Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.5% → 99.2%

Time: 7.9s

Alternatives: 12

Speedup: 3.5×

• 62.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 94.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.7e+38)
   (pow a 4.0)
   (if (<= a 2.2e+16) (- (fma (* 4.0 b) b (pow b 4.0)) 1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.7e+38) {
		tmp = pow(a, 4.0);
	} else if (a <= 2.2e+16) {
		tmp = fma((4.0 * b), b, pow(b, 4.0)) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.7e+38)
		tmp = a ^ 4.0;
	elseif (a <= 2.2e+16)
		tmp = Float64(fma(Float64(4.0 * b), b, (b ^ 4.0)) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.7e+38], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 2.2e+16], N[(N[(N[(4.0 * b), $MachinePrecision] * b + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.6999999999999999e38 or 2.2e16 < a

   1. Initial program 43.3%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 92.7

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

   4. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 92.7

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

   6. Applied rewrites 92.7%

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

   if -4.6999999999999999e38 < a < 2.2e16

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 95.8

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

   6. Applied rewrites 95.8%

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

Alternative 3: 94.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.7e+38)
   (pow a 4.0)
   (if (<= a 2.2e+16) (fma (* (fma b b 4.0) b) b -1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.7e+38) {
		tmp = pow(a, 4.0);
	} else if (a <= 2.2e+16) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.7e+38)
		tmp = a ^ 4.0;
	elseif (a <= 2.2e+16)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.7e+38], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 2.2e+16], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.6999999999999999e38 or 2.2e16 < a

   1. Initial program 43.3%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 92.7

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

   4. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 92.7

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

   6. Applied rewrites 92.7%

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

   if -4.6999999999999999e38 < a < 2.2e16

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 95.8

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied rewrites 95.7%

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

Alternative 4: 94.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 1.62 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right) \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.7e+38)
   (* (* a a) (* a a))
   (if (<= a 1.62e+16)
     (fma (* (fma b b 4.0) b) b -1.0)
     (* (* (* (- a -4.0) a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.7e+38) {
		tmp = (a * a) * (a * a);
	} else if (a <= 1.62e+16) {
		tmp = fma((fma(b, b, 4.0) * b), b, -1.0);
	} else {
		tmp = (((a - -4.0) * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.7e+38)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 1.62e+16)
		tmp = fma(Float64(fma(b, b, 4.0) * b), b, -1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.7e+38], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.62e+16], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision], N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.6999999999999999e38

   1. Initial program 23.2%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 94.2

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

   4. Applied rewrites 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 94.1

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

   6. Applied rewrites 94.1%

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

   if -4.6999999999999999e38 < a < 1.62e16

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 95.8

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

   6. Applied rewrites 95.8%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied rewrites 95.7%

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

   if 1.62e16 < a

   1. Initial program 61.7%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

         \[\leadsto {a}^{3} \cdot 4 + {a}^{4} \]

      9. *-commutative N/A

         \[\leadsto 4 \cdot {a}^{3} + {\color{blue}{a}}^{4} \]

      10. cube-mult N/A

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

      11. pow2 N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

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

      14. pow-prod-up N/A

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

      15. distribute-rgt-in N/A

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

      16. pow2 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 91.3%

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

Alternative 5: 82.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8500:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8500.0) (- (* (* (* a a) a) a) 1.0) (* (* (fma b b 4.0) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8500.0) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = (fma(b, b, 4.0) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8500.0)
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(fma(b, b, 4.0) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8500.0], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8500

   1. Initial program 76.4%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} - 1 \]

      3. lower-*.f64 N/A

         \[\leadsto {a}^{3} \cdot \color{blue}{a} - 1 \]

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

   if 8500 < b

   1. Initial program 64.7%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 89.9

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

   4. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 90.0

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

   6. Applied rewrites 90.0%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied rewrites 89.9%

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

   10. Taylor expanded in b around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. *-lft-identity N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. pow-flip N/A

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

       8. pow-prod-up N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

           \[\leadsto {b}^{2} \cdot 4 + {b}^{4} \]

       11. *-commutative N/A

           \[\leadsto 4 \cdot {b}^{2} + {b}^{4} \]

       12. metadata-eval N/A

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

       13. pow-prod-up N/A

           \[\leadsto 4 \cdot {b}^{2} + {b}^{2} \cdot {b}^{\color{blue}{2}} \]

       14. pow2 N/A

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

       15. sqr-neg-rev N/A

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

       16. associate-*r* N/A

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

       17. pow2 N/A

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

       18. sqr-neg-rev N/A

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

       19. pow3 N/A

           \[\leadsto 4 \cdot {b}^{2} + {\left(\mathsf{neg}\left(b\right)\right)}^{3} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

       20. cube-neg-rev N/A

           \[\leadsto 4 \cdot {b}^{2} + \left(\mathsf{neg}\left({b}^{3}\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   12. Applied rewrites 89.9%

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

Alternative 6: 82.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 20000000.0) (- (* (* a a) 4.0) 1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000000.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	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 ((b * b) <= 20000000.0d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000000.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 20000000.0:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 20000000.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 20000000.0)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 20000000.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e7

   1. Initial program 82.2%

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

   2. 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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-out N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   if 2e7 < (*.f64 b b)

   1. Initial program 64.7%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. cube-unmult N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

Alternative 7: 82.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 20000000.0)
   (- (* (* a a) 4.0) 1.0)
   (* (* (fma b b 4.0) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000000.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (fma(b, b, 4.0) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 20000000.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(fma(b, b, 4.0) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 20000000.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e7

   1. Initial program 82.2%

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

   2. 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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-out N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   if 2e7 < (*.f64 b b)

   1. Initial program 64.7%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 89.6

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

   4. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 89.7

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

   6. Applied rewrites 89.7%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied rewrites 89.6%

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

   10. Taylor expanded in b around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. *-lft-identity N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. pow-flip N/A

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

       8. pow-prod-up N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

           \[\leadsto {b}^{2} \cdot 4 + {b}^{4} \]

       11. *-commutative N/A

           \[\leadsto 4 \cdot {b}^{2} + {b}^{4} \]

       12. metadata-eval N/A

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

       13. pow-prod-up N/A

           \[\leadsto 4 \cdot {b}^{2} + {b}^{2} \cdot {b}^{\color{blue}{2}} \]

       14. pow2 N/A

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

       15. sqr-neg-rev N/A

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

       16. associate-*r* N/A

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

       17. pow2 N/A

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

       18. sqr-neg-rev N/A

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

       19. pow3 N/A

           \[\leadsto 4 \cdot {b}^{2} + {\left(\mathsf{neg}\left(b\right)\right)}^{3} \cdot \left(\mathsf{neg}\left(b\right)\right) \]

       20. cube-neg-rev N/A

           \[\leadsto 4 \cdot {b}^{2} + \left(\mathsf{neg}\left({b}^{3}\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right) \]

   12. Applied rewrites 89.6%

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

Alternative 8: 82.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 20000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 20000000.0) (- (* (* a a) 4.0) 1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000000.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	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 ((b * b) <= 20000000.0d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 20000000.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 20000000.0:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 20000000.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 20000000.0)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 20000000.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2e7

   1. Initial program 82.2%

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

   2. 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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-out N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 76.0

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

   7. Applied rewrites 76.0%

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

   if 2e7 < (*.f64 b b)

   1. Initial program 64.7%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 89.5

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

   7. Applied rewrites 89.5%

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

Alternative 9: 81.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -8 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.62 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -8e+19) t_0 (if (<= a 1.62e+16) (fma (* 4.0 b) b -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -8e+19) {
		tmp = t_0;
	} else if (a <= 1.62e+16) {
		tmp = fma((4.0 * b), b, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -8e+19)
		tmp = t_0;
	elseif (a <= 1.62e+16)
		tmp = fma(Float64(4.0 * b), b, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -8e+19], t$95$0, If[LessEqual[a, 1.62e+16], N[(N[(4.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -8e19 or 1.62e16 < a

   1. Initial program 45.1%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 91.5

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

   4. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

   if -8e19 < a < 1.62e16

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 97.2

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

   6. Applied rewrites 97.2%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 74.3%

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

Alternative 10: 60.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+125}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e+125) (- (* (* a a) 4.0) 1.0) (fma (* 4.0 b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e+125) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = fma((4.0 * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e+125)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = fma(Float64(4.0 * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e+125], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.79999999999999986e125

   1. Initial program 75.9%

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

   2. 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. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow3 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. distribute-lft-out N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 57.9

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

   4. Applied rewrites 57.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if 5.79999999999999986e125 < b

   1. Initial program 60.2%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow2 N/A

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

      8. associate-*r* N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. cube-unmult N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

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

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 86.2%

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

Alternative 11: 51.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

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. pow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. pow2 N/A

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

   8. associate-*r* N/A

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

   9. pow-plus N/A

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

   10. metadata-eval N/A

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

   11. cube-unmult N/A

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

   12. pow2 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. cube-unmult N/A

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

   16. metadata-eval N/A

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

   17. pow-plus N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 69.5

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

4. Applied rewrites 69.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow3 N/A

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

   5. pow-plus N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 69.5

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

6. Applied rewrites 69.5%

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

7. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. associate-*l* N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. metadata-eval N/A

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

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

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

9. Applied rewrites 69.5%

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

10. Taylor expanded in b around 0

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

12. Applied rewrites 51.3%

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

Alternative 12: 24.5% accurate, 56.6× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Java

public static double code(double a, double b) {
	return -1.0;
}

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 73.5%

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

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. pow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. pow2 N/A

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

   8. associate-*r* N/A

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

   9. pow-plus N/A

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

   10. metadata-eval N/A

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

   11. cube-unmult N/A

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

   12. pow2 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. cube-unmult N/A

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

   16. metadata-eval N/A

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

   17. pow-plus N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 69.5

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

4. Applied rewrites 69.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow3 N/A

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

   5. pow-plus N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 69.5

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

6. Applied rewrites 69.5%

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

7. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. associate-*l* N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. metadata-eval N/A

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

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

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

9. Applied rewrites 69.5%

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

10. Taylor expanded in b around 0

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

12. Applied rewrites 24.5%

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

Reproduce

herbie shell --seed 2025132 
(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))