Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.3% → 96.6%

Time: 8.3s

Alternatives: 12

Speedup: 5.9×

• 61.1% 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: 74.3% 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: 96.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.8 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - -4, a, 4\right), a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot b\_m, 4, \left(\mathsf{fma}\left(a \cdot a, 2, b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m - 1\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.8e+24)
   (fma (fma (- a -4.0) a 4.0) (* a a) -1.0)
   (fma
    (* b_m b_m)
    4.0
    (- (* (* (fma (* a a) 2.0 (* 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 <= 1.8e+24) {
		tmp = fma(fma((a - -4.0), a, 4.0), (a * a), -1.0);
	} else {
		tmp = fma((b_m * b_m), 4.0, (((fma((a * a), 2.0, (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 <= 1.8e+24)
		tmp = fma(fma(Float64(a - -4.0), a, 4.0), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b_m * b_m), 4.0, Float64(Float64(Float64(fma(Float64(a * a), 2.0, Float64(b_m * b_m)) * b_m) * b_m) - 1.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.8e+24], N[(N[(N[(a - -4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * 4.0 + N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0 + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.79999999999999992e24

   1. Initial program 72.8%

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

   3. Applied rewrites 73.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 84.1%

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

   if 1.79999999999999992e24 < b

   1. Initial program 64.6%

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

   3. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b, b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 100.0%

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

Alternative 2: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b\_m \cdot b\_m\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b\_m \cdot b\_m\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b_m b_m)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b_m b_m) (- 1.0 (* 3.0 a))))))
       1.0)
      5.0)
   (fma (* 4.0 a) a -1.0)
   (* (* a a) (* a a))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (((pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= 5.0) {
		tmp = fma((4.0 * a), a, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b_m * b_m)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b_m * b_m) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= 5.0)
		tmp = fma(Float64(4.0 * a), a, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[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[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], 5.0], N[(N[(4.0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < 5

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.2%

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

   if 5 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 62.6%

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

   3. Applied rewrites 66.1%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 63.7%

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

   9. Applied rewrites 63.6%

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

Alternative 3: 51.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b\_m \cdot b\_m\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b\_m \cdot b\_m\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b_m b_m)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b_m b_m) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.5)
   -1.0
   (* (* 4.0 a) a)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (((pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = (4.0 * a) * a;
	}
	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 * a) + (b_m * b_m)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b_m * b_m) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.5d0)) then
        tmp = -1.0d0
    else
        tmp = (4.0d0 * a) * a
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (((Math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = (4.0 * a) * a;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if ((math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5:
		tmp = -1.0
	else:
		tmp = (4.0 * a) * a
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b_m * b_m)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b_m * b_m) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.5)
		tmp = -1.0;
	else
		tmp = Float64(Float64(4.0 * a) * a);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if ((((((a * a) + (b_m * b_m)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5)
		tmp = -1.0;
	else
		tmp = (4.0 * a) * a;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[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[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], -1.0, N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.6%

       \[\leadsto -1 \]

   if -0.5 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 62.8%

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

   3. Applied rewrites 66.2%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 40.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 35.3%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 35.7%

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

Alternative 4: 94.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{+50} \lor \neg \left(a \leq 2 \cdot 10^{-56}\right):\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot b\_m, 4, \mathsf{fma}\left(b\_m, b\_m, a \cdot a\right) \cdot \left(a \cdot a\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b\_m, b\_m, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\_m, b\_m, -1\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (or (<= a -3.05e+50) (not (<= a 2e-56)))
   (fma (* b_m b_m) 4.0 (- (* (fma b_m b_m (* a a)) (* a a)) 1.0))
   (fma (* (fma b_m b_m (fma -12.0 a 4.0)) b_m) b_m -1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if ((a <= -3.05e+50) || !(a <= 2e-56)) {
		tmp = fma((b_m * b_m), 4.0, ((fma(b_m, b_m, (a * a)) * (a * a)) - 1.0));
	} else {
		tmp = fma((fma(b_m, b_m, fma(-12.0, a, 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 <= -3.05e+50) || !(a <= 2e-56))
		tmp = fma(Float64(b_m * b_m), 4.0, Float64(Float64(fma(b_m, b_m, Float64(a * a)) * Float64(a * a)) - 1.0));
	else
		tmp = fma(Float64(fma(b_m, b_m, fma(-12.0, a, 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, -3.05e+50], N[Not[LessEqual[a, 2e-56]], $MachinePrecision]], N[(N[(b$95$m * b$95$m), $MachinePrecision] * 4.0 + N[(N[(N[(b$95$m * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.05000000000000013e50 or 2.0000000000000001e-56 < a

   1. Initial program 47.1%

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

   3. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b, b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 52.0

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

   6. Applied rewrites 52.0%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 98.6%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 97.3%

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

   if -3.05000000000000013e50 < a < 2.0000000000000001e-56

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 96.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{+50} \lor \neg \left(a \leq 2 \cdot 10^{-56}\right):\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \mathsf{fma}\left(b, b, a \cdot a\right) \cdot \left(a \cdot a\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.8 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - -4, a, 4\right), a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_m \cdot b\_m, 4, \mathsf{fma}\left(b\_m, b\_m, a \cdot a\right) \cdot \left(b\_m \cdot b\_m\right) - 1\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.8e+24)
   (fma (fma (- a -4.0) a 4.0) (* a a) -1.0)
   (fma (* b_m b_m) 4.0 (- (* (fma b_m b_m (* a a)) (* b_m b_m)) 1.0))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.8e+24) {
		tmp = fma(fma((a - -4.0), a, 4.0), (a * a), -1.0);
	} else {
		tmp = fma((b_m * b_m), 4.0, ((fma(b_m, b_m, (a * a)) * (b_m * b_m)) - 1.0));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 1.8e+24)
		tmp = fma(fma(Float64(a - -4.0), a, 4.0), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(b_m * b_m), 4.0, Float64(Float64(fma(b_m, b_m, Float64(a * a)) * Float64(b_m * b_m)) - 1.0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.8e+24], N[(N[(N[(a - -4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * 4.0 + N[(N[(N[(b$95$m * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.79999999999999992e24

   1. Initial program 72.8%

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

   3. Applied rewrites 73.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 84.1%

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

   if 1.79999999999999992e24 < b

   1. Initial program 64.6%

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

   3. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b, b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 71.9

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

   6. Applied rewrites 71.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 99.9%

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

Alternative 6: 98.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Applied rewrites 73.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-3, a, 1\right) \cdot b, b, \mathsf{fma}\left(a, a, a\right) \cdot a\right), 4, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 73.4

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

6. Applied rewrites 73.4%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 98.6%

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

Alternative 7: 79.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{elif}\;b\_m \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.6e-45)
   (fma (* 4.0 a) a -1.0)
   (if (<= b_m 4.5e+49) (* (* (* a a) a) a) (* (* (* b_m b_m) b_m) b_m))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.6e-45) {
		tmp = fma((4.0 * a), a, -1.0);
	} else if (b_m <= 4.5e+49) {
		tmp = ((a * a) * a) * a;
	} else {
		tmp = ((b_m * b_m) * b_m) * b_m;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 1.6e-45)
		tmp = fma(Float64(4.0 * a), a, -1.0);
	elseif (b_m <= 4.5e+49)
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * b_m) * b_m);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.6e-45], N[(N[(4.0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[b$95$m, 4.5e+49], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.60000000000000004e-45

   1. Initial program 73.8%

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

   3. Applied rewrites 74.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 58.6%

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

   if 1.60000000000000004e-45 < b < 4.49999999999999982e49

   1. Initial program 59.7%

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

   3. Applied rewrites 63.7%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 63.5%

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

   9. Applied rewrites 63.4%

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

   if 4.49999999999999982e49 < b

   1. Initial program 66.0%

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

   3. Applied rewrites 72.8%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.4%

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

   9. Applied rewrites 95.4%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{elif}\;b\_m \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.6e-45)
   (fma (* 4.0 a) a -1.0)
   (if (<= b_m 4.5e+49) (* (* a a) (* a a)) (* (* (* b_m b_m) b_m) b_m))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.6e-45) {
		tmp = fma((4.0 * a), a, -1.0);
	} else if (b_m <= 4.5e+49) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b_m * b_m) * b_m) * b_m;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 1.6e-45)
		tmp = fma(Float64(4.0 * a), a, -1.0);
	elseif (b_m <= 4.5e+49)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * b_m) * b_m);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.6e-45], N[(N[(4.0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[b$95$m, 4.5e+49], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.60000000000000004e-45

   1. Initial program 73.8%

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

   3. Applied rewrites 74.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 58.6%

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

   if 1.60000000000000004e-45 < b < 4.49999999999999982e49

   1. Initial program 59.7%

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

   3. Applied rewrites 63.7%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 63.5%

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

   9. Applied rewrites 63.4%

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

   if 4.49999999999999982e49 < b

   1. Initial program 66.0%

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

   3. Applied rewrites 72.8%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.4%

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

   9. Applied rewrites 95.4%

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

Alternative 9: 79.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.6 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot a, a, -1\right)\\ \mathbf{elif}\;b\_m \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.6e-45)
   (fma (* 4.0 a) a -1.0)
   (if (<= b_m 4.5e+49) (* (* a a) (* a a)) (* (* b_m b_m) (* b_m b_m)))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.6e-45) {
		tmp = fma((4.0 * a), a, -1.0);
	} else if (b_m <= 4.5e+49) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b_m * b_m) * (b_m * b_m);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 1.6e-45)
		tmp = fma(Float64(4.0 * a), a, -1.0);
	elseif (b_m <= 4.5e+49)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b_m * b_m) * Float64(b_m * b_m));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.6e-45], N[(N[(4.0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision], If[LessEqual[b$95$m, 4.5e+49], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.60000000000000004e-45

   1. Initial program 73.8%

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

   3. Applied rewrites 74.9%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 65.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 58.6%

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

   if 1.60000000000000004e-45 < b < 4.49999999999999982e49

   1. Initial program 59.7%

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

   3. Applied rewrites 63.7%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 63.5%

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

   9. Applied rewrites 63.4%

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

   if 4.49999999999999982e49 < b

   1. Initial program 66.0%

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

   3. Applied rewrites 72.8%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.4%

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

   9. Applied rewrites 95.4%

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

Alternative 10: 93.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 4.5 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a - -4, a, 4\right), a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 4.5e+49)
   (fma (fma (- a -4.0) a 4.0) (* a a) -1.0)
   (* (* (* b_m b_m) b_m) b_m)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 4.5e+49) {
		tmp = fma(fma((a - -4.0), a, 4.0), (a * a), -1.0);
	} else {
		tmp = ((b_m * b_m) * b_m) * b_m;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 4.5e+49)
		tmp = fma(fma(Float64(a - -4.0), a, 4.0), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * b_m) * b_m);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 4.5e+49], N[(N[(N[(a - -4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.49999999999999982e49

   1. Initial program 72.0%

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

   3. Applied rewrites 73.5%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 62.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 82.8%

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

   if 4.49999999999999982e49 < b

   1. Initial program 66.0%

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

   3. Applied rewrites 72.8%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.4%

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

   9. Applied rewrites 95.4%

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

Alternative 11: 51.1% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Applied rewrites 73.4%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 53.1%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 48.7%

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

Alternative 12: 25.0% accurate, 160.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ -1 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := -1.0

Derivation

1. Initial program 70.6%

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

3. Applied rewrites 73.4%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 53.1%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 21.4%

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

Reproduce

herbie shell --seed 2025019 
(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))