Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 100.0%

Time: 4.1s

Alternatives: 13

Speedup: 2.8×

• 61.9% 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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. pow-prod-up N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. pow-prod-up N/A

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

   4. metadata-eval N/A

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

   5. associate-+r+ N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 99.9

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

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot a, 2, 4\right)\right) \cdot b, b, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1 \]
8. Add Preprocessing

Alternative 3: 99.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   5. lift-*.f64 N/A

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

   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

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. pow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 4: 99.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   5. lift-*.f64 N/A

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

   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

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. pow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in b around 0

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

9. Applied rewrites 99.6%

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

Alternative 5: 86.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

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

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

5. Applied rewrites 86.3%

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

Alternative 6: 77.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5e+36)
   (fma (* b b) (fma b b 4.0) -1.0)
   (fma (* (* (* a a) 2.0) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5e+36) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = fma((((a * a) * 2.0) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5e+36)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = fma(Float64(Float64(Float64(a * a) * 2.0) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5e+36], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999977e36

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. 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 82.0

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

   5. Applied rewrites 82.0%

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

   if 4.99999999999999977e36 < 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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 74.8

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

   8. Applied rewrites 74.8%

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

Alternative 7: 74.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot 2, b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 5e+36)
   (fma (* b b) (fma b b 4.0) -1.0)
   (fma (* (* a a) 2.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 5e+36) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = fma(((a * a) * 2.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 5e+36)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = fma(Float64(Float64(a * a) * 2.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 5e+36], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * 2.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.99999999999999977e36

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. 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 82.0

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

   5. Applied rewrites 82.0%

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

   if 4.99999999999999977e36 < 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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. metadata-eval N/A

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

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

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 74.8

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

   8. Applied rewrites 74.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(a \cdot a\right) \cdot 2\right) \cdot b\right) \cdot b + -1 \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot 2\right) \cdot \left(b \cdot b\right) + -1 \]

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 61.8

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

   10. Applied rewrites 61.8%

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

Alternative 8: 85.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(b, b, \left(a \cdot a\right) \cdot 2\right) \cdot b, b, -1\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma (* (fma b b (* (* a a) 2.0)) b) b -1.0))

C

double code(double a, double b) {
	return fma((fma(b, b, ((a * a) * 2.0)) * b), b, -1.0);
}

Julia

function code(a, b)
	return fma(Float64(fma(b, b, Float64(Float64(a * a) * 2.0)) * b), b, -1.0)
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

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

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

5. Applied rewrites 86.3%

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

6. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 86.1

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

8. Applied rewrites 86.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, \left(a \cdot a\right) \cdot 2\right) \cdot b, b, -1\right) \]
9. Add Preprocessing

Alternative 9: 69.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. 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 70.4

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

5. Applied rewrites 70.4%

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

Alternative 10: 68.8% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

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

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

5. Applied rewrites 86.3%

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

6. Taylor expanded in b around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 70.2

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

8. Applied rewrites 70.2%

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

Alternative 11: 68.8% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. 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 70.4

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

5. Applied rewrites 70.4%

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

6. Taylor expanded in b around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 70.2

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

8. Applied rewrites 70.2%

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

Alternative 12: 51.4% accurate, 10.9× 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 99.8%

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

3. Taylor expanded in a around 0

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

   1. 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 70.4

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

5. Applied rewrites 70.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 53.9%

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

Alternative 13: 25.0% accurate, 131.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

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

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

5. Applied rewrites 86.3%

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

6. Taylor expanded in b around 0

   \[\leadsto -1 \]
7. Step-by-step derivation

8. Applied rewrites 25.3%

   \[\leadsto -1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))