Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 2.9s

Alternatives: 11

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in b around 0

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate--l+ N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 3: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate--l+ N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 75.7

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

6. Applied rewrites 75.7%

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

7. Taylor expanded in b around 0

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

9. Applied rewrites 99.4%

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

Alternative 4: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 14:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 4\right) \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 2, a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 14.0)
   (fma b (* (fma b b 4.0) b) -1.0)
   (* (fma (* b b) 2.0 (* a a)) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 14.0) {
		tmp = fma(b, (fma(b, b, 4.0) * b), -1.0);
	} else {
		tmp = fma((b * b), 2.0, (a * a)) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 14.0)
		tmp = fma(b, Float64(fma(b, b, 4.0) * b), -1.0);
	else
		tmp = Float64(fma(Float64(b * b), 2.0, Float64(a * a)) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 14.0], N[(b * N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 2.0 + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 14

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lift-fma.f64 69.8

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

   6. Applied rewrites 69.8%

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

   if 14 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow-prod-up N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 46.9

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

Alternative 5: 84.2% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 14.0)
   (fma b (* (fma b b 4.0) b) -1.0)
   (* (* (fma b (+ b b) (* a a)) a) a)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 14.0) {
		tmp = fma(b, (fma(b, b, 4.0) * b), -1.0);
	} else {
		tmp = (fma(b, (b + b), (a * a)) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 14.0)
		tmp = fma(b, Float64(fma(b, b, 4.0) * b), -1.0);
	else
		tmp = Float64(Float64(fma(b, Float64(b + b), Float64(a * a)) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 14.0], N[(b * N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * N[(b + b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 14

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lift-fma.f64 69.8

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

   6. Applied rewrites 69.8%

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

   if 14 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow-prod-up N/A

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

      15. pow2 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 46.9

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot 2 + a \cdot a\right) \cdot \left(a \cdot a\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot 2 + a \cdot a\right) \cdot \left(a \cdot a\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot 2 + a \cdot a\right) \cdot \left(a \cdot a\right) \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\left(b \cdot b\right) \cdot 2 + a \cdot a\right) \cdot a\right) \cdot a \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b\right) \cdot 2 + a \cdot a\right) \cdot a\right) \cdot a \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(b \cdot b\right) \cdot 2 + a \cdot a\right) \cdot a\right) \cdot a \]

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 62.7

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

   9. Applied rewrites 62.7%

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

Alternative 6: 82.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.9e+52) (fma (* a a) (* a a) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.9e+52) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.9e+52)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.9e+52], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9e52

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 1.9e52 < b

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow3 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 46.2

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

   9. Applied rewrites 46.2%

      \[\leadsto {b}^{\color{blue}{4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 82.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.9e+52) (fma (* a a) (* a a) -1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.9e+52) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.9e+52)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.9e+52], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9e52

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 1.9e52 < b

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

Alternative 8: 82.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 4\right) \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.9e+52)
   (fma (* a a) (* a a) -1.0)
   (fma b (* (fma b b 4.0) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.9e+52) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma(b, (fma(b, b, 4.0) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.9e+52)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(b, Float64(fma(b, b, 4.0) * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.9e+52], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(b * N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9e52

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 1.9e52 < b

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*l* N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lift-fma.f64 69.8

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

   6. Applied rewrites 69.8%

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

Alternative 9: 67.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.00385:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{+54}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.00385)
   (fma (* b b) 4.0 -1.0)
   (if (<= a 1.52e+54) (* (* b b) (* b b)) (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.00385) {
		tmp = fma((b * b), 4.0, -1.0);
	} else if (a <= 1.52e+54) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.00385)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	elseif (a <= 1.52e+54)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.00385], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], If[LessEqual[a, 1.52e+54], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 0.0038500000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 51.4%

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

   if 0.0038500000000000001 < a < 1.51999999999999999e54

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 46.2

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

   7. Applied rewrites 46.2%

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

   if 1.51999999999999999e54 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 46.6

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

   4. Applied rewrites 46.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

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

Alternative 10: 66.7% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 14.0) (fma (* b b) 4.0 -1.0) (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 14.0) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 14.0)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 14.0], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 14

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 51.4%

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

   if 14 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 46.6

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

   4. Applied rewrites 46.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

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

Alternative 11: 51.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lower-fma.f64 69.8

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

4. Applied rewrites 69.8%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 51.4%

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

Reproduce

herbie shell --seed 2025136 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))