Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 3.0s

Alternatives: 13

Speedup: 1.0×

• 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a\_m \cdot a\_m\right), a\_m \cdot a\_m, \mathsf{fma}\left(\mathsf{fma}\left(a\_m, a\_m, \mathsf{fma}\left(b, b, 4\right)\right) \cdot b, b, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1e-33)
   (fma (* b b) 4.0 (- (pow b 4.0) 1.0))
   (fma
    (fma b b (* a_m a_m))
    (* a_m a_m)
    (fma (* (fma a_m a_m (fma b b 4.0)) b) b -1.0))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1e-33) {
		tmp = fma((b * b), 4.0, (pow(b, 4.0) - 1.0));
	} else {
		tmp = fma(fma(b, b, (a_m * a_m)), (a_m * a_m), fma((fma(a_m, a_m, fma(b, b, 4.0)) * b), b, -1.0));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1e-33)
		tmp = fma(Float64(b * b), 4.0, Float64((b ^ 4.0) - 1.0));
	else
		tmp = fma(fma(b, b, Float64(a_m * a_m)), Float64(a_m * a_m), fma(Float64(fma(a_m, a_m, fma(b, b, 4.0)) * b), b, -1.0));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1e-33], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[Power[b, 4.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * b + N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision] + N[(N[(N[(a$95$m * a$95$m + N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.0000000000000001e-33

   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

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

      1. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 69.4

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-pow.f64 69.4

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

   9. Applied rewrites 69.4%

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

   if 1.0000000000000001e-33 < 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. 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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 94.0

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

      4. lift--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. associate--l+ N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. add-flip-rev N/A

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

      12. lower-fma.f64 94.0

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

   7. Applied rewrites 94.0%

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

Alternative 2: 99.9% accurate, 0.9× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (let* ((t_0 (fma b b (* a_m a_m))))
   (- (+ (fma (* 4.0 b) b (* (* t_0 b) b)) (* (* t_0 a_m) a_m)) 1.0)))

C

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

Julia

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := Block[{t$95$0 = N[(b * b + N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(4.0 * b), $MachinePrecision] * b + N[(N[(t$95$0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision] - 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. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-+r+ N/A

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

   9. lower-+.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (- (+ (pow (+ (* a_m a_m) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(N[(N[Power[N[(N[(a$95$m * a$95$m), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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. Add Preprocessing

Alternative 4: 94.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, {b}^{4} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{4}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1.7e+35) (fma (* b b) 4.0 (- (pow b 4.0) 1.0)) (pow a_m 4.0)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.7e+35) {
		tmp = fma((b * b), 4.0, (pow(b, 4.0) - 1.0));
	} else {
		tmp = pow(a_m, 4.0);
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1.7e+35)
		tmp = fma(Float64(b * b), 4.0, Float64((b ^ 4.0) - 1.0));
	else
		tmp = a_m ^ 4.0;
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1.7e+35], N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[Power[b, 4.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[Power[a$95$m, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7000000000000001e35

   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

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

      1. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 69.4

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-pow.f64 69.4

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

   9. Applied rewrites 69.4%

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

   if 1.7000000000000001e35 < 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. lower-pow.f64 46.1

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

   4. Applied rewrites 46.1%

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

Alternative 5: 94.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.7 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, \left(\left(b \cdot b\right) \cdot b\right) \cdot b - 1\right)\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{4}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1.7e+35)
   (fma (* 4.0 b) b (- (* (* (* b b) b) b) 1.0))
   (pow a_m 4.0)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.7e+35) {
		tmp = fma((4.0 * b), b, ((((b * b) * b) * b) - 1.0));
	} else {
		tmp = pow(a_m, 4.0);
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1.7e+35)
		tmp = fma(Float64(4.0 * b), b, Float64(Float64(Float64(Float64(b * b) * b) * b) - 1.0));
	else
		tmp = a_m ^ 4.0;
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1.7e+35], N[(N[(4.0 * b), $MachinePrecision] * b + N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[Power[a$95$m, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7000000000000001e35

   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

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

      1. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 69.4

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.4

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if 1.7000000000000001e35 < 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. lower-pow.f64 46.1

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

   4. Applied rewrites 46.1%

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

Alternative 6: 82.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 95000000000000:\\ \;\;\;\;{a\_m}^{4} - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= b 95000000000000.0) (- (pow a_m 4.0) 1.0) (pow b 4.0)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (b <= 95000000000000.0) {
		tmp = pow(a_m, 4.0) - 1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 95000000000000.0d0) then
        tmp = (a_m ** 4.0d0) - 1.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if (b <= 95000000000000.0) {
		tmp = Math.pow(a_m, 4.0) - 1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if b <= 95000000000000.0:
		tmp = math.pow(a_m, 4.0) - 1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (b <= 95000000000000.0)
		tmp = Float64((a_m ^ 4.0) - 1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if (b <= 95000000000000.0)
		tmp = (a_m ^ 4.0) - 1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[b, 95000000000000.0], N[(N[Power[a$95$m, 4.0], $MachinePrecision] - 1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5e13

   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. lower--.f64 N/A

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

      2. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

   if 9.5e13 < 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. 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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

Alternative 7: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \cdot a\_m \leq 10^{-309}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a\_m \cdot a\_m \leq 2 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a\_m \cdot a\_m \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;{a\_m}^{4}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* a_m a_m) 1e-309)
   (pow b 4.0)
   (if (<= (* a_m a_m) 2e-152)
     -1.0
     (if (<= (* a_m a_m) 5e+65) (pow b 4.0) (pow a_m 4.0)))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 1e-309) {
		tmp = pow(b, 4.0);
	} else if ((a_m * a_m) <= 2e-152) {
		tmp = -1.0;
	} else if ((a_m * a_m) <= 5e+65) {
		tmp = pow(b, 4.0);
	} else {
		tmp = pow(a_m, 4.0);
	}
	return tmp;
}

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a_m * a_m) <= 1d-309) then
        tmp = b ** 4.0d0
    else if ((a_m * a_m) <= 2d-152) then
        tmp = -1.0d0
    else if ((a_m * a_m) <= 5d+65) then
        tmp = b ** 4.0d0
    else
        tmp = a_m ** 4.0d0
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 1e-309) {
		tmp = Math.pow(b, 4.0);
	} else if ((a_m * a_m) <= 2e-152) {
		tmp = -1.0;
	} else if ((a_m * a_m) <= 5e+65) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = Math.pow(a_m, 4.0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (a_m * a_m) <= 1e-309:
		tmp = math.pow(b, 4.0)
	elif (a_m * a_m) <= 2e-152:
		tmp = -1.0
	elif (a_m * a_m) <= 5e+65:
		tmp = math.pow(b, 4.0)
	else:
		tmp = math.pow(a_m, 4.0)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (Float64(a_m * a_m) <= 1e-309)
		tmp = b ^ 4.0;
	elseif (Float64(a_m * a_m) <= 2e-152)
		tmp = -1.0;
	elseif (Float64(a_m * a_m) <= 5e+65)
		tmp = b ^ 4.0;
	else
		tmp = a_m ^ 4.0;
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((a_m * a_m) <= 1e-309)
		tmp = b ^ 4.0;
	elseif ((a_m * a_m) <= 2e-152)
		tmp = -1.0;
	elseif ((a_m * a_m) <= 5e+65)
		tmp = b ^ 4.0;
	else
		tmp = a_m ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 1e-309], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 2e-152], -1.0, If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 5e+65], N[Power[b, 4.0], $MachinePrecision], N[Power[a$95$m, 4.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a a) < 1.000000000000002e-309 or 2.00000000000000013e-152 < (*.f64 a a) < 4.99999999999999973e65

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

   if 1.000000000000002e-309 < (*.f64 a a) < 2.00000000000000013e-152

   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. lower--.f64 N/A

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

      2. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 24.6%

      \[\leadsto -1 \]

   if 4.99999999999999973e65 < (*.f64 a 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. lower-pow.f64 46.1

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

   4. Applied rewrites 46.1%

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

Alternative 8: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \cdot a\_m \leq 10^{-309}:\\ \;\;\;\;{b}^{4}\\ \mathbf{elif}\;a\_m \cdot a\_m \leq 2 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a\_m \cdot a\_m \leq 5 \cdot 10^{+65}:\\ \;\;\;\;{b}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a\_m \cdot a\_m\right) \cdot a\_m\right) \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* a_m a_m) 1e-309)
   (pow b 4.0)
   (if (<= (* a_m a_m) 2e-152)
     -1.0
     (if (<= (* a_m a_m) 5e+65) (pow b 4.0) (* (* (* a_m a_m) a_m) a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 1e-309) {
		tmp = pow(b, 4.0);
	} else if ((a_m * a_m) <= 2e-152) {
		tmp = -1.0;
	} else if ((a_m * a_m) <= 5e+65) {
		tmp = pow(b, 4.0);
	} else {
		tmp = ((a_m * a_m) * a_m) * a_m;
	}
	return tmp;
}

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a_m * a_m) <= 1d-309) then
        tmp = b ** 4.0d0
    else if ((a_m * a_m) <= 2d-152) then
        tmp = -1.0d0
    else if ((a_m * a_m) <= 5d+65) then
        tmp = b ** 4.0d0
    else
        tmp = ((a_m * a_m) * a_m) * a_m
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 1e-309) {
		tmp = Math.pow(b, 4.0);
	} else if ((a_m * a_m) <= 2e-152) {
		tmp = -1.0;
	} else if ((a_m * a_m) <= 5e+65) {
		tmp = Math.pow(b, 4.0);
	} else {
		tmp = ((a_m * a_m) * a_m) * a_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (a_m * a_m) <= 1e-309:
		tmp = math.pow(b, 4.0)
	elif (a_m * a_m) <= 2e-152:
		tmp = -1.0
	elif (a_m * a_m) <= 5e+65:
		tmp = math.pow(b, 4.0)
	else:
		tmp = ((a_m * a_m) * a_m) * a_m
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (Float64(a_m * a_m) <= 1e-309)
		tmp = b ^ 4.0;
	elseif (Float64(a_m * a_m) <= 2e-152)
		tmp = -1.0;
	elseif (Float64(a_m * a_m) <= 5e+65)
		tmp = b ^ 4.0;
	else
		tmp = Float64(Float64(Float64(a_m * a_m) * a_m) * a_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((a_m * a_m) <= 1e-309)
		tmp = b ^ 4.0;
	elseif ((a_m * a_m) <= 2e-152)
		tmp = -1.0;
	elseif ((a_m * a_m) <= 5e+65)
		tmp = b ^ 4.0;
	else
		tmp = ((a_m * a_m) * a_m) * a_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 1e-309], N[Power[b, 4.0], $MachinePrecision], If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 2e-152], -1.0, If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 5e+65], N[Power[b, 4.0], $MachinePrecision], N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 a a) < 1.000000000000002e-309 or 2.00000000000000013e-152 < (*.f64 a a) < 4.99999999999999973e65

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

   if 1.000000000000002e-309 < (*.f64 a a) < 2.00000000000000013e-152

   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. lower--.f64 N/A

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

      2. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 24.6%

      \[\leadsto -1 \]

   if 4.99999999999999973e65 < (*.f64 a 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. lower-pow.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.0

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

   6. Applied rewrites 46.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.0

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

   8. Applied rewrites 46.0%

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

Alternative 9: 70.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \cdot a\_m \leq 10^{-309}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \mathbf{elif}\;a\_m \cdot a\_m \leq 2 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a\_m \cdot a\_m \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a\_m \cdot a\_m\right) \cdot a\_m\right) \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (* a_m a_m) 1e-309)
   (* (* (* b b) b) b)
   (if (<= (* a_m a_m) 2e-152)
     -1.0
     (if (<= (* a_m a_m) 5e+65)
       (* (* b b) (* b b))
       (* (* (* a_m a_m) a_m) a_m)))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 1e-309) {
		tmp = ((b * b) * b) * b;
	} else if ((a_m * a_m) <= 2e-152) {
		tmp = -1.0;
	} else if ((a_m * a_m) <= 5e+65) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = ((a_m * a_m) * a_m) * a_m;
	}
	return tmp;
}

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a_m * a_m) <= 1d-309) then
        tmp = ((b * b) * b) * b
    else if ((a_m * a_m) <= 2d-152) then
        tmp = -1.0d0
    else if ((a_m * a_m) <= 5d+65) then
        tmp = (b * b) * (b * b)
    else
        tmp = ((a_m * a_m) * a_m) * a_m
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if ((a_m * a_m) <= 1e-309) {
		tmp = ((b * b) * b) * b;
	} else if ((a_m * a_m) <= 2e-152) {
		tmp = -1.0;
	} else if ((a_m * a_m) <= 5e+65) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = ((a_m * a_m) * a_m) * a_m;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if (a_m * a_m) <= 1e-309:
		tmp = ((b * b) * b) * b
	elif (a_m * a_m) <= 2e-152:
		tmp = -1.0
	elif (a_m * a_m) <= 5e+65:
		tmp = (b * b) * (b * b)
	else:
		tmp = ((a_m * a_m) * a_m) * a_m
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (Float64(a_m * a_m) <= 1e-309)
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	elseif (Float64(a_m * a_m) <= 2e-152)
		tmp = -1.0;
	elseif (Float64(a_m * a_m) <= 5e+65)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(a_m * a_m) * a_m) * a_m);
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if ((a_m * a_m) <= 1e-309)
		tmp = ((b * b) * b) * b;
	elseif ((a_m * a_m) <= 2e-152)
		tmp = -1.0;
	elseif ((a_m * a_m) <= 5e+65)
		tmp = (b * b) * (b * b);
	else
		tmp = ((a_m * a_m) * a_m) * a_m;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 1e-309], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 2e-152], -1.0, If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 5e+65], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 a a) < 1.000000000000002e-309

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

   if 1.000000000000002e-309 < (*.f64 a a) < 2.00000000000000013e-152

   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. lower--.f64 N/A

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

      2. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 24.6%

      \[\leadsto -1 \]

   if 2.00000000000000013e-152 < (*.f64 a a) < 4.99999999999999973e65

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

   if 4.99999999999999973e65 < (*.f64 a 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. lower-pow.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.0

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

   6. Applied rewrites 46.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.0

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

   8. Applied rewrites 46.0%

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

Alternative 10: 70.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.75 \cdot 10^{-152}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \mathbf{elif}\;a\_m \leq 1.28 \cdot 10^{-76}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a\_m \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a\_m \cdot a\_m\right) \cdot \left(a\_m \cdot a\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1.75e-152)
   (* (* (* b b) b) b)
   (if (<= a_m 1.28e-76)
     -1.0
     (if (<= a_m 1.8e+35) (* (* b b) (* b b)) (* (* a_m a_m) (* a_m a_m))))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.75e-152) {
		tmp = ((b * b) * b) * b;
	} else if (a_m <= 1.28e-76) {
		tmp = -1.0;
	} else if (a_m <= 1.8e+35) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = (a_m * a_m) * (a_m * a_m);
	}
	return tmp;
}

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a_m <= 1.75d-152) then
        tmp = ((b * b) * b) * b
    else if (a_m <= 1.28d-76) then
        tmp = -1.0d0
    else if (a_m <= 1.8d+35) then
        tmp = (b * b) * (b * b)
    else
        tmp = (a_m * a_m) * (a_m * a_m)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.75e-152) {
		tmp = ((b * b) * b) * b;
	} else if (a_m <= 1.28e-76) {
		tmp = -1.0;
	} else if (a_m <= 1.8e+35) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = (a_m * a_m) * (a_m * a_m);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if a_m <= 1.75e-152:
		tmp = ((b * b) * b) * b
	elif a_m <= 1.28e-76:
		tmp = -1.0
	elif a_m <= 1.8e+35:
		tmp = (b * b) * (b * b)
	else:
		tmp = (a_m * a_m) * (a_m * a_m)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1.75e-152)
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	elseif (a_m <= 1.28e-76)
		tmp = -1.0;
	elseif (a_m <= 1.8e+35)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	else
		tmp = Float64(Float64(a_m * a_m) * Float64(a_m * a_m));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if (a_m <= 1.75e-152)
		tmp = ((b * b) * b) * b;
	elseif (a_m <= 1.28e-76)
		tmp = -1.0;
	elseif (a_m <= 1.8e+35)
		tmp = (b * b) * (b * b);
	else
		tmp = (a_m * a_m) * (a_m * a_m);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1.75e-152], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[a$95$m, 1.28e-76], -1.0, If[LessEqual[a$95$m, 1.8e+35], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < 1.7500000000000001e-152

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

   if 1.7500000000000001e-152 < a < 1.28e-76

   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. lower--.f64 N/A

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

      2. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 24.6%

      \[\leadsto -1 \]

   if 1.28e-76 < a < 1.8e35

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

   if 1.8e35 < 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. lower-pow.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.0

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

   6. Applied rewrites 46.0%

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

Alternative 11: 68.5% accurate, 0.7× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (- (+ (pow (+ (* a_m a_m) (* b b)) 2.0) (* 4.0 (* b b))) 1.0) -1.0)
   -1.0
   (* (* (* b b) b) b)))

C

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

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a_m * a_m) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0) <= (-1.0d0)) then
        tmp = -1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a$95$m * a$95$m), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -1.0], -1.0, N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. lower--.f64 N/A

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

      2. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 24.6%

      \[\leadsto -1 \]

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

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

Alternative 12: 68.5% accurate, 0.7× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= (- (+ (pow (+ (* a_m a_m) (* b b)) 2.0) (* 4.0 (* b b))) 1.0) -1.0)
   -1.0
   (* (* b b) (* b b))))

C

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

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a_m * a_m) + (b * b)) ** 2.0d0) + (4.0d0 * (b * b))) - 1.0d0) <= (-1.0d0)) then
        tmp = -1.0d0
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a$95$m * a$95$m), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -1.0], -1.0, N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. lower--.f64 N/A

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

      2. lower-pow.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 24.6%

      \[\leadsto -1 \]

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

   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. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-flip-reverse N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.7

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

   10. Applied rewrites 45.7%

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

Alternative 13: 24.6% accurate, 36.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ -1 \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 -1.0)

C

a_m = fabs(a);
double code(double a_m, double b) {
	return -1.0;
}

Fortran

a_m =     private
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_m, b)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = -1.0d0
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return -1.0;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return -1.0

Julia

a_m = abs(a)
function code(a_m, b)
	return -1.0
end

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := -1.0

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

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

   1. lower--.f64 N/A

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

   2. lower-pow.f64 69.4

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 24.6%

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

Reproduce

herbie shell --seed 2025162 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))