual[a, 1.2e+14]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.8e9 or 1.2e14 < a

   1. Initial program 52.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.4%

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

   if -7.8e9 < a < 1.2e14

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

         \[\leadsto \left({b}^{\color{blue}{\left(2 \cdot 2\right)}} + 12 \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      5. pow-sqr N/A

         \[\leadsto \left(\color{blue}{{b}^{2} \cdot {b}^{2}} + 12 \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      6. distribute-rgt-out N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left({b}^{2} + 12\right)} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7800000000 \lor \neg \left(a \leq 1.2 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 12\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.118 \lor \neg \left(a \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(4 \cdot a\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.118) (not (<= a 1.15e+14)))
   (* (* a a) (* a a))
   (fma (* (* 4.0 a) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.118) || !(a <= 1.15e+14)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = fma(((4.0 * a) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.118) || !(a <= 1.15e+14))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = fma(Float64(Float64(4.0 * a) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.118], N[Not[LessEqual[a, 1.15e+14]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * a), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.11799999999999999 or 1.15e14 < a

   1. Initial program 51.9%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 91.2%

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

   if -0.11799999999999999 < a < 1.15e14

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 52.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.118 \lor \neg \left(a \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(4 \cdot a\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 77.8%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out-- N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 80.4%

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

   if 1 < b

   1. Initial program 75.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.1%

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

Alternative 10: 81.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 77.8%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out-- N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.4%

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

   7. Applied rewrites 80.4%

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

   if 1 < b

   1. Initial program 75.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.1%

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

Alternative 11: 54.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e+106) (fma (* a a) 4.0 -1.0) (* (* (* b b) a) 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e+106) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = ((b * b) * a) * 4.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.7e+106)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * a) * 4.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.7e+106], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.69999999999999995e106

   1. Initial program 77.3%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out-- N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 57.8%

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

   if 3.69999999999999995e106 < b

   1. Initial program 76.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 47.5%

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

Alternative 12: 53.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot a\right) \cdot 4\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e+106) (fma (* a a) 4.0 -1.0) (* (* (* b a) 4.0) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e+106) {
		tmp = fma((a * a), 4.0, -1.0);
	} else {
		tmp = ((b * a) * 4.0) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.7e+106)
		tmp = fma(Float64(a * a), 4.0, -1.0);
	else
		tmp = Float64(Float64(Float64(b * a) * 4.0) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.7e+106], N[(N[(a * a), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.69999999999999995e106

   1. Initial program 77.3%

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

   3. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. distribute-lft-out-- N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. distribute-rgt-out N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 57.8%

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

   if 3.69999999999999995e106 < b

   1. Initial program 76.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-in N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 23.1%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 23.1%

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

Alternative 13: 51.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a \cdot a, 4, -1\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in b around 0

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

   1. metadata-eval N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. distribute-lft-out-- N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. distribute-rgt-out N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 68.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 50.6%

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

Alternative 14: 25.1% accurate, 155.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in b around 0

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

   1. metadata-eval N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. distribute-lft-out-- N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. distribute-rgt-out N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

5. Applied rewrites 68.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 24.4%

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

Reproduce

herbie shell --seed 2025017 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))