Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.5% → 99.1%

Time: 3.4s

Alternatives: 8

Speedup: 2.4×

• 61.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(a, a, \left(3 \cdot b\right) \cdot b\right), 4, \mathsf{fma}\left(t\_0, t\_0, -1\right)\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (fma (fma a a (* (* 3.0 b) b)) 4.0 (fma t_0 t_0 -1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	return fma(fma(a, a, ((3.0 * b) * b)), 4.0, fma(t_0, t_0, -1.0));
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(fma(a, a, Float64(Float64(3.0 * b) * b)), 4.0, fma(t_0, t_0, -1.0))
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * a + N[(N[(3.0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 4.0 + N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 75.1%

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

4. Taylor expanded in a around 0

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

6. Applied rewrites 90.0%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 99.1%

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

Alternative 2: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(\left(\frac{\frac{\mathsf{fma}\left(b + b, b, 4\right)}{a} - 4}{a} - -1\right) \cdot a\right) \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -1800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (*
          (* (* (* (- (/ (- (/ (fma (+ b b) b 4.0) a) 4.0) a) -1.0) a) a) a)
          a)))
   (if (<= a -1800000000.0)
     t_0
     (if (<= a 1.9e+47) (- (fma 12.0 (* b b) (pow b 4.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (((((((fma((b + b), b, 4.0) / a) - 4.0) / a) - -1.0) * a) * a) * a) * a;
	double tmp;
	if (a <= -1800000000.0) {
		tmp = t_0;
	} else if (a <= 1.9e+47) {
		tmp = fma(12.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(fma(Float64(b + b), b, 4.0) / a) - 4.0) / a) - -1.0) * a) * a) * a) * a)
	tmp = 0.0
	if (a <= -1800000000.0)
		tmp = t_0;
	elseif (a <= 1.9e+47)
		tmp = Float64(fma(12.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(b + b), $MachinePrecision] * b + 4.0), $MachinePrecision] / a), $MachinePrecision] - 4.0), $MachinePrecision] / a), $MachinePrecision] - -1.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1800000000.0], t$95$0, If[LessEqual[a, 1.9e+47], N[(N[(12.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e9 or 1.9000000000000002e47 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 53.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.6

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

   6. Applied rewrites 53.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   8. Applied rewrites 67.5%

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

   if -1.8e9 < a < 1.9000000000000002e47

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.3

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      2. pow2 N/A

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

      3. lower-*.f64 69.3

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

   6. Applied rewrites 69.3%

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

Alternative 3: 94.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0078:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1\\ \mathbf{elif}\;a \leq 1.28 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0078)
   (- (fma (- 1.0 a) (* (* a a) 4.0) (* (* a a) (* a a))) 1.0)
   (if (<= a 1.28e+54)
     (- (fma 12.0 (* b b) (pow b 4.0)) 1.0)
     (* (* (* a a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.0078) {
		tmp = fma((1.0 - a), ((a * a) * 4.0), ((a * a) * (a * a))) - 1.0;
	} else if (a <= 1.28e+54) {
		tmp = fma(12.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.0078)
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), Float64(Float64(a * a) * Float64(a * a))) - 1.0);
	elseif (a <= 1.28e+54)
		tmp = Float64(fma(12.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.0078], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1.28e+54], N[(N[(12.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0077999999999999996

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 56.9

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 56.8

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

   6. Applied rewrites 56.8%

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

   if -0.0077999999999999996 < a < 1.28e54

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.3

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      2. pow2 N/A

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

      3. lower-*.f64 69.3

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

   6. Applied rewrites 69.3%

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

   if 1.28e54 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 46.0

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

   4. Applied rewrites 46.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 45.9

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

   6. Applied rewrites 45.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.9

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

   8. Applied rewrites 45.9%

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

Alternative 4: 94.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0078:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) - 1\\ \mathbf{elif}\;a \leq 1.28 \cdot 10^{+54}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0078)
   (- (fma (- 1.0 a) (* (* a a) 4.0) (* (* a a) (* a a))) 1.0)
   (if (<= a 1.28e+54)
     (- (* b (* b (fma b b 12.0))) 1.0)
     (* (* (* a a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.0078) {
		tmp = fma((1.0 - a), ((a * a) * 4.0), ((a * a) * (a * a))) - 1.0;
	} else if (a <= 1.28e+54) {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.0078)
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), Float64(Float64(a * a) * Float64(a * a))) - 1.0);
	elseif (a <= 1.28e+54)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.0078], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1.28e+54], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0077999999999999996

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 56.9

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

      13. lift-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow2 N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 56.8

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

   6. Applied rewrites 56.8%

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

   if -0.0077999999999999996 < a < 1.28e54

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.3

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 69.2

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

      15. lift-pow.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 69.2

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lower-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) - 1 \]

      10. lower-fma.f64 69.2

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

   8. Applied rewrites 69.2%

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

   if 1.28e54 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 46.0

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

   4. Applied rewrites 46.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 45.9

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

   6. Applied rewrites 45.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.9

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

   8. Applied rewrites 45.9%

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

Alternative 5: 93.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 1.28 \cdot 10^{+54}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.4e+55)
   (pow a 4.0)
   (if (<= a 1.28e+54)
     (- (* b (* b (fma b b 12.0))) 1.0)
     (* (* (* a a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.4e+55) {
		tmp = pow(a, 4.0);
	} else if (a <= 1.28e+54) {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.4e+55)
		tmp = a ^ 4.0;
	elseif (a <= 1.28e+54)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.4e+55], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 1.28e+54], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.40000000000000021e55

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 46.0

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

   4. Applied rewrites 46.0%

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

   if -4.40000000000000021e55 < a < 1.28e54

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.3

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 69.2

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

      15. lift-pow.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 69.2

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lower-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) - 1 \]

      10. lower-fma.f64 69.2

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

   8. Applied rewrites 69.2%

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

   if 1.28e54 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 46.0

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

   4. Applied rewrites 46.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 45.9

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

   6. Applied rewrites 45.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.9

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

   8. Applied rewrites 45.9%

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

Alternative 6: 93.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -4.4 \cdot 10^{+55}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 1.28 \cdot 10^{+54}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.4e+55)
   (* (* a a) (* a a))
   (if (<= a 1.28e+54)
     (- (* b (* b (fma b b 12.0))) 1.0)
     (* (* (* a a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.4e+55) {
		tmp = (a * a) * (a * a);
	} else if (a <= 1.28e+54) {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.4e+55)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 1.28e+54)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.4e+55], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.28e+54], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.40000000000000021e55

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 46.0

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

   4. Applied rewrites 46.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 45.9

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

   6. Applied rewrites 45.9%

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

   if -4.40000000000000021e55 < a < 1.28e54

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 69.3

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 69.2

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

      15. lift-pow.f64 N/A

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

      16. pow2 N/A

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

      17. lower-*.f64 69.2

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lower-*.f64 69.2

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

   6. Applied rewrites 69.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

         \[\leadsto b \cdot \left(b \cdot \left(b \cdot b + 12\right)\right) - 1 \]

      10. lower-fma.f64 69.2

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

   8. Applied rewrites 69.2%

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

   if 1.28e54 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 46.0

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

   4. Applied rewrites 46.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 45.9

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

   6. Applied rewrites 45.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.9

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

   8. Applied rewrites 45.9%

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

Alternative 7: 45.9% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* (* (* a a) a) a))

C

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

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) * a) * a
end function

Java

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

Python

def code(a, b):
	return ((a * a) * a) * a

Julia

function code(a, b)
	return Float64(Float64(Float64(a * a) * a) * a)
end

MATLAB

function tmp = code(a, b)
	tmp = ((a * a) * a) * a;
end

Wolfram

code[a_, b_] := N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in a around inf

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

   1. lower-pow.f64 46.0

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

4. Applied rewrites 46.0%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 45.9

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

6. Applied rewrites 45.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 45.9

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

8. Applied rewrites 45.9%

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

Alternative 8: 45.9% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* (* a a) (* a a)))

C

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

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) * (a * a)
end function

Java

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

Python

def code(a, b):
	return (a * a) * (a * a)

Julia

function code(a, b)
	return Float64(Float64(a * a) * Float64(a * a))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) * (a * a);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in a around inf

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

   1. lower-pow.f64 46.0

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

4. Applied rewrites 46.0%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 45.9

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

6. Applied rewrites 45.9%

   \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025178 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))