Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.9% → 98.1%

Time: 8.3s

Alternatives: 7

Speedup: 6.0×

• 61.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 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) * (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) * (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) * (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) * (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(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) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 0.00045:\\ \;\;\;\;\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b\_m, b\_m, a \cdot a\right), b\_m \cdot b\_m, \left(b\_m \cdot b\_m\right) \cdot 12\right) - 1\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 0.00045)
   (- (* (* (fma (- a 4.0) a 4.0) a) a) 1.0)
   (- (fma (fma b_m b_m (* a a)) (* b_m b_m) (* (* b_m b_m) 12.0)) 1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 0.00045) {
		tmp = ((fma((a - 4.0), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma(fma(b_m, b_m, (a * a)), (b_m * b_m), ((b_m * b_m) * 12.0)) - 1.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 0.00045)
		tmp = Float64(Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(fma(fma(b_m, b_m, Float64(a * a)), Float64(b_m * b_m), Float64(Float64(b_m * b_m) * 12.0)) - 1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 0.00045], N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4999999999999999e-4

   1. Initial program 73.2%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 84.1

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

   8. Applied rewrites 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 84.2

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

   10. Applied rewrites 84.2%

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

   if 4.4999999999999999e-4 < b

   1. Initial program 62.2%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 96.6

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

   10. Applied rewrites 96.6%

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

Alternative 2: 68.9% accurate, 0.9× speedup

Math

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

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) (+ 3.0 a)))))
       1.0)
      10.0)
   (- (* (* a a) 4.0) 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) * (3.0 + a))))) - 1.0) <= 10.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (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) * (3.0d0 + a))))) - 1.0d0) <= 10.0d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (a * a) * (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) * (3.0 + a))))) - 1.0) <= 10.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (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) * (3.0 + a))))) - 1.0) <= 10.0:
		tmp = ((a * a) * 4.0) - 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(3.0 + a))))) - 1.0) <= 10.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(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) * (3.0 + a))))) - 1.0) <= 10.0)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (a * a) * (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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], 10.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 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 3 binary64) a))))) #s(literal 1 binary64)) < 10

   1. Initial program 100.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. pow2 N/A

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

      4. lift-*.f64 95.2

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

   8. Applied rewrites 95.2%

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

   if 10 < (-.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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 58.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 61.3

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

   5. Applied rewrites 61.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 61.2

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

   7. Applied rewrites 61.2%

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

Alternative 3: 99.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b\_m, b\_m, a \cdot a\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(b\_m \cdot b\_m\right) \cdot 12\right) - 1 \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (let* ((t_0 (fma b_m b_m (* a a))))
   (- (fma t_0 t_0 (* (* b_m b_m) 12.0)) 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(t_0, t_0, ((b_m * b_m) * 12.0)) - 1.0;
}

Julia

b_m = abs(b)
function code(a, b_m)
	t_0 = fma(b_m, b_m, Float64(a * a))
	return Float64(fma(t_0, t_0, Float64(Float64(b_m * b_m) * 12.0)) - 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[(t$95$0 * t$95$0 + N[(N[(b$95$m * b$95$m), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Initial program 70.2%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

4. Applied rewrites 71.8%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 98.6

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

7. Applied rewrites 98.6%

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

Alternative 4: 93.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.6 \cdot 10^{+39}:\\ \;\;\;\;\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 2.6e+39)
   (- (* (* (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 <= 2.6e+39) {
		tmp = ((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 <= 2.6e+39)
		tmp = Float64(Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * a) * a) - 1.0);
	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, 2.6e+39], N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e39

   1. Initial program 72.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 82.1

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

   8. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 82.1

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

   10. Applied rewrites 82.1%

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

   if 2.6e39 < b

   1. Initial program 63.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 5: 79.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;b\_m \leq 2.6 \cdot 10^{+39}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 3.1e-47)
   (- (* (* a a) 4.0) 1.0)
   (if (<= b_m 2.6e+39) (* (* 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 <= 3.1e-47) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b_m <= 2.6e+39) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b_m * b_m) * (b_m * b_m);
	}
	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 <= 3.1d-47) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else if (b_m <= 2.6d+39) then
        tmp = (a * a) * (a * a)
    else
        tmp = (b_m * b_m) * (b_m * b_m)
    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 <= 3.1e-47) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b_m <= 2.6e+39) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b_m * b_m) * (b_m * b_m);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if b_m <= 3.1e-47:
		tmp = ((a * a) * 4.0) - 1.0
	elif b_m <= 2.6e+39:
		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 <= 3.1e-47)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (b_m <= 2.6e+39)
		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

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if (b_m <= 3.1e-47)
		tmp = ((a * a) * 4.0) - 1.0;
	elseif (b_m <= 2.6e+39)
		tmp = (a * a) * (a * a);
	else
		tmp = (b_m * b_m) * (b_m * b_m);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 3.1e-47], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[b$95$m, 2.6e+39], 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 < 3.0999999999999998e-47

   1. Initial program 72.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. pow2 N/A

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

      4. lift-*.f64 63.2

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

   8. Applied rewrites 63.2%

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

   if 3.0999999999999998e-47 < b < 2.6e39

   1. Initial program 65.1%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   if 2.6e39 < b

   1. Initial program 63.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 6: 93.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.6 \cdot 10^{+39}:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 2.6e+39)
   (- (* (* (fma a 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 <= 2.6e+39) {
		tmp = ((fma(a, 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 <= 2.6e+39)
		tmp = Float64(Float64(Float64(fma(a, a, 4.0) * a) * a) - 1.0);
	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, 2.6e+39], N[(N[(N[(N[(a * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e39

   1. Initial program 72.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
   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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-pow.f64 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 82.1

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

   8. Applied rewrites 82.1%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 80.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 80.8

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

   13. Applied rewrites 80.8%

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

   if 2.6e39 < b

   1. Initial program 63.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 7: 50.7% accurate, 11.1× 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 70.2%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
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. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lower-pow.f64 56.1

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

5. Applied rewrites 56.1%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

      \[\leadsto {a}^{2} \cdot 4 - 1 \]

   2. lower-*.f64 N/A

      \[\leadsto {a}^{2} \cdot 4 - 1 \]

   3. pow2 N/A

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

   4. lift-*.f64 53.1

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

8. Applied rewrites 53.1%

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

Reproduce

herbie shell --seed 2025045 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))