Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.4% → 99.7%

Time: 3.7s

Alternatives: 10

Speedup: 2.6×

• 61.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.4% 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: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;a \leq -3.3 \cdot 10^{+73}:\\ \;\;\;\;{a}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot \mathsf{fma}\left(a, a, a\right)\right), 4, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<= a -3.3e+73)
     (pow a 4.0)
     (fma t_0 t_0 (fma (fma b b (* a (fma a a a))) 4.0 -1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (a <= -3.3e+73) {
		tmp = pow(a, 4.0);
	} else {
		tmp = fma(t_0, t_0, fma(fma(b, b, (a * fma(a, a, a))), 4.0, -1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (a <= -3.3e+73)
		tmp = a ^ 4.0;
	else
		tmp = fma(t_0, t_0, fma(fma(b, b, Float64(a * fma(a, a, a))), 4.0, -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.3e+73], N[Power[a, 4.0], $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(b * b + N[(a * N[(a * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3000000000000003e73

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   if -3.3000000000000003e73 < a

   1. Initial program 74.4%

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

   3. Applied rewrites 75.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 84.3%

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

Alternative 2: 94.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+22)
   (pow a 4.0)
   (if (<= a 4.6e-11)
     (- (fma 4.0 (* b b) (pow b 4.0)) 1.0)
     (- (fma (* a a) (* a a) (* (* (fma a a a) a) 4.0)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+22) {
		tmp = pow(a, 4.0);
	} else if (a <= 4.6e-11) {
		tmp = fma(4.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = fma((a * a), (a * a), ((fma(a, a, a) * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+22)
		tmp = a ^ 4.0;
	elseif (a <= 4.6e-11)
		tmp = Float64(fma(4.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(a * a), Float64(a * a), Float64(Float64(fma(a, a, a) * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+22], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 4.6e-11], N[(N[(4.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e22

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   if -1.5e22 < a < 4.60000000000000027e-11

   1. Initial program 74.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

   if 4.60000000000000027e-11 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. distribute-lft1-in N/A

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

      23. lift-fma.f64 N/A

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

      24. lower-*.f64 53.9

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

   6. Applied rewrites 53.9%

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

Alternative 3: 94.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 4.6 \cdot 10^{-11}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, \left(\mathsf{fma}\left(a, a, a\right) \cdot a\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+22)
   (pow a 4.0)
   (if (<= a 4.6e-11)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (- (fma (* a a) (* a a) (* (* (fma a a a) a) 4.0)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+22) {
		tmp = pow(a, 4.0);
	} else if (a <= 4.6e-11) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = fma((a * a), (a * a), ((fma(a, a, a) * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+22)
		tmp = a ^ 4.0;
	elseif (a <= 4.6e-11)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(fma(Float64(a * a), Float64(a * a), Float64(Float64(fma(a, a, a) * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+22], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 4.6e-11], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(a * a + a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e22

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   if -1.5e22 < a < 4.60000000000000027e-11

   1. Initial program 74.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 69.5

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

   8. Applied rewrites 69.5%

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

   if 4.60000000000000027e-11 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 54.0

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

   4. Applied rewrites 54.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lift-+.f64 N/A

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

      21. +-commutative N/A

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

      22. distribute-lft1-in N/A

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

      23. lift-fma.f64 N/A

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

      24. lower-*.f64 53.9

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

   6. Applied rewrites 53.9%

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

Alternative 4: 94.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2000:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(a \cdot a\right) \cdot a} \cdot \sqrt{a}\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+22)
   (pow a 4.0)
   (if (<= a 2000.0)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (* (* (sqrt (* (* a a) a)) (sqrt a)) (fma a a (* 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+22) {
		tmp = pow(a, 4.0);
	} else if (a <= 2000.0) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = (sqrt(((a * a) * a)) * sqrt(a)) * fma(a, a, (4.0 * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+22)
		tmp = a ^ 4.0;
	elseif (a <= 2000.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(a * a) * a)) * sqrt(a)) * fma(a, a, Float64(4.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+22], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 2000.0], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * N[(a * a + N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e22

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   if -1.5e22 < a < 2e3

   1. Initial program 74.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 69.5

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

   8. Applied rewrites 69.5%

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

   if 2e3 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. inv-pow N/A

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

      19. pow-prod-up N/A

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

      20. metadata-eval N/A

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

      21. cube-unmult N/A

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

      22. lower-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lower-*.f64 N/A

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

      25. lift-*.f64 30.5

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

   6. Applied rewrites 30.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 46.0

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

   8. Applied rewrites 46.0%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-prod N/A

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

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

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

      8. lower-unsound-sqrt.f64 N/A

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

      9. lower-unsound-sqrt.f64 24.4

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

   10. Applied rewrites 24.4%

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

Alternative 5: 94.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 2000:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+22)
   (pow a 4.0)
   (if (<= a 2000.0)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (* (* a a) (fma a a (* 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+22) {
		tmp = pow(a, 4.0);
	} else if (a <= 2000.0) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = (a * a) * fma(a, a, (4.0 * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+22)
		tmp = a ^ 4.0;
	elseif (a <= 2000.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(a * a) * fma(a, a, Float64(4.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+22], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 2000.0], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e22

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

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

   if -1.5e22 < a < 2e3

   1. Initial program 74.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 69.5

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

   8. Applied rewrites 69.5%

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

   if 2e3 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. inv-pow N/A

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

      19. pow-prod-up N/A

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

      20. metadata-eval N/A

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

      21. cube-unmult N/A

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

      22. lower-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lower-*.f64 N/A

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

      25. lift-*.f64 30.5

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

   6. Applied rewrites 30.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 46.0

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

   8. Applied rewrites 46.0%

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

Alternative 6: 93.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{elif}\;a \leq 2000:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, 4 \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.5e+22)
   (* (* (* a a) a) a)
   (if (<= a 2000.0)
     (- (* b (* b (fma b b 4.0))) 1.0)
     (* (* a a) (fma a a (* 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.5e+22) {
		tmp = ((a * a) * a) * a;
	} else if (a <= 2000.0) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = (a * a) * fma(a, a, (4.0 * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.5e+22)
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	elseif (a <= 2000.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(a * a) * fma(a, a, Float64(4.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.5e+22], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 2000.0], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e22

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 45.6

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

   6. Applied rewrites 45.6%

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

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 45.6

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

   8. Applied rewrites 45.6%

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

   if -1.5e22 < a < 2e3

   1. Initial program 74.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 69.5

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

   8. Applied rewrites 69.5%

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

   if 2e3 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. inv-pow N/A

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

      19. pow-prod-up N/A

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

      20. metadata-eval N/A

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

      21. cube-unmult N/A

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

      22. lower-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lower-*.f64 N/A

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

      25. lift-*.f64 30.5

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

   6. Applied rewrites 30.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 46.0

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

   8. Applied rewrites 46.0%

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

Alternative 7: 93.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0 \cdot a\\ \mathbf{elif}\;a \leq 2000:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a - -4\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) a)))
   (if (<= a -1.5e+22)
     (* t_0 a)
     (if (<= a 2000.0) (- (* b (* b (fma b b 4.0))) 1.0) (* (- a -4.0) t_0)))))

C

double code(double a, double b) {
	double t_0 = (a * a) * a;
	double tmp;
	if (a <= -1.5e+22) {
		tmp = t_0 * a;
	} else if (a <= 2000.0) {
		tmp = (b * (b * fma(b, b, 4.0))) - 1.0;
	} else {
		tmp = (a - -4.0) * t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * a)
	tmp = 0.0
	if (a <= -1.5e+22)
		tmp = Float64(t_0 * a);
	elseif (a <= 2000.0)
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 4.0))) - 1.0);
	else
		tmp = Float64(Float64(a - -4.0) * t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.5e+22], N[(t$95$0 * a), $MachinePrecision], If[LessEqual[a, 2000.0], N[(N[(b * N[(b * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a - -4.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e22

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.7

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

   4. Applied rewrites 45.7%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 45.6

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

   6. Applied rewrites 45.6%

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

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 45.6

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

   8. Applied rewrites 45.6%

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

   if -1.5e22 < a < 2e3

   1. Initial program 74.4%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 69.5

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

   6. Applied rewrites 69.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 69.5

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

   8. Applied rewrites 69.5%

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

   if 2e3 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 46.1

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

   4. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lift-pow.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. inv-pow N/A

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

      19. pow-prod-up N/A

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

      20. metadata-eval N/A

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

      21. cube-unmult N/A

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

      22. lower-*.f64 N/A

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

      23. associate-*l* N/A

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

      24. lower-*.f64 N/A

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

      25. lift-*.f64 30.5

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

   6. Applied rewrites 30.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 46.0

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

   8. Applied rewrites 46.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. *-commutative N/A

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

      7. sum-to-mult N/A

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

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

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

      9. lower-unsound-+.f32 N/A

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

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

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

      11. lower-+.f32 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-flip N/A

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

      15. lift--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

   10. Applied rewrites 46.1%

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

Alternative 8: 46.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left(a - -4\right) \cdot \left(\left(a \cdot a\right) \cdot a\right) \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return (a - -4.0) * ((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 - (-4.0d0)) * ((a * a) * a)
end function

Java

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

Python

def code(a, b):
	return (a - -4.0) * ((a * a) * a)

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = (a - -4.0) * ((a * a) * a);
end

Wolfram

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

Derivation

1. Initial program 74.4%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 46.1

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

4. Applied rewrites 46.1%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lift-pow.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. inv-pow N/A

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

   19. pow-prod-up N/A

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

   20. metadata-eval N/A

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

   21. cube-unmult N/A

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

   22. lower-*.f64 N/A

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

   23. associate-*l* N/A

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

   24. lower-*.f64 N/A

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

   25. lift-*.f64 30.5

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

6. Applied rewrites 30.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 46.0

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

8. Applied rewrites 46.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. sum-to-mult N/A

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

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

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

   9. lower-unsound-+.f32 N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. +-commutative N/A

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

   13. metadata-eval N/A

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

   14. sub-flip N/A

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

   15. lift--.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. associate-*r* N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

10. Applied rewrites 46.1%

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

Alternative 9: 45.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot a\right) \cdot \left(a \cdot a\right) \end{array} \]

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 74.4%

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

2. Taylor expanded in a around inf

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

   1. lower-pow.f64 45.7

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

4. Applied rewrites 45.7%

   \[\leadsto \color{blue}{{a}^{4}} \]
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 45.6

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

6. Applied rewrites 45.6%

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

Alternative 10: 19.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot a\right) \cdot \left(4 \cdot a\right) \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return (a * a) * (4.0 * 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) * (4.0d0 * a)
end function

Java

public static double code(double a, double b) {
	return (a * a) * (4.0 * a);
}

Python

def code(a, b):
	return (a * a) * (4.0 * a)

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = (a * a) * (4.0 * a);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.4%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 46.1

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

4. Applied rewrites 46.1%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lift-pow.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. inv-pow N/A

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

   19. pow-prod-up N/A

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

   20. metadata-eval N/A

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

   21. cube-unmult N/A

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

   22. lower-*.f64 N/A

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

   23. associate-*l* N/A

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

   24. lower-*.f64 N/A

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

   25. lift-*.f64 30.5

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

6. Applied rewrites 30.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 46.0

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

8. Applied rewrites 46.0%

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

9. Taylor expanded in a around 0

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

    1. lower-*.f64 19.9

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

11. Applied rewrites 19.9%

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

Reproduce

herbie shell --seed 2025157 
(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))