Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.5% → 99.0%

Time: 3.3s

Alternatives: 14

Speedup: 2.6×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(b \cdot b\right) \cdot 12\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* (* b b) 12.0)) 1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + ((b * b) * 12.0)) - 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) + (b * b)) ** 2.0d0) + ((b * b) * 12.0d0)) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + ((b * b) * 12.0)) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + ((b * b) * 12.0)) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(Float64(b * b) * 12.0)) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + ((b * b) * 12.0)) - 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[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 74.5%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.0

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

4. Applied rewrites 99.0%

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

Alternative 2: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(a - 4, a, \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (+ (fma (- a 4.0) a (* (* b b) 2.0)) 4.0) (* a a))))
   (if (<= a -7.5e+28)
     t_0
     (if (<= a 3.4) (- (fma (* b b) 12.0 (pow b 4.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (fma((a - 4.0), a, ((b * b) * 2.0)) + 4.0) * (a * a);
	double tmp;
	if (a <= -7.5e+28) {
		tmp = t_0;
	} else if (a <= 3.4) {
		tmp = fma((b * b), 12.0, pow(b, 4.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(fma(Float64(a - 4.0), a, Float64(Float64(b * b) * 2.0)) + 4.0) * Float64(a * a))
	tmp = 0.0
	if (a <= -7.5e+28)
		tmp = t_0;
	elseif (a <= 3.4)
		tmp = Float64(fma(Float64(b * b), 12.0, (b ^ 4.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.5e+28], t$95$0, If[LessEqual[a, 3.4], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.4999999999999998e28 or 3.39999999999999991 < a

   1. Initial program 74.5%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if -7.4999999999999998e28 < a < 3.39999999999999991

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot \color{blue}{12}\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      10. pow2 N/A

          \[\leadsto \left({b}^{3} \cdot b + {b}^{2} \cdot 12\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \left({b}^{3} \cdot b + 12 \cdot \color{blue}{{b}^{2}}\right) - 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({b}^{3}, \color{blue}{b}, 12 \cdot {b}^{2}\right) - 1 \]

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      4. pow3 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(\color{blue}{b} \cdot b\right) \cdot 12\right) - 1 \]

      5. pow-plus N/A

         \[\leadsto \left({b}^{\left(3 + 1\right)} + \color{blue}{\left(b \cdot b\right)} \cdot 12\right) - 1 \]

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 69.2

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

   10. Applied rewrites 69.2%

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

Alternative 3: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(a - 4, a, \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (+ (fma (- a 4.0) a (* (* b b) 2.0)) 4.0) (* a a))))
   (if (<= a -7.5e+28)
     t_0
     (if (<= a 3.4) (- (fma (* (* b b) b) b (* (* b b) 12.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (fma((a - 4.0), a, ((b * b) * 2.0)) + 4.0) * (a * a);
	double tmp;
	if (a <= -7.5e+28) {
		tmp = t_0;
	} else if (a <= 3.4) {
		tmp = fma(((b * b) * b), b, ((b * b) * 12.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(fma(Float64(a - 4.0), a, Float64(Float64(b * b) * 2.0)) + 4.0) * Float64(a * a))
	tmp = 0.0
	if (a <= -7.5e+28)
		tmp = t_0;
	elseif (a <= 3.4)
		tmp = Float64(fma(Float64(Float64(b * b) * b), b, Float64(Float64(b * b) * 12.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.5e+28], t$95$0, If[LessEqual[a, 3.4], N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.4999999999999998e28 or 3.39999999999999991 < a

   1. Initial program 74.5%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if -7.4999999999999998e28 < a < 3.39999999999999991

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot \color{blue}{12}\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      10. pow2 N/A

          \[\leadsto \left({b}^{3} \cdot b + {b}^{2} \cdot 12\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \left({b}^{3} \cdot b + 12 \cdot \color{blue}{{b}^{2}}\right) - 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({b}^{3}, \color{blue}{b}, 12 \cdot {b}^{2}\right) - 1 \]

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

Alternative 4: 93.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(a \cdot a, 2, b \cdot b\right) \cdot \left(b \cdot b\right) + \left(b \cdot b\right) \cdot 12\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8e-5)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (- (+ (* (fma (* a a) 2.0 (* b b)) (* b b)) (* (* b b) 12.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8e-5) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = ((fma((a * a), 2.0, (b * b)) * (b * b)) + ((b * b) * 12.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8e-5)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(a * a), 2.0, Float64(b * b)) * Float64(b * b)) + Float64(Float64(b * b) * 12.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8e-5], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0 + N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000065e-5

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
   6. 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. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 69.0

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

   10. Applied rewrites 69.0%

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

   if 8.00000000000000065e-5 < b

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 79.5

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

   7. Applied rewrites 79.5%

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

Alternative 5: 93.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{+61}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 36000000000000:\\ \;\;\;\;\mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.08e+61)
   (pow a 4.0)
   (if (<= a 36000000000000.0)
     (- (* (fma b b 12.0) (* b b)) 1.0)
     (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.08e+61) {
		tmp = pow(a, 4.0);
	} else if (a <= 36000000000000.0) {
		tmp = (fma(b, b, 12.0) * (b * b)) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.08e+61)
		tmp = a ^ 4.0;
	elseif (a <= 36000000000000.0)
		tmp = Float64(Float64(fma(b, b, 12.0) * Float64(b * b)) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.08e+61], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 36000000000000.0], N[(N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.08e61 or 3.6e13 < a

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 45.6

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

   6. Applied rewrites 45.6%

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

   if -1.08e61 < a < 3.6e13

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 69.1

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

   6. Applied rewrites 69.1%

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

Alternative 6: 93.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.08 \cdot 10^{+61}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a}}\\ \mathbf{elif}\;a \leq 36000000000000:\\ \;\;\;\;\mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.08e+61)
   (/ 1.0 (/ 1.0 (* (* (* a a) a) a)))
   (if (<= a 36000000000000.0)
     (- (* (fma b b 12.0) (* b b)) 1.0)
     (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.08e+61) {
		tmp = 1.0 / (1.0 / (((a * a) * a) * a));
	} else if (a <= 36000000000000.0) {
		tmp = (fma(b, b, 12.0) * (b * b)) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.08e+61)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(Float64(a * a) * a) * a)));
	elseif (a <= 36000000000000.0)
		tmp = Float64(Float64(fma(b, b, 12.0) * Float64(b * b)) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.08e+61], N[(1.0 / N[(1.0 / N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 36000000000000.0], N[(N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.08e61

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

         \[\leadsto {a}^{\left(\mathsf{neg}\left(-4\right)\right)} \]

      9. pow-neg N/A

         \[\leadsto \frac{1}{\color{blue}{{a}^{-4}}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{{a}^{\left(\mathsf{neg}\left(4\right)\right)}} \]

      11. pow-flip N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{{a}^{4}}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{{a}^{4}}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{{a}^{4}}}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{1}{{a}^{\left(3 + \color{blue}{1}\right)}}} \]

      15. pow-plus N/A

          \[\leadsto \frac{1}{\frac{1}{{a}^{3} \cdot \color{blue}{a}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{1}{{a}^{3} \cdot \color{blue}{a}}} \]

      17. unpow3 N/A

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

      18. pow2 N/A

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

      19. lower-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 45.6

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

   6. Applied rewrites 45.6%

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

   if -1.08e61 < a < 3.6e13

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 69.1

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

   6. Applied rewrites 69.1%

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

   if 3.6e13 < a

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

Alternative 7: 83.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -1.08 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 36000000000000:\\ \;\;\;\;\mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -1.08e+61)
     t_0
     (if (<= a 36000000000000.0) (- (* (fma b b 12.0) (* b b)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -1.08e+61) {
		tmp = t_0;
	} else if (a <= 36000000000000.0) {
		tmp = (fma(b, b, 12.0) * (b * b)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -1.08e+61)
		tmp = t_0;
	elseif (a <= 36000000000000.0)
		tmp = Float64(Float64(fma(b, b, 12.0) * Float64(b * b)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.08e+61], t$95$0, If[LessEqual[a, 36000000000000.0], N[(N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.08e61 or 3.6e13 < a

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

   if -1.08e61 < a < 3.6e13

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 69.1

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

   6. Applied rewrites 69.1%

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

Alternative 8: 82.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00013:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 12\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00013)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (- (fma (* (* b b) b) b (* (* b b) 12.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00013) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma(((b * b) * b), b, ((b * b) * 12.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00013)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(b * b) * b), b, Float64(Float64(b * b) * 12.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00013], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.29999999999999989e-4

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
   6. 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. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 69.0

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

   10. Applied rewrites 69.0%

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

   if 1.29999999999999989e-4 < b

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot \color{blue}{12}\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      10. pow2 N/A

          \[\leadsto \left({b}^{3} \cdot b + {b}^{2} \cdot 12\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \left({b}^{3} \cdot b + 12 \cdot \color{blue}{{b}^{2}}\right) - 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({b}^{3}, \color{blue}{b}, 12 \cdot {b}^{2}\right) - 1 \]

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

Alternative 9: 82.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00013:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 12, \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00013)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (- (fma (* b b) 12.0 (* (* b b) (* b b))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00013) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma((b * b), 12.0, ((b * b) * (b * b))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00013)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(fma(Float64(b * b), 12.0, Float64(Float64(b * b) * Float64(b * b))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00013], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 12.0 + N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.29999999999999989e-4

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
   6. 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. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 69.0

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

   10. Applied rewrites 69.0%

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

   if 1.29999999999999989e-4 < b

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot \color{blue}{12}\right) - 1 \]

      9. lift-*.f64 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      10. pow2 N/A

          \[\leadsto \left({b}^{3} \cdot b + {b}^{2} \cdot 12\right) - 1 \]

      11. *-commutative N/A

          \[\leadsto \left({b}^{3} \cdot b + 12 \cdot \color{blue}{{b}^{2}}\right) - 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({b}^{3}, \color{blue}{b}, 12 \cdot {b}^{2}\right) - 1 \]

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b + \left(b \cdot b\right) \cdot 12\right) - 1 \]

      4. pow3 N/A

         \[\leadsto \left({b}^{3} \cdot b + \left(\color{blue}{b} \cdot b\right) \cdot 12\right) - 1 \]

      5. pow-plus N/A

         \[\leadsto \left({b}^{\left(3 + 1\right)} + \color{blue}{\left(b \cdot b\right)} \cdot 12\right) - 1 \]

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 69.1

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

   8. Applied rewrites 69.1%

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

Alternative 10: 82.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00013:\\ \;\;\;\;\mathsf{fma}\left(a - 4, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00013)
   (- (* (fma (- a 4.0) a 4.0) (* a a)) 1.0)
   (- (* (fma b b 12.0) (* b b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00013) {
		tmp = (fma((a - 4.0), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = (fma(b, b, 12.0) * (b * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00013)
		tmp = Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(fma(b, b, 12.0) * Float64(b * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00013], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 12.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.29999999999999989e-4

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
   6. 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. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 69.0

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

   10. Applied rewrites 69.0%

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

   if 1.29999999999999989e-4 < b

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-out N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 69.1

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

   6. Applied rewrites 69.1%

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

Alternative 11: 68.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
   6. 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. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 41.7

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

   10. Applied rewrites 41.7%

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

   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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. unpow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      8. lower-*.f64 N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 45.6

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

   6. Applied rewrites 45.6%

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

Alternative 12: 68.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0) <= -0.5)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * 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) * (3.0 + a))))) - 1.0) <= -0.5)
		tmp = ((4.0 * a) * a) - 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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
   6. 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. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 51.0

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

   10. Applied rewrites 51.0%

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

   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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. unpow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      8. lower-*.f64 N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 45.6

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

   6. Applied rewrites 45.6%

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

Alternative 13: 68.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0) <= -0.5)
		tmp = Float64(Float64(Float64(4.0 * a) * a) - 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) * (3.0 + a))))) - 1.0) <= -0.5)
		tmp = ((4.0 * a) * a) - 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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(4.0 * a), $MachinePrecision] * a), $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 3 binary64) a))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 74.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
   6. 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. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 51.0

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

   10. Applied rewrites 51.0%

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

   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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 74.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

Alternative 14: 51.0% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\left(4 \cdot \left({a}^{2} \cdot \left(1 - a\right)\right) + {a}^{4}\right)} - 1 \]
6. 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. metadata-eval N/A

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

   8. pow-plus N/A

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

   9. lower-*.f64 N/A

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

   10. unpow3 N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 56.8

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

7. Applied rewrites 56.8%

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

8. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 51.0

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

10. Applied rewrites 51.0%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))