Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.5% → 99.9%

Time: 8.9s

Alternatives: 11

Speedup: 5.7×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 99.8%

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

   if +inf.0 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 0.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

Alternative 2: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} + 4 \cdot \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+47)
   (fma (* b b) 4.0 (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0))
   (- (+ (pow (fma b b (* a a)) 2.0) (* 4.0 (* (fma a a a) a))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+47) {
		tmp = fma((b * b), 4.0, (((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0));
	} else {
		tmp = (pow(fma(b, b, (a * a)), 2.0) + (4.0 * (fma(a, a, a) * a))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+47)
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0));
	else
		tmp = Float64(Float64((fma(b, b, Float64(a * a)) ^ 2.0) + Float64(4.0 * Float64(fma(a, a, a) * a))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+47], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5000000000000001e47

   1. Initial program 24.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 24.1

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

   5. Applied rewrites 24.1%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 100.0%

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

   if -1.5000000000000001e47 < a

   1. Initial program 91.3%

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

   3. Taylor expanded in a around inf

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

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. lft-mult-inverse N/A

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

      7. *-lft-identity N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 3: 98.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{if}\;a \leq -380:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, t\_0\right)\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right), \left(\mathsf{fma}\left(4, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right) \cdot a, 4, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)))
   (if (<= a -380.0)
     (fma (* b b) 4.0 t_0)
     (if (<= a 4.3e-7)
       (-
        (fma
         (* b b)
         (fma b b (fma -12.0 a 4.0))
         (* (* (fma 4.0 a (fma (* b b) 2.0 4.0)) a) a))
        1.0)
       (fma (* (fma a a a) a) 4.0 t_0)))))

C

double code(double a, double b) {
	double t_0 = ((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0;
	double tmp;
	if (a <= -380.0) {
		tmp = fma((b * b), 4.0, t_0);
	} else if (a <= 4.3e-7) {
		tmp = fma((b * b), fma(b, b, fma(-12.0, a, 4.0)), ((fma(4.0, a, fma((b * b), 2.0, 4.0)) * a) * a)) - 1.0;
	} else {
		tmp = fma((fma(a, a, a) * a), 4.0, t_0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0)
	tmp = 0.0
	if (a <= -380.0)
		tmp = fma(Float64(b * b), 4.0, t_0);
	elseif (a <= 4.3e-7)
		tmp = Float64(fma(Float64(b * b), fma(b, b, fma(-12.0, a, 4.0)), Float64(Float64(fma(4.0, a, fma(Float64(b * b), 2.0, 4.0)) * a) * a)) - 1.0);
	else
		tmp = fma(Float64(fma(a, a, a) * a), 4.0, t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -380.0], N[(N[(b * b), $MachinePrecision] * 4.0 + t$95$0), $MachinePrecision], If[LessEqual[a, 4.3e-7], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(4.0 * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -380

   1. Initial program 34.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.0

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

   8. Applied rewrites 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 97.0%

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

   if -380 < a < 4.3000000000000001e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 99.8%

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

   if 4.3000000000000001e-7 < a

   1. Initial program 72.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.4

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

   8. Applied rewrites 97.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 97.4%

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

   11. Taylor expanded in a around inf

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. unpow2 N/A

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

       10. lft-mult-inverse N/A

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

       11. +-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. distribute-lft-in N/A

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

       17. *-rgt-identity N/A

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

       18. lower-fma.f64 99.9

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

   13. Applied rewrites 99.9%

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

Alternative 4: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{if}\;a \leq -400:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, t\_0\right)\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right), \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right) \cdot a, 4, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)))
   (if (<= a -400.0)
     (fma (* b b) 4.0 t_0)
     (if (<= a 4.3e-7)
       (-
        (fma
         (* b b)
         (fma b b (fma -12.0 a 4.0))
         (* (* (fma (* b b) 2.0 4.0) a) a))
        1.0)
       (fma (* (fma a a a) a) 4.0 t_0)))))

C

double code(double a, double b) {
	double t_0 = ((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0;
	double tmp;
	if (a <= -400.0) {
		tmp = fma((b * b), 4.0, t_0);
	} else if (a <= 4.3e-7) {
		tmp = fma((b * b), fma(b, b, fma(-12.0, a, 4.0)), ((fma((b * b), 2.0, 4.0) * a) * a)) - 1.0;
	} else {
		tmp = fma((fma(a, a, a) * a), 4.0, t_0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0)
	tmp = 0.0
	if (a <= -400.0)
		tmp = fma(Float64(b * b), 4.0, t_0);
	elseif (a <= 4.3e-7)
		tmp = Float64(fma(Float64(b * b), fma(b, b, fma(-12.0, a, 4.0)), Float64(Float64(fma(Float64(b * b), 2.0, 4.0) * a) * a)) - 1.0);
	else
		tmp = fma(Float64(fma(a, a, a) * a), 4.0, t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -400.0], N[(N[(b * b), $MachinePrecision] * 4.0 + t$95$0), $MachinePrecision], If[LessEqual[a, 4.3e-7], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -400

   1. Initial program 34.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.0

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

   8. Applied rewrites 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 97.0%

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

   if -400 < a < 4.3000000000000001e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 99.4%

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

   if 4.3000000000000001e-7 < a

   1. Initial program 72.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.4

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

   8. Applied rewrites 97.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 97.4%

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

   11. Taylor expanded in a around inf

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. unpow2 N/A

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

       10. lft-mult-inverse N/A

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

       11. +-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. distribute-lft-in N/A

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

       17. *-rgt-identity N/A

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

       18. lower-fma.f64 99.9

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

   13. Applied rewrites 99.9%

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

Alternative 5: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{if}\;a \leq -0.00014:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, t\_0\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, a, a\right) \cdot a, 4, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0)))
   (if (<= a -0.00014)
     (fma (* b b) 4.0 t_0)
     (if (<= a 3.5e-7)
       (- (* (* (fma b b 4.0) b) b) 1.0)
       (fma (* (fma a a a) a) 4.0 t_0)))))

C

double code(double a, double b) {
	double t_0 = ((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0;
	double tmp;
	if (a <= -0.00014) {
		tmp = fma((b * b), 4.0, t_0);
	} else if (a <= 3.5e-7) {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	} else {
		tmp = fma((fma(a, a, a) * a), 4.0, t_0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0)
	tmp = 0.0
	if (a <= -0.00014)
		tmp = fma(Float64(b * b), 4.0, t_0);
	elseif (a <= 3.5e-7)
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	else
		tmp = fma(Float64(fma(a, a, a) * a), 4.0, t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -0.00014], N[(N[(b * b), $MachinePrecision] * 4.0 + t$95$0), $MachinePrecision], If[LessEqual[a, 3.5e-7], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3999999999999999e-4

   1. Initial program 36.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 34.8

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 95.7%

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

   if -1.3999999999999999e-4 < a < 3.49999999999999984e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   if 3.49999999999999984e-7 < a

   1. Initial program 72.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.4

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

   8. Applied rewrites 97.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 97.4%

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

   11. Taylor expanded in a around inf

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

       1. distribute-rgt-in N/A

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

       2. *-lft-identity N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. unpow2 N/A

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

       10. lft-mult-inverse N/A

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

       11. +-commutative N/A

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

       12. associate-*r* N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. +-commutative N/A

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

       16. distribute-lft-in N/A

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

       17. *-rgt-identity N/A

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

       18. lower-fma.f64 99.9

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

   13. Applied rewrites 99.9%

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

Alternative 6: 97.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00014 \lor \neg \left(a \leq 4.3 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -0.00014) (not (<= a 4.3e-7)))
   (fma (* b b) 4.0 (- (* (* (fma (* b b) 2.0 (* a a)) a) a) 1.0))
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -0.00014) || !(a <= 4.3e-7)) {
		tmp = fma((b * b), 4.0, (((fma((b * b), 2.0, (a * a)) * a) * a) - 1.0));
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -0.00014) || !(a <= 4.3e-7))
		tmp = fma(Float64(b * b), 4.0, Float64(Float64(Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * a) * a) - 1.0));
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -0.00014], N[Not[LessEqual[a, 4.3e-7]], $MachinePrecision]], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3999999999999999e-4 or 4.3000000000000001e-7 < a

   1. Initial program 53.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.5

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

   8. Applied rewrites 96.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

   10. Applied rewrites 96.5%

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

   if -1.3999999999999999e-4 < a < 4.3000000000000001e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00014 \lor \neg \left(a \leq 4.3 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot a\right) \cdot a - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+32} \lor \neg \left(a \leq 22500\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7.2e+32) (not (<= a 22500.0)))
   (* (* a a) (* a a))
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7.2e+32) || !(a <= 22500.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7.2e+32) || !(a <= 22500.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7.2e+32], N[Not[LessEqual[a, 22500.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.1999999999999994e32 or 22500 < a

   1. Initial program 49.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 94.7

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

   8. Applied rewrites 94.7%

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

   10. Applied rewrites 94.6%

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

   if -7.1999999999999994e32 < a < 22500

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   7. Applied rewrites 96.2%

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

   9. Applied rewrites 96.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+32} \lor \neg \left(a \leq 22500\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+32} \lor \neg \left(a \leq 22500\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -7.2e+32) (not (<= a 22500.0)))
   (* (* a a) (* a a))
   (- (* (* b b) (fma b b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -7.2e+32) || !(a <= 22500.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -7.2e+32) || !(a <= 22500.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -7.2e+32], N[Not[LessEqual[a, 22500.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.1999999999999994e32 or 22500 < a

   1. Initial program 49.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 94.7

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

   8. Applied rewrites 94.7%

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

   10. Applied rewrites 94.6%

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

   if -7.1999999999999994e32 < a < 22500

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   7. Applied rewrites 96.2%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+32} \lor \neg \left(a \leq 22500\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.38 \cdot 10^{+32} \lor \neg \left(a \leq 21500\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.38e+32) (not (<= a 21500.0)))
   (* (* a a) (* a a))
   (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.38e+32) || !(a <= 21500.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-1.38d+32)) .or. (.not. (a <= 21500.0d0))) then
        tmp = (a * a) * (a * a)
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -1.38e+32) || !(a <= 21500.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -1.38e+32) or not (a <= 21500.0):
		tmp = (a * a) * (a * a)
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.38e+32) || !(a <= 21500.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -1.38e+32) || ~((a <= 21500.0)))
		tmp = (a * a) * (a * a);
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.38e+32], N[Not[LessEqual[a, 21500.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.38e32 or 21500 < a

   1. Initial program 49.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 68.4%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 94.7

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

   8. Applied rewrites 94.7%

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

   10. Applied rewrites 94.6%

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

   if -1.38e32 < a < 21500

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 96.3

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 72.2%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.38 \cdot 10^{+32} \lor \neg \left(a \leq 21500\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+134}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.6e+134) (- (* (* a a) 4.0) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.6e+134) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 6.6d+134) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 6.6e+134) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 6.6e+134:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.6e+134)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 6.6e+134)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.6e+134], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.6e134

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 54.7%

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

   if 6.6e134 < b

   1. Initial program 68.6%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.4%

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

Alternative 11: 51.3% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.0%

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

3. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. pow-plus N/A

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

   8. metadata-eval N/A

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

   9. cube-unmult N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. cube-unmult N/A

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

   16. metadata-eval N/A

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

   17. pow-plus N/A

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

   18. associate-*r* N/A

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

   19. unpow2 N/A

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

   20. pow-sqr N/A

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

   21. metadata-eval N/A

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

   22. lower-fma.f64 N/A

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

   23. unpow2 N/A

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

   24. lower-*.f64 N/A

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

   25. lower-pow.f64 65.5

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

5. Applied rewrites 65.5%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 48.3%

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

Reproduce

herbie shell --seed 2024363 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))