Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.3% → 99.8%

Time: 5.2s

Alternatives: 9

Speedup: 3.0×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a + 12 \cdot {b}^{2}\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (-
          (+
           (pow (+ (* a a) (* b b)) 2.0)
           (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
          1.0)))
   (if (<= t_0 INFINITY)
     t_0
     (- (+ (* (* (* a a) a) a) (* 12.0 (pow b 2.0))) 1.0))))

C

double code(double a, double b) {
	double t_0 = (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = ((((a * a) * a) * a) + (12.0 * pow(b, 2.0))) - 1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = ((((a * a) * a) * a) + (12.0 * Math.pow(b, 2.0))) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = ((((a * a) * a) * a) + (12.0 * math.pow(b, 2.0))) - 1.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * a) * a) + Float64(12.0 * (b ^ 2.0))) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = ((((a * a) * a) * a) + (12.0 * (b ^ 2.0))) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(12.0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 73.3%

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

   if +inf.0 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 59.4%

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

   4. Applied rewrites 59.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.4%

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

   6. Applied rewrites 59.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 84.8%

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

   9. Applied rewrites 84.8%

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

Alternative 2: 95.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -57000:\\ \;\;\;\;\left({a}^{4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot 3\right)\right) - 1\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - 4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -57000.0)
   (- (+ (pow a 4.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) 3.0)))) 1.0)
   (if (<= a 7.4e+65)
     (- (fma 12.0 (* b b) (pow b 4.0)) 1.0)
     (- (* (* (- a 4.0) a) (* a a)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -57000.0) {
		tmp = (pow(a, 4.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * 3.0)))) - 1.0;
	} else if (a <= 7.4e+65) {
		tmp = fma(12.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = (((a - 4.0) * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -57000.0)
		tmp = Float64(Float64((a ^ 4.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * 3.0)))) - 1.0);
	elseif (a <= 7.4e+65)
		tmp = Float64(fma(12.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a - 4.0) * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -57000.0], N[(N[(N[Power[a, 4.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 7.4e+65], N[(N[(12.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -57000

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 59.4%

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 68.4%

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

   if -57000 < a < 7.3999999999999999e65

   1. Initial program 73.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

   if 7.3999999999999999e65 < a

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.3%

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.3%

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 69.3%

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

   8. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.3%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. sub-to-mult-rev N/A

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

      16. lower--.f64 69.3%

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

   10. Applied rewrites 69.3%

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

Alternative 3: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -57000:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot 3\right)\right) - 1\\ \mathbf{elif}\;a \leq 7.4 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - 4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -57000.0)
   (-
    (+ (* (* (* a a) a) a) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) 3.0))))
    1.0)
   (if (<= a 7.4e+65)
     (- (fma 12.0 (* b b) (pow b 4.0)) 1.0)
     (- (* (* (- a 4.0) a) (* a a)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -57000.0) {
		tmp = ((((a * a) * a) * a) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * 3.0)))) - 1.0;
	} else if (a <= 7.4e+65) {
		tmp = fma(12.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = (((a - 4.0) * a) * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -57000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * a) * a) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * 3.0)))) - 1.0);
	elseif (a <= 7.4e+65)
		tmp = Float64(fma(12.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a - 4.0) * a) * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -57000.0], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 7.4e+65], N[(N[(12.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -57000

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 59.4%

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

   4. Applied rewrites 59.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.4%

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

   6. Applied rewrites 59.4%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 68.3%

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

   if -57000 < a < 7.3999999999999999e65

   1. Initial program 73.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

   if 7.3999999999999999e65 < a

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.3%

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.3%

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 69.3%

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

   8. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.3%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. sub-to-mult-rev N/A

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

      16. lower--.f64 69.3%

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

   10. Applied rewrites 69.3%

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

Alternative 4: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 5.2 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a + 12 \cdot {\left(\left|b\right|\right)}^{2}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12, \left|b\right| \cdot \left|b\right|, {\left(\left|b\right|\right)}^{4}\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fabs b) 5.2e+29)
   (- (+ (* (* (* a a) a) a) (* 12.0 (pow (fabs b) 2.0))) 1.0)
   (- (fma 12.0 (* (fabs b) (fabs b)) (pow (fabs b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 5.2e+29) {
		tmp = ((((a * a) * a) * a) + (12.0 * pow(fabs(b), 2.0))) - 1.0;
	} else {
		tmp = fma(12.0, (fabs(b) * fabs(b)), pow(fabs(b), 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 5.2e+29)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * a) * a) + Float64(12.0 * (abs(b) ^ 2.0))) - 1.0);
	else
		tmp = Float64(fma(12.0, Float64(abs(b) * abs(b)), (abs(b) ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 5.2e+29], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(12.0 * N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(12.0 * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2e29

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 59.4%

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

   4. Applied rewrites 59.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. cube-unmult N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.4%

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

   6. Applied rewrites 59.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 84.8%

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

   9. Applied rewrites 84.8%

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

   if 5.2e29 < b

   1. Initial program 73.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

Alternative 5: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 2.55 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(a - 4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(12, \left|b\right| \cdot \left|b\right|, {\left(\left|b\right|\right)}^{4}\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fabs b) 2.55e+24)
   (- (* (* (- a 4.0) a) (* a a)) 1.0)
   (- (fma 12.0 (* (fabs b) (fabs b)) (pow (fabs b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 2.55e+24) {
		tmp = (((a - 4.0) * a) * (a * a)) - 1.0;
	} else {
		tmp = fma(12.0, (fabs(b) * fabs(b)), pow(fabs(b), 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 2.55e+24)
		tmp = Float64(Float64(Float64(Float64(a - 4.0) * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(fma(12.0, Float64(abs(b) * abs(b)), (abs(b) ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 2.55e+24], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(12.0 * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5499999999999998e24

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.3%

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.3%

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 69.3%

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

   8. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.3%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. sub-to-mult-rev N/A

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

      16. lower--.f64 69.3%

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

   10. Applied rewrites 69.3%

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

   if 2.5499999999999998e24 < b

   1. Initial program 73.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

Alternative 6: 93.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 2.55 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(a - 4\right) \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(\left|b\right| \cdot \mathsf{fma}\left(\left|b\right|, \left|b\right|, 12\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (fabs b) 2.55e+24)
   (- (* (* (- a 4.0) a) (* a a)) 1.0)
   (- (* (fabs b) (* (fabs b) (fma (fabs b) (fabs b) 12.0))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (fabs(b) <= 2.55e+24) {
		tmp = (((a - 4.0) * a) * (a * a)) - 1.0;
	} else {
		tmp = (fabs(b) * (fabs(b) * fma(fabs(b), fabs(b), 12.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (abs(b) <= 2.55e+24)
		tmp = Float64(Float64(Float64(Float64(a - 4.0) * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(abs(b) * Float64(abs(b) * fma(abs(b), abs(b), 12.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Abs[b], $MachinePrecision], 2.55e+24], N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision] + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5499999999999998e24

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.3%

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.3%

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 69.3%

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

   8. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 69.3%

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift--.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. sub-to-mult-rev N/A

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

      16. lower--.f64 69.3%

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

   10. Applied rewrites 69.3%

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

   if 2.5499999999999998e24 < b

   1. Initial program 73.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-add N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 69.6%

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

   8. Applied rewrites 69.6%

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

Alternative 7: 79.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-4 \cdot a\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \mathsf{fma}\left(b, b, 12\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+97)
   (- (* a (* a (* -4.0 a))) 1.0)
   (- (* b (* b (fma b b 12.0))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+97) {
		tmp = (a * (a * (-4.0 * a))) - 1.0;
	} else {
		tmp = (b * (b * fma(b, b, 12.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+97)
		tmp = Float64(Float64(a * Float64(a * Float64(-4.0 * a))) - 1.0);
	else
		tmp = Float64(Float64(b * Float64(b * fma(b, b, 12.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.5e+97], N[(N[(a * N[(a * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * N[(b * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4999999999999999e97

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.3%

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.3%

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 69.3%

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 41.9%

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

   11. Applied rewrites 41.9%

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

   if -6.4999999999999999e97 < a

   1. Initial program 73.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-add N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 69.6%

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

   8. Applied rewrites 69.6%

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

Alternative 8: 62.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-4 \cdot a\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+97) (- (* a (* a (* -4.0 a))) 1.0) (- (* b (* b 12.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+97) {
		tmp = (a * (a * (-4.0 * a))) - 1.0;
	} else {
		tmp = (b * (b * 12.0)) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.5d+97)) then
        tmp = (a * (a * ((-4.0d0) * a))) - 1.0d0
    else
        tmp = (b * (b * 12.0d0)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+97) {
		tmp = (a * (a * (-4.0 * a))) - 1.0;
	} else {
		tmp = (b * (b * 12.0)) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.5e+97:
		tmp = (a * (a * (-4.0 * a))) - 1.0
	else:
		tmp = (b * (b * 12.0)) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+97)
		tmp = Float64(Float64(a * Float64(a * Float64(-4.0 * a))) - 1.0);
	else
		tmp = Float64(Float64(b * Float64(b * 12.0)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.5e+97)
		tmp = (a * (a * (-4.0 * a))) - 1.0;
	else
		tmp = (b * (b * 12.0)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.5e+97], N[(N[(a * N[(a * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4999999999999999e97

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.4%

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

   4. Applied rewrites 69.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 69.3%

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.3%

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 69.3%

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

   8. Applied rewrites 69.3%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 41.9%

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

   11. Applied rewrites 41.9%

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

   if -6.4999999999999999e97 < a

   1. Initial program 73.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 69.6%

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

   4. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 69.6%

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

   6. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-add N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. distribute-rgt-out N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 69.6%

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in b around 0

      \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 50.9%

       \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.9% accurate, 5.7× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (- (* b (* b 12.0)) 1.0))

C

double code(double a, double b) {
	return (b * (b * 12.0)) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * (b * 12.0d0)) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (b * (b * 12.0)) - 1.0;
}

Python

def code(a, b):
	return (b * (b * 12.0)) - 1.0

Julia

function code(a, b)
	return Float64(Float64(b * Float64(b * 12.0)) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = (b * (b * 12.0)) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-pow.f64 69.6%

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

4. Applied rewrites 69.6%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 69.6%

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

6. Applied rewrites 69.6%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-add N/A

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

   4. unpow-prod-down N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. distribute-rgt-out N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 69.6%

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

8. Applied rewrites 69.6%

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

9. Taylor expanded in b around 0

   \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]
10. Step-by-step derivation

11. Applied rewrites 50.9%

    \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025200 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))