Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.4% → 99.5%

Time: 3.8s

Alternatives: 10

Speedup: 3.1×

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 73.4% accurate, 1.0× speedup

Math

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

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: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;a \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\left(\left(a - -4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot \mathsf{fma}\left(a, a, a\right)\right), 4, \mathsf{fma}\left(t\_0 \cdot b, b, \mathsf{fma}\left(t\_0 \cdot a, a, -1\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<= a -2e+109)
     (- (* (* (- a -4.0) a) (* a a)) 1.0)
     (fma
      (fma b b (* a (fma a a a)))
      4.0
      (fma (* t_0 b) b (fma (* t_0 a) a -1.0))))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (a <= -2e+109) {
		tmp = (((a - -4.0) * a) * (a * a)) - 1.0;
	} else {
		tmp = fma(fma(b, b, (a * fma(a, a, a))), 4.0, fma((t_0 * b), b, fma((t_0 * a), a, -1.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (a <= -2e+109)
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(fma(b, b, Float64(a * fma(a, a, a))), 4.0, fma(Float64(t_0 * b), b, fma(Float64(t_0 * a), a, -1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2e+109], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b + N[(a * N[(a * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(N[(t$95$0 * b), $MachinePrecision] * b + N[(N[(t$95$0 * a), $MachinePrecision] * a + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2e109

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.0%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 69.0%

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 69.0%

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.0%

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-+.f64 N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. sum-to-mult N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 69.0%

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

   8. Applied rewrites 69.0%

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

   if -2e109 < a

   1. Initial program 73.4%

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 82.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 82.8%

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

   8. Applied rewrites 82.8%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;a \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\left(\left(a - -4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot \mathsf{fma}\left(a, a, a\right)\right), 4, \mathsf{fma}\left(t\_0, t\_0, -1\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<= a -2e+109)
     (- (* (* (- a -4.0) a) (* a a)) 1.0)
     (fma (fma b b (* a (fma a a a))) 4.0 (fma t_0 t_0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (a <= -2e+109) {
		tmp = (((a - -4.0) * a) * (a * a)) - 1.0;
	} else {
		tmp = fma(fma(b, b, (a * fma(a, a, a))), 4.0, fma(t_0, t_0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (a <= -2e+109)
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * a) * Float64(a * a)) - 1.0);
	else
		tmp = fma(fma(b, b, Float64(a * fma(a, a, a))), 4.0, fma(t_0, t_0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2e+109], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b + N[(a * N[(a * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2e109

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.0%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 69.0%

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 69.0%

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.0%

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-+.f64 N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. sum-to-mult N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 69.0%

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

   8. Applied rewrites 69.0%

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

   if -2e109 < a

   1. Initial program 73.4%

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

   3. Applied rewrites 74.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 82.7%

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

Alternative 3: 94.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;\left(\left(a - -4\right) \cdot \left(a \cdot a\right)\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 1950000:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - -4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.2e+39)
   (- (* (* (- a -4.0) (* a a)) a) 1.0)
   (if (<= a 1950000.0)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (- (* (* (- a -4.0) a) (* a a)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.2e+39) {
		tmp = (((a - -4.0) * (a * a)) * a) - 1.0;
	} else if (a <= 1950000.0) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = (((a - -4.0) * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.2e+39)
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * Float64(a * a)) * a) - 1.0);
	elseif (a <= 1950000.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.2e+39], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1950000.0], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.2000000000000001e39

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.0%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 69.0%

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-+.f64 N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. sum-to-mult N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lower-*.f64 N/A

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

   8. Applied rewrites 69.0%

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

   if -2.2000000000000001e39 < a < 1.95e6

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 68.9%

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

   8. Applied rewrites 68.9%

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

   if 1.95e6 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.0%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 69.0%

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 69.0%

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.0%

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-+.f64 N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. sum-to-mult N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 69.0%

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

   8. Applied rewrites 69.0%

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

Alternative 4: 94.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(a - -4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1950000:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* (- a -4.0) a) (* a a)) 1.0)))
   (if (<= a -2.2e+39)
     t_0
     (if (<= a 1950000.0) (- (* b (* b (fma b b 4.0))) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (((a - -4.0) * a) * (a * a)) - 1.0;
	double tmp;
	if (a <= -2.2e+39) {
		tmp = t_0;
	} else if (a <= 1950000.0) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(Float64(a - -4.0) * a) * Float64(a * a)) - 1.0)
	tmp = 0.0
	if (a <= -2.2e+39)
		tmp = t_0;
	elseif (a <= 1950000.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -2.2e+39], t$95$0, If[LessEqual[a, 1950000.0], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2000000000000001e39 or 1.95e6 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.0%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 69.0%

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 69.0%

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.0%

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-+.f64 N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. sum-to-mult N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 69.0%

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

   8. Applied rewrites 69.0%

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

   if -2.2000000000000001e39 < a < 1.95e6

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 68.9%

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

   8. Applied rewrites 68.9%

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

Alternative 5: 93.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a, \left(a \cdot a\right) \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|b\right| \cdot \left|b\right|, 4, {\left(\left|b\right|\right)}^{4}\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fabs b) 1.08e+49)
   (- (* a (fma (fma 4.0 a 4.0) a (* (* a a) a))) 1.0)
   (- (fma (* (fabs b) (fabs b)) 4.0 (pow (fabs b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 1.08e+49) {
		tmp = (a * fma(fma(4.0, a, 4.0), a, ((a * a) * a))) - 1.0;
	} else {
		tmp = fma((fabs(b) * fabs(b)), 4.0, pow(fabs(b), 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 1.08e+49)
		tmp = Float64(Float64(a * fma(fma(4.0, a, 4.0), a, Float64(Float64(a * a) * a))) - 1.0);
	else
		tmp = Float64(fma(Float64(abs(b) * abs(b)), 4.0, (abs(b) ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.08e+49], N[(N[(a * N[(N[(4.0 * a + 4.0), $MachinePrecision] * a + N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * 4.0 + N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.08e49

   1. Initial program 73.4%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 52.5%

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

   4. Applied rewrites 52.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 56.6%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 56.6%

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow-prod-down N/A

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

      21. lift-*.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 56.5%

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   if 1.08e49 < b

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-prod-down N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 69.0%

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

   8. Applied rewrites 69.0%

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

Alternative 6: 93.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a, \left(a \cdot a\right) \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(\left|b\right| \cdot \mathsf{fma}\left(\left|b\right|, \left|b\right|, 4\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fabs b) 1.08e+49)
   (- (* a (fma (fma 4.0 a 4.0) a (* (* a a) a))) 1.0)
   (- (* (fabs b) (* (fabs b) (fma (fabs b) (fabs b) 4.0))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 1.08e+49) {
		tmp = (a * fma(fma(4.0, a, 4.0), a, ((a * a) * a))) - 1.0;
	} else {
		tmp = (fabs(b) * (fabs(b) * fma(fabs(b), fabs(b), 4.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 1.08e+49)
		tmp = Float64(Float64(a * fma(fma(4.0, a, 4.0), a, Float64(Float64(a * a) * a))) - 1.0);
	else
		tmp = Float64(Float64(abs(b) * Float64(abs(b) * fma(abs(b), abs(b), 4.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.08e+49], N[(N[(a * N[(N[(4.0 * a + 4.0), $MachinePrecision] * a + N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.08e49

   1. Initial program 73.4%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 52.5%

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

   4. Applied rewrites 52.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 56.6%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 56.6%

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow-prod-down N/A

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

      21. lift-*.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 56.5%

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fma.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   if 1.08e49 < b

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 68.9%

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

   8. Applied rewrites 68.9%

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

Alternative 7: 93.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(\left|b\right| \cdot \mathsf{fma}\left(\left|b\right|, \left|b\right|, 4\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fabs b) 1.08e+49)
   (- (* (fma a a (fma 4.0 a 4.0)) (* a a)) 1.0)
   (- (* (fabs b) (* (fabs b) (fma (fabs b) (fabs b) 4.0))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 1.08e+49) {
		tmp = (fma(a, a, fma(4.0, a, 4.0)) * (a * a)) - 1.0;
	} else {
		tmp = (fabs(b) * (fabs(b) * fma(fabs(b), fabs(b), 4.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 1.08e+49)
		tmp = Float64(Float64(fma(a, a, fma(4.0, a, 4.0)) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(abs(b) * Float64(abs(b) * fma(abs(b), abs(b), 4.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.08e+49], N[(N[(N[(a * a + N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.08e49

   1. Initial program 73.4%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 52.5%

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

   4. Applied rewrites 52.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft1-in N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 56.6%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 56.6%

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

      17. lift-pow.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-prod-up N/A

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

      20. pow-prod-down N/A

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

      21. lift-*.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 56.5%

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   if 1.08e49 < b

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 68.9%

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

   8. Applied rewrites 68.9%

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

Alternative 8: 76.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 1950000:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1950000.0)
   (- (* b (* b (fma b b 4.0))) 1.0)
   (- (* (* 4.0 a) (* a a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1950000.0) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1950000.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1950000.0], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.95e6

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 68.9%

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

   8. Applied rewrites 68.9%

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

   if 1.95e6 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.0%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 69.0%

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 69.0%

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.0%

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-+.f64 N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. sum-to-mult N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 69.0%

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 41.7%

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

Alternative 9: 60.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 1950000:\\ \;\;\;\;b \cdot \left(4 \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1950000.0) (- (* b (* 4.0 b)) 1.0) (- (* (* 4.0 a) (* a a)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1950000.0) {
		tmp = (b * (4.0 * b)) - 1.0;
	} else {
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= 1950000.0d0) then
        tmp = (b * (4.0d0 * b)) - 1.0d0
    else
        tmp = ((4.0d0 * a) * (a * a)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1950000.0) {
		tmp = (b * (4.0 * b)) - 1.0;
	} else {
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1950000.0:
		tmp = (b * (4.0 * b)) - 1.0
	else:
		tmp = ((4.0 * a) * (a * a)) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1950000.0)
		tmp = Float64(Float64(b * Float64(4.0 * b)) - 1.0);
	else
		tmp = Float64(Float64(Float64(4.0 * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1950000.0)
		tmp = (b * (4.0 * b)) - 1.0;
	else
		tmp = ((4.0 * a) * (a * a)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1950000.0], N[(N[(b * N[(4.0 * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(4.0 * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.95e6

   1. Initial program 73.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 68.9%

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in b around 0

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

       1. lower-*.f64 49.7%

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

   11. Applied rewrites 49.7%

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

   if 1.95e6 < a

   1. Initial program 73.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.0%

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

   4. Applied rewrites 69.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.0%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 69.0%

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 69.0%

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow-prod-down N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.9%

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

   6. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 69.0%

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

          \[\leadsto \left(\left(\left(\frac{4}{a} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot a\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 69.0%

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

      14. lower-unsound-/.f64 N/A

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

      15. lower-unsound-+.f64 N/A

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

      16. lower-unsound-*.f64 N/A

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

      17. sum-to-mult N/A

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

      18. add-flip N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 69.0%

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

   8. Applied rewrites 69.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(4 \cdot a\right) \cdot \left(a \cdot a\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 41.7%

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

Alternative 10: 49.7% accurate, 6.0× speedup

Math

\[b \cdot \left(4 \cdot b\right) - 1 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, b):
	return (b * (4.0 * b)) - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-pow.f64 69.0%

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

4. Applied rewrites 69.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. *-commutative N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow-prod-down N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 68.9%

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

6. Applied rewrites 68.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 68.9%

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

8. Applied rewrites 68.9%

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

9. Taylor expanded in b around 0

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

    1. lower-*.f64 49.7%

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

11. Applied rewrites 49.7%

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

Reproduce

herbie shell --seed 2025201 
(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))