Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.6% → 99.0%

Time: 4.3s

Alternatives: 9

Speedup: 6.2×

• 60.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(t_0, t_0, Float64(Float64(Float64(b * b) * 4.0) - 1.0))
end

Wolfram

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

Derivation

1. Initial program 74.6%

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

4. Applied rewrites 76.0%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 99.0

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

7. Applied rewrites 99.0%

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

Alternative 2: 68.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	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 * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.5d0)) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (a * a) * (a * a)
    end if
    code = tmp
end function

Java

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

Python

def code(a, b):
	tmp = 0
	if ((math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (a * a) * (a * a)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-pow.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

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

   1. Initial program 65.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 58.8

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

   5. Applied rewrites 58.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 58.7

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

   7. Applied rewrites 58.7%

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

Alternative 3: 84.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5400:\\ \;\;\;\;\mathsf{fma}\left(a, 4, \left(\left(4 + a\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), b \cdot b, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5400.0)
   (- (* (fma a 4.0 (* (* (+ 4.0 a) a) a)) a) 1.0)
   (fma (fma b b (* a a)) (* b b) (- (* (* b b) 4.0) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5400.0) {
		tmp = (fma(a, 4.0, (((4.0 + a) * a) * a)) * a) - 1.0;
	} else {
		tmp = fma(fma(b, b, (a * a)), (b * b), (((b * b) * 4.0) - 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5400.0)
		tmp = Float64(Float64(fma(a, 4.0, Float64(Float64(Float64(4.0 + a) * a) * a)) * a) - 1.0);
	else
		tmp = fma(fma(b, b, Float64(a * a)), Float64(b * b), Float64(Float64(Float64(b * b) * 4.0) - 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5400.0], N[(N[(N[(a * 4.0 + N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5400

   1. Initial program 77.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-pow.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 79.7

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

   8. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. unpow3 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 79.7%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-+.f64 79.7

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

   12. Applied rewrites 79.7%

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

   if 5400 < b

   1. Initial program 65.5%

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

   4. Applied rewrites 68.4%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 97.4

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

   10. Applied rewrites 97.4%

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

Alternative 4: 81.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;b \cdot b \leq 10^{+131}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-6)
   (- (* (* a a) 4.0) 1.0)
   (if (<= (* b b) 1e+131) (* (* a a) (* a a)) (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-6) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if ((b * b) <= 1e+131) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 5d-6) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else if ((b * b) <= 1d+131) then
        tmp = (a * a) * (a * a)
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-6) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if ((b * b) <= 1e+131) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-6:
		tmp = ((a * a) * 4.0) - 1.0
	elif (b * b) <= 1e+131:
		tmp = (a * a) * (a * a)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-6)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (Float64(b * b) <= 1e+131)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e-6)
		tmp = ((a * a) * 4.0) - 1.0;
	elseif ((b * b) <= 1e+131)
		tmp = (a * a) * (a * a);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-6], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 1e+131], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 5.00000000000000041e-6

   1. Initial program 83.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-pow.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 77.2

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

   8. Applied rewrites 77.2%

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

   if 5.00000000000000041e-6 < (*.f64 b b) < 9.9999999999999991e130

   1. Initial program 79.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 44.3

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

   5. Applied rewrites 44.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 44.3

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

   7. Applied rewrites 44.3%

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

   if 9.9999999999999991e130 < (*.f64 b b)

   1. Initial program 61.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 5: 82.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(a, 4, \left(\left(4 + a\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e+65)
   (- (* (fma a 4.0 (* (* (+ 4.0 a) a) a)) a) 1.0)
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e+65) {
		tmp = (fma(a, 4.0, (((4.0 + a) * a) * a)) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e+65)
		tmp = Float64(Float64(fma(a, 4.0, Float64(Float64(Float64(4.0 + a) * a) * a)) * a) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e+65], N[(N[(N[(a * 4.0 + N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e65

   1. Initial program 77.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-pow.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. unpow3 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 77.7%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. distribute-rgt-in N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-+.f64 77.7

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

   12. Applied rewrites 77.7%

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

   if 3.5000000000000001e65 < b

   1. Initial program 62.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 6: 82.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e+65)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e+65) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e+65)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e+65], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e65

   1. Initial program 77.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-pow.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. unpow3 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 77.7%

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

   if 3.5000000000000001e65 < b

   1. Initial program 62.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 7: 82.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e+65)
   (- (* (fma (+ 4.0 a) a 4.0) (* a a)) 1.0)
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e+65) {
		tmp = (fma((4.0 + a), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e+65)
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e+65], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e65

   1. Initial program 77.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-pow.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

   if 3.5000000000000001e65 < b

   1. Initial program 62.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 8: 81.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e+65) (- (* (* (fma a a 4.0) a) a) 1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e+65) {
		tmp = ((fma(a, a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e+65)
		tmp = Float64(Float64(Float64(fma(a, a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e+65], N[(N[(N[(N[(a * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e65

   1. Initial program 77.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-pow.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 77.7

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

   8. Applied rewrites 77.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 76.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 76.7

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

   13. Applied rewrites 76.7%

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

   if 3.5000000000000001e65 < b

   1. Initial program 62.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 9: 51.2% accurate, 11.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Taylor expanded in b around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lower-pow.f64 57.7

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

5. Applied rewrites 57.7%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 51.2

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

8. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2025091 
(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))