Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.1% → 98.0%

Time: 8.1s

Alternatives: 11

Speedup: 6.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.5 \cdot 10^{+17}:\\ \;\;\;\;\left({\left(a \cdot a + b\_m \cdot b\_m\right)}^{2} + 4 \cdot \left(\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b\_m\right) \cdot b\_m\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot b\_m, \mathsf{fma}\left(b\_m, b\_m, \mathsf{fma}\left(-12, a, 4\right)\right), \left(\mathsf{fma}\left(b\_m \cdot b\_m, 2, 4\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 2.5e+17)
   (-
    (+
     (pow (+ (* a a) (* b_m b_m)) 2.0)
     (* 4.0 (* (* (fma -3.0 a 1.0) b_m) b_m)))
    1.0)
   (-
    (fma
     (* b_m b_m)
     (fma b_m b_m (fma -12.0 a 4.0))
     (* (* (fma (* b_m b_m) 2.0 4.0) a) a))
    1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 2.5e+17) {
		tmp = (pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * ((fma(-3.0, a, 1.0) * b_m) * b_m))) - 1.0;
	} else {
		tmp = fma((b_m * b_m), fma(b_m, b_m, fma(-12.0, a, 4.0)), ((fma((b_m * b_m), 2.0, 4.0) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 2.5e+17)
		tmp = Float64(Float64((Float64(Float64(a * a) + Float64(b_m * b_m)) ^ 2.0) + Float64(4.0 * Float64(Float64(fma(-3.0, a, 1.0) * b_m) * b_m))) - 1.0);
	else
		tmp = Float64(fma(Float64(b_m * b_m), fma(b_m, b_m, fma(-12.0, a, 4.0)), Float64(Float64(fma(Float64(b_m * b_m), 2.0, 4.0) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 2.5e+17], N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(-3.0 * a + 1.0), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5e17

   1. Initial program 77.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot {b}^{2} + {b}^{2}\right)\right) - 1 \]

      3. distribute-lft1-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right) \cdot {b}^{2}\right)}\right) - 1 \]

      4. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot {b}^{2}\right)\right) - 1 \]

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

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

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      11. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      12. metadata-eval N/A

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

      13. lower-fma.f64 92.2

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(-3, a, 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

   5. Applied rewrites 92.2%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b\right) \cdot b\right)}\right) - 1 \]

   if 2.5e17 < b

   1. Initial program 57.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 99.9%

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

Alternative 2: 98.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot b\_m, \mathsf{fma}\left(b\_m, b\_m, \mathsf{fma}\left(-12, a, 4\right)\right), \left(\mathsf{fma}\left(b\_m \cdot b\_m, 2, 4\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 9.2e-7)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (-
    (fma
     (* b_m b_m)
     (fma b_m b_m (fma -12.0 a 4.0))
     (* (* (fma (* b_m b_m) 2.0 4.0) a) a))
    1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 9.2e-7) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma((b_m * b_m), fma(b_m, b_m, fma(-12.0, a, 4.0)), ((fma((b_m * b_m), 2.0, 4.0) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 9.2e-7)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(fma(Float64(b_m * b_m), fma(b_m, b_m, fma(-12.0, a, 4.0)), Float64(Float64(fma(Float64(b_m * b_m), 2.0, 4.0) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 9.2e-7], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.1999999999999998e-7

   1. Initial program 76.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot {b}^{2} + {b}^{2}\right)\right) - 1 \]

      3. distribute-lft1-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right) \cdot {b}^{2}\right)}\right) - 1 \]

      4. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot {b}^{2}\right)\right) - 1 \]

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

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

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      11. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      12. metadata-eval N/A

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

      13. lower-fma.f64 91.9

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(-3, a, 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

   5. Applied rewrites 91.9%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b\right) \cdot b\right)}\right) - 1 \]

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-out N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-+.f64 76.8

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

   8. Applied rewrites 76.8%

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

   if 9.1999999999999998e-7 < b

   1. Initial program 62.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 97.2%

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

Alternative 3: 94.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-5} \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b\_m, b\_m, 4\right) \cdot b\_m\right) \cdot b\_m - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (or (<= a -5.8e-5) (not (<= a 3.3e+25)))
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (- (* (* (fma b_m b_m 4.0) b_m) b_m) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if ((a <= -5.8e-5) || !(a <= 3.3e+25)) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b_m, b_m, 4.0) * b_m) * b_m) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if ((a <= -5.8e-5) || !(a <= 3.3e+25))
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b_m, b_m, 4.0) * b_m) * b_m) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[Or[LessEqual[a, -5.8e-5], N[Not[LessEqual[a, 3.3e+25]], $MachinePrecision]], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b$95$m * b$95$m + 4.0), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8e-5 or 3.3000000000000001e25 < a

   1. Initial program 47.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot {b}^{2} + {b}^{2}\right)\right) - 1 \]

      3. distribute-lft1-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right) \cdot {b}^{2}\right)}\right) - 1 \]

      4. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot {b}^{2}\right)\right) - 1 \]

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

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

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      11. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      12. metadata-eval N/A

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

      13. lower-fma.f64 80.0

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(-3, a, 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

   5. Applied rewrites 80.0%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b\right) \cdot b\right)}\right) - 1 \]

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-out N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-+.f64 93.0

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

   8. Applied rewrites 93.0%

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

   if -5.8e-5 < a < 3.3000000000000001e25

   1. Initial program 95.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{-5} \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -2400 \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b\_m, b\_m, 4\right) \cdot b\_m\right) \cdot b\_m - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (or (<= a -2400.0) (not (<= a 3.3e+25)))
   (- (* (* (* (- a -4.0) a) a) a) 1.0)
   (- (* (* (fma b_m b_m 4.0) b_m) b_m) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if ((a <= -2400.0) || !(a <= 3.3e+25)) {
		tmp = ((((a - -4.0) * a) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b_m, b_m, 4.0) * b_m) * b_m) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if ((a <= -2400.0) || !(a <= 3.3e+25))
		tmp = Float64(Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b_m, b_m, 4.0) * b_m) * b_m) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[Or[LessEqual[a, -2400.0], N[Not[LessEqual[a, 3.3e+25]], $MachinePrecision]], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b$95$m * b$95$m + 4.0), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2400 or 3.3000000000000001e25 < a

   1. Initial program 47.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot {b}^{2} + {b}^{2}\right)\right) - 1 \]

      3. distribute-lft1-in N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right) \cdot {b}^{2}\right)}\right) - 1 \]

      4. +-commutative N/A

         \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot {b}^{2}\right)\right) - 1 \]

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

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

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(1 + \left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      11. +-commutative N/A

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a + 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

      12. metadata-eval N/A

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

      13. lower-fma.f64 80.2

          \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\color{blue}{\mathsf{fma}\left(-3, a, 1\right)} \cdot b\right) \cdot b\right)\right) - 1 \]

   5. Applied rewrites 80.2%

      \[\leadsto \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b\right) \cdot b\right)}\right) - 1 \]

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. distribute-lft-out N/A

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

      9. associate-+r+ N/A

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

      10. unpow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-+.f64 92.9

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

   8. Applied rewrites 92.9%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 92.5%

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

   if -2400 < a < 3.3000000000000001e25

   1. Initial program 95.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2400 \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -2400 \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b\_m, b\_m, 4\right) \cdot b\_m\right) \cdot b\_m - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (or (<= a -2400.0) (not (<= a 3.3e+25)))
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (fma b_m b_m 4.0) b_m) b_m) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if ((a <= -2400.0) || !(a <= 3.3e+25)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b_m, b_m, 4.0) * b_m) * b_m) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if ((a <= -2400.0) || !(a <= 3.3e+25))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b_m, b_m, 4.0) * b_m) * b_m) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[Or[LessEqual[a, -2400.0], N[Not[LessEqual[a, 3.3e+25]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b$95$m * b$95$m + 4.0), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2400 or 3.3000000000000001e25 < a

   1. Initial program 47.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.3%

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

   if -2400 < a < 3.3000000000000001e25

   1. Initial program 95.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2400 \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -2400 \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \mathsf{fma}\left(b\_m, b\_m, 4\right) - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (or (<= a -2400.0) (not (<= a 3.3e+25)))
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* b_m b_m) (fma b_m b_m 4.0)) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if ((a <= -2400.0) || !(a <= 3.3e+25)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b_m * b_m) * fma(b_m, b_m, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if ((a <= -2400.0) || !(a <= 3.3e+25))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * fma(b_m, b_m, 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[Or[LessEqual[a, -2400.0], N[Not[LessEqual[a, 3.3e+25]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2400 or 3.3000000000000001e25 < a

   1. Initial program 47.4%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 92.4

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

   5. Applied rewrites 92.4%

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

   7. Applied rewrites 92.3%

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

   if -2400 < a < 3.3000000000000001e25

   1. Initial program 95.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2400 \lor \neg \left(a \leq 3.3 \cdot 10^{+25}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -1600 \lor \neg \left(a \leq -1.35 \cdot 10^{-297}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot a\right) \cdot -12 - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (or (<= a -1600.0) (not (<= a -1.35e-297)))
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (* b_m b_m) a) -12.0) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if ((a <= -1600.0) || !(a <= -1.35e-297)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0;
	}
	return tmp;
}

Fortran

b_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, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if ((a <= (-1600.0d0)) .or. (.not. (a <= (-1.35d-297)))) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = (((b_m * b_m) * a) * (-12.0d0)) - 1.0d0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if ((a <= -1600.0) || !(a <= -1.35e-297)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if (a <= -1600.0) or not (a <= -1.35e-297):
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if ((a <= -1600.0) || !(a <= -1.35e-297))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(b_m * b_m) * a) * -12.0) - 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if ((a <= -1600.0) || ~((a <= -1.35e-297)))
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[Or[LessEqual[a, -1600.0], N[Not[LessEqual[a, -1.35e-297]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * a), $MachinePrecision] * -12.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1600 or -1.3500000000000001e-297 < a

   1. Initial program 64.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 72.5%

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

   if -1600 < a < -1.3500000000000001e-297

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 71.7%

      \[\leadsto \left(\left(b \cdot b\right) \cdot a\right) \cdot \color{blue}{-12} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1600 \lor \neg \left(a \leq -1.35 \cdot 10^{-297}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+145} \lor \neg \left(a \leq -1.35 \cdot 10^{-297}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot a\right) \cdot -12 - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (or (<= a -5.8e+145) (not (<= a -1.35e-297)))
   (- (* (* a a) 4.0) 1.0)
   (- (* (* (* b_m b_m) a) -12.0) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if ((a <= -5.8e+145) || !(a <= -1.35e-297)) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0;
	}
	return tmp;
}

Fortran

b_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, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if ((a <= (-5.8d+145)) .or. (.not. (a <= (-1.35d-297)))) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (((b_m * b_m) * a) * (-12.0d0)) - 1.0d0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if ((a <= -5.8e+145) || !(a <= -1.35e-297)) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if (a <= -5.8e+145) or not (a <= -1.35e-297):
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if ((a <= -5.8e+145) || !(a <= -1.35e-297))
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(b_m * b_m) * a) * -12.0) - 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if ((a <= -5.8e+145) || ~((a <= -1.35e-297)))
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (((b_m * b_m) * a) * -12.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[Or[LessEqual[a, -5.8e+145], N[Not[LessEqual[a, -1.35e-297]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * a), $MachinePrecision] * -12.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8000000000000001e145 or -1.3500000000000001e-297 < a

   1. Initial program 61.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 58.2%

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

   if -5.8000000000000001e145 < a < -1.3500000000000001e-297

   1. Initial program 94.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 61.1%

      \[\leadsto \left(\left(b \cdot b\right) \cdot a\right) \cdot \color{blue}{-12} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+145} \lor \neg \left(a \leq -1.35 \cdot 10^{-297}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot a\right) \cdot -12 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{+28}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.6e+28)
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (* b_m b_m) b_m) b_m) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.6e+28) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	}
	return tmp;
}

Fortran

b_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, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if (b_m <= 1.6d+28) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = (((b_m * b_m) * b_m) * b_m) - 1.0d0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.6e+28) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if b_m <= 1.6e+28:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 1.6e+28)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(b_m * b_m) * b_m) * b_m) - 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if (b_m <= 1.6e+28)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.6e+28], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6e28

   1. Initial program 77.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.4%

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

   if 1.6e28 < b

   1. Initial program 57.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 96.9%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 93.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{+28}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right) - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.6e+28)
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* b_m b_m) (* b_m b_m)) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.6e+28) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b_m * b_m) * (b_m * b_m)) - 1.0;
	}
	return tmp;
}

Fortran

b_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, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if (b_m <= 1.6d+28) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = ((b_m * b_m) * (b_m * b_m)) - 1.0d0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.6e+28) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b_m * b_m) * (b_m * b_m)) - 1.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if b_m <= 1.6e+28:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = ((b_m * b_m) * (b_m * b_m)) - 1.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 1.6e+28)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * Float64(b_m * b_m)) - 1.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if (b_m <= 1.6e+28)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = ((b_m * b_m) * (b_m * b_m)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.6e+28], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6e28

   1. Initial program 77.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 74.4%

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

   if 1.6e28 < b

   1. Initial program 57.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 96.7

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

   5. Applied rewrites 96.7%

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 96.9%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot b - 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 96.9%

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

Alternative 11: 51.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(a \cdot a\right) \cdot 4 - 1 \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m) :precision binary64 (- (* (* a a) 4.0) 1.0))

C

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

Fortran

b_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, b_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    code = ((a * a) * 4.0d0) - 1.0d0
end function

Java

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

Python

b_m = math.fabs(b)
def code(a, b_m):
	return ((a * a) * 4.0) - 1.0

Julia

b_m = abs(b)
function code(a, b_m)
	return Float64(Float64(Float64(a * a) * 4.0) - 1.0)
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m)
	tmp = ((a * a) * 4.0) - 1.0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. *-rgt-identity N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-pow.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 49.8%

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

Reproduce

herbie shell --seed 2024358 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))