Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.4% → 99.8%

Time: 4.0s

Alternatives: 11

Speedup: 2.4×

• 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: 73.4% 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: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;a \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, \mathsf{fma}\left(12, b \cdot b, t\_0 \cdot t\_0\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<= a 5e+76)
     (- (fma (- 1.0 a) (* (* a a) 4.0) (fma 12.0 (* b b) (* t_0 t_0))) 1.0)
     (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (a <= 5e+76) {
		tmp = fma((1.0 - a), ((a * a) * 4.0), fma(12.0, (b * b), (t_0 * t_0))) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (a <= 5e+76)
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), fma(12.0, Float64(b * b), Float64(t_0 * t_0))) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5e+76], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(12.0 * N[(b * b), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999991e76

   1. Initial program 73.4%

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

   3. Applied rewrites 81.1%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 87.2%

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

   if 4.99999999999999991e76 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.4

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

   6. Applied rewrites 45.4%

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

Alternative 2: 94.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+71}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 21000:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \mathsf{fma}\left(-4, \mathsf{fma}\left(3 \cdot b, b, a \cdot a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+71)
   (* (* a a) (* a a))
   (if (<= a 21000.0)
     (- (* (* b b) (* b b)) (fma -4.0 (fma (* 3.0 b) b (* a a)) 1.0))
     (* (pow a 4.0) (- 1.0 (/ 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+71) {
		tmp = (a * a) * (a * a);
	} else if (a <= 21000.0) {
		tmp = ((b * b) * (b * b)) - fma(-4.0, fma((3.0 * b), b, (a * a)), 1.0);
	} else {
		tmp = pow(a, 4.0) * (1.0 - (4.0 / a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+71)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 21000.0)
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) - fma(-4.0, fma(Float64(3.0 * b), b, Float64(a * a)), 1.0));
	else
		tmp = Float64((a ^ 4.0) * Float64(1.0 - Float64(4.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+71], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 21000.0], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(N[(3.0 * b), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.8000000000000002e71

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.4

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

   6. Applied rewrites 45.4%

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

   if -7.8000000000000002e71 < a < 21000

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-pow.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 65.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 73.5%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 85.9%

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

   if 21000 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 45.8

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

   6. Applied rewrites 45.8%

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

Alternative 3: 93.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+71}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 600:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+71)
   (* (* a a) (* a a))
   (if (<= a 600.0)
     (- (* b (* b (fma b b 12.0))) 1.0)
     (* (pow a 4.0) (- 1.0 (/ 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+71) {
		tmp = (a * a) * (a * a);
	} else if (a <= 600.0) {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	} else {
		tmp = pow(a, 4.0) * (1.0 - (4.0 / a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+71)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 600.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	else
		tmp = Float64((a ^ 4.0) * Float64(1.0 - Float64(4.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+71], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 600.0], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.8000000000000002e71

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.4

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

   6. Applied rewrites 45.4%

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

   if -7.8000000000000002e71 < a < 600

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 69.8

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

   6. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 69.8

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

   8. Applied rewrites 69.8%

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

   if 600 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 45.8

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

   6. Applied rewrites 45.8%

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

Alternative 4: 93.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+71}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 600:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{-4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot \left(-a\right)\right) \cdot \left(-a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+71)
   (* (* a a) (* a a))
   (if (<= a 600.0)
     (- (* b (* b (fma b b 12.0))) 1.0)
     (* (* (* (* (- (/ -4.0 a) -1.0) a) a) (- a)) (- a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+71) {
		tmp = (a * a) * (a * a);
	} else if (a <= 600.0) {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	} else {
		tmp = (((((-4.0 / a) - -1.0) * a) * a) * -a) * -a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+71)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 600.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-4.0 / a) - -1.0) * a) * a) * Float64(-a)) * Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+71], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 600.0], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(-4.0 / a), $MachinePrecision] - -1.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * (-a)), $MachinePrecision] * (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.8000000000000002e71

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.4

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

   6. Applied rewrites 45.4%

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

   if -7.8000000000000002e71 < a < 600

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 69.8

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

   6. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 69.8

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

   8. Applied rewrites 69.8%

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

   if 600 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. lift-/.f64 N/A

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

      23. mult-flip-rev N/A

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

   6. Applied rewrites 26.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-neg.f64 45.8

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

   10. Applied rewrites 45.8%

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

Alternative 5: 93.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+71}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 600:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{-4}{a} - -1\right) \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+71)
   (* (* a a) (* a a))
   (if (<= a 600.0)
     (- (* b (* b (fma b b 12.0))) 1.0)
     (* (* (* (- (/ -4.0 a) -1.0) a) (* a a)) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+71) {
		tmp = (a * a) * (a * a);
	} else if (a <= 600.0) {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	} else {
		tmp = ((((-4.0 / a) - -1.0) * a) * (a * a)) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+71)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 600.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-4.0 / a) - -1.0) * a) * Float64(a * a)) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+71], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 600.0], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(-4.0 / a), $MachinePrecision] - -1.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.8000000000000002e71

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.4

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

   6. Applied rewrites 45.4%

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

   if -7.8000000000000002e71 < a < 600

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 69.8

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

   6. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 69.8

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

   8. Applied rewrites 69.8%

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

   if 600 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. lift-/.f64 N/A

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

      23. mult-flip-rev N/A

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

   6. Applied rewrites 26.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 45.8

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 45.8

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

   10. Applied rewrites 45.8%

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

Alternative 6: 93.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+71}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 600:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a - 4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+71)
   (* (* a a) (* a a))
   (if (<= a 600.0)
     (- (* b (* b (fma b b 12.0))) 1.0)
     (* (* a a) (* a (- a 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+71) {
		tmp = (a * a) * (a * a);
	} else if (a <= 600.0) {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	} else {
		tmp = (a * a) * (a * (a - 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+71)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 600.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * Float64(a - 4.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+71], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 600.0], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.8000000000000002e71

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.5

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

   4. Applied rewrites 45.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.4

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

   6. Applied rewrites 45.4%

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

   if -7.8000000000000002e71 < a < 600

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 69.8

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

   6. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 69.8

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

   8. Applied rewrites 69.8%

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

   if 600 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-lft-in N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. lift-/.f64 N/A

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

      23. mult-flip-rev N/A

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

   6. Applied rewrites 26.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 45.8

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 45.8

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

   11. Applied rewrites 45.8%

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

Alternative 7: 92.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 600:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \mathsf{fma}\left(-4, \mathsf{fma}\left(3 \cdot b, b, \left(\left(1 - a\right) \cdot a\right) \cdot a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4} \cdot \left(1 - \frac{4}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 600.0)
   (-
    (* (* b b) (* b b))
    (fma -4.0 (fma (* 3.0 b) b (* (* (- 1.0 a) a) a)) 1.0))
   (* (pow a 4.0) (- 1.0 (/ 4.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 600.0) {
		tmp = ((b * b) * (b * b)) - fma(-4.0, fma((3.0 * b), b, (((1.0 - a) * a) * a)), 1.0);
	} else {
		tmp = pow(a, 4.0) * (1.0 - (4.0 / a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 600.0)
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) - fma(-4.0, fma(Float64(3.0 * b), b, Float64(Float64(Float64(1.0 - a) * a) * a)), 1.0));
	else
		tmp = Float64((a ^ 4.0) * Float64(1.0 - Float64(4.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 600.0], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(N[(3.0 * b), $MachinePrecision] * b + N[(N[(N[(1.0 - a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 4.0], $MachinePrecision] * N[(1.0 - N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 600

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-pow.f64 64.5

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

   4. Applied rewrites 64.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 65.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 73.5%

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

   if 600 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 45.8

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

   6. Applied rewrites 45.8%

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

Alternative 8: 45.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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) * (a * (a - 4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 45.8

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

4. Applied rewrites 45.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow-prod-down N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. pow-prod-up N/A

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

   16. pow-prod-down N/A

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

   17. lift-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. lift-/.f64 N/A

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

   23. mult-flip-rev N/A

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

6. Applied rewrites 26.1%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 45.8

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

8. Applied rewrites 45.8%

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

9. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. lower--.f64 45.8

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

11. Applied rewrites 45.8%

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

Alternative 9: 45.4% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* (* (* a a) a) a))

C

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

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) * a) * a
end function

Java

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

Python

def code(a, b):
	return ((a * a) * a) * a

Julia

function code(a, b)
	return Float64(Float64(Float64(a * a) * a) * a)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in a around inf

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

   1. lower-pow.f64 45.5

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

4. Applied rewrites 45.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. lift-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 45.4

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

6. Applied rewrites 45.4%

   \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
7. 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. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 45.4

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

8. Applied rewrites 45.4%

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

Alternative 10: 45.4% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* (* a a) (* a a)))

C

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

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) * (a * a)
end function

Java

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

Python

def code(a, b):
	return (a * a) * (a * a)

Julia

function code(a, b)
	return Float64(Float64(a * a) * Float64(a * a))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in a around inf

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

   1. lower-pow.f64 45.5

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

4. Applied rewrites 45.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. lift-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 45.4

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

6. Applied rewrites 45.4%

   \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
7. Add Preprocessing

Alternative 11: 18.3% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 45.8

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

4. Applied rewrites 45.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

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

   4. distribute-lft-in N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow-prod-down N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. pow-prod-up N/A

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

   16. pow-prod-down N/A

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

   17. lift-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. lift-/.f64 N/A

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

   23. mult-flip-rev N/A

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

6. Applied rewrites 26.1%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 45.8

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

8. Applied rewrites 45.8%

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

9. Taylor expanded in a around 0

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

    1. lower-*.f64 18.3

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

11. Applied rewrites 18.3%

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

Reproduce

herbie shell --seed 2025154 
(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))