Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.0% → 99.9%

Time: 4.0s

Alternatives: 15

Speedup: 3.2×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], Infinity], N[(t$95$0 * t$95$0 + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(-3.0 * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(2.0 * N[(b * b), $MachinePrecision] + N[(a * N[(4.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 56.4

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

   7. Applied rewrites 56.4%

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

Alternative 2: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, \left(b \cdot b\right) \cdot 2\right) + 4, a, \left(b \cdot b\right) \cdot -12\right) \cdot a\\ \mathbf{elif}\;a \leq 27:\\ \;\;\;\;\mathsf{fma}\left({b}^{3}, b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\frac{\left(-\frac{\mathsf{fma}\left(2 - \frac{12}{a}, b \cdot b, 4\right)}{a}\right) - 4}{a}\right) + 1\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6)
   (* (fma (+ (fma (+ 4.0 a) a (* (* b b) 2.0)) 4.0) a (* (* b b) -12.0)) a)
   (if (<= a 27.0)
     (- (fma (pow b 3.0) b (* (* b b) 4.0)) 1.0)
     (*
      (+ (- (/ (- (- (/ (fma (- 2.0 (/ 12.0 a)) (* b b) 4.0) a)) 4.0) a)) 1.0)
      (* (* a a) (* a a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6) {
		tmp = fma((fma((4.0 + a), a, ((b * b) * 2.0)) + 4.0), a, ((b * b) * -12.0)) * a;
	} else if (a <= 27.0) {
		tmp = fma(pow(b, 3.0), b, ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = (-((-(fma((2.0 - (12.0 / a)), (b * b), 4.0) / a) - 4.0) / a) + 1.0) * ((a * a) * (a * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6)
		tmp = Float64(fma(Float64(fma(Float64(4.0 + a), a, Float64(Float64(b * b) * 2.0)) + 4.0), a, Float64(Float64(b * b) * -12.0)) * a);
	elseif (a <= 27.0)
		tmp = Float64(fma((b ^ 3.0), b, Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(fma(Float64(2.0 - Float64(12.0 / a)), Float64(b * b), 4.0) / a)) - 4.0) / a)) + 1.0) * Float64(Float64(a * a) * Float64(a * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 27.0], N[(N[(N[Power[b, 3.0], $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[((-N[(N[((-N[(N[(N[(2.0 - N[(12.0 / a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + 4.0), $MachinePrecision] / a), $MachinePrecision]) - 4.0), $MachinePrecision] / a), $MachinePrecision]) + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.59999999999999964

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 49.2%

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

   if -8.59999999999999964 < a < 27

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if 27 < a

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\left(-\frac{\left(-\frac{\mathsf{fma}\left(2 - \frac{12}{a}, b \cdot b, 4\right)}{a}\right) - 4}{a}\right) + 1\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \]

      9. lift-*.f64 51.3

         \[\leadsto \left(\left(-\frac{\left(-\frac{\mathsf{fma}\left(2 - \frac{12}{a}, b \cdot b, 4\right)}{a}\right) - 4}{a}\right) + 1\right) \cdot \left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \]

   7. Applied rewrites 51.3%

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

Alternative 3: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4 + a, a, \left(b \cdot b\right) \cdot 2\right) + 4, a, \left(b \cdot b\right) \cdot -12\right) \cdot a\\ \mathbf{elif}\;a \leq 27:\\ \;\;\;\;\mathsf{fma}\left({b}^{3}, b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + \mathsf{fma}\left(2, b \cdot b, a \cdot \left(4 + a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6)
   (* (fma (+ (fma (+ 4.0 a) a (* (* b b) 2.0)) 4.0) a (* (* b b) -12.0)) a)
   (if (<= a 27.0)
     (- (fma (pow b 3.0) b (* (* b b) 4.0)) 1.0)
     (* (* a a) (+ 4.0 (fma 2.0 (* b b) (* a (+ 4.0 a))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6) {
		tmp = fma((fma((4.0 + a), a, ((b * b) * 2.0)) + 4.0), a, ((b * b) * -12.0)) * a;
	} else if (a <= 27.0) {
		tmp = fma(pow(b, 3.0), b, ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = (a * a) * (4.0 + fma(2.0, (b * b), (a * (4.0 + a))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6)
		tmp = Float64(fma(Float64(fma(Float64(4.0 + a), a, Float64(Float64(b * b) * 2.0)) + 4.0), a, Float64(Float64(b * b) * -12.0)) * a);
	elseif (a <= 27.0)
		tmp = Float64(fma((b ^ 3.0), b, Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(4.0 + fma(2.0, Float64(b * b), Float64(a * Float64(4.0 + a)))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 27.0], N[(N[(N[Power[b, 3.0], $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(2.0 * N[(b * b), $MachinePrecision] + N[(a * N[(4.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.59999999999999964

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 49.2%

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

   if -8.59999999999999964 < a < 27

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if 27 < a

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 56.4

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

   7. Applied rewrites 56.4%

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

Alternative 4: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(4 + \mathsf{fma}\left(2, b \cdot b, a \cdot \left(4 + a\right)\right)\right)\\ \mathbf{if}\;a \leq -8.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 27:\\ \;\;\;\;\mathsf{fma}\left({b}^{3}, b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (+ 4.0 (fma 2.0 (* b b) (* a (+ 4.0 a)))))))
   (if (<= a -8.6)
     t_0
     (if (<= a 27.0) (- (fma (pow b 3.0) b (* (* b b) 4.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (4.0 + fma(2.0, (b * b), (a * (4.0 + a))));
	double tmp;
	if (a <= -8.6) {
		tmp = t_0;
	} else if (a <= 27.0) {
		tmp = fma(pow(b, 3.0), b, ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(4.0 + fma(2.0, Float64(b * b), Float64(a * Float64(4.0 + a)))))
	tmp = 0.0
	if (a <= -8.6)
		tmp = t_0;
	elseif (a <= 27.0)
		tmp = Float64(fma((b ^ 3.0), b, Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(2.0 * N[(b * b), $MachinePrecision] + N[(a * N[(4.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -8.6], t$95$0, If[LessEqual[a, 27.0], N[(N[(N[Power[b, 3.0], $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.59999999999999964 or 27 < a

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 56.4

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

   7. Applied rewrites 56.4%

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

   if -8.59999999999999964 < a < 27

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

Alternative 5: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(4 + \mathsf{fma}\left(2, b \cdot b, a \cdot \left(4 + a\right)\right)\right)\\ \mathbf{if}\;a \leq -8.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 27:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (+ 4.0 (fma 2.0 (* b b) (* a (+ 4.0 a)))))))
   (if (<= a -8.6)
     t_0
     (if (<= a 27.0) (- (fma (* (* b b) b) b (* (* b b) 4.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (4.0 + fma(2.0, (b * b), (a * (4.0 + a))));
	double tmp;
	if (a <= -8.6) {
		tmp = t_0;
	} else if (a <= 27.0) {
		tmp = fma(((b * b) * b), b, ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(4.0 + fma(2.0, Float64(b * b), Float64(a * Float64(4.0 + a)))))
	tmp = 0.0
	if (a <= -8.6)
		tmp = t_0;
	elseif (a <= 27.0)
		tmp = Float64(fma(Float64(Float64(b * b) * b), b, Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(4.0 + N[(2.0 * N[(b * b), $MachinePrecision] + N[(a * N[(4.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -8.6], t$95$0, If[LessEqual[a, 27.0], N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -8.59999999999999964 or 27 < a

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 56.4

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

   7. Applied rewrites 56.4%

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

   if -8.59999999999999964 < a < 27

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

Alternative 6: 93.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+78}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 47:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(1 + \frac{4}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e+78)
   (* (* a a) (* a a))
   (if (<= a 47.0)
     (- (fma (* (* b b) b) b (* (* b b) 4.0)) 1.0)
     (* (* (* (* a a) a) a) (+ 1.0 (/ 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.9e+78) {
		tmp = (a * a) * (a * a);
	} else if (a <= 47.0) {
		tmp = fma(((b * b) * b), b, ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = (((a * a) * a) * a) * (1.0 + (4.0 / a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.9e+78)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 47.0)
		tmp = Float64(fma(Float64(Float64(b * b) * b), b, Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) * Float64(1.0 + Float64(4.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.9e+78], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 47.0], N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.9e78

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

   if -5.9e78 < a < 47

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

   if 47 < a

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. unpow3 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 46.0

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

   6. Applied rewrites 46.0%

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

Alternative 7: 93.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+78}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 47:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(1 + \frac{4}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e+78)
   (* (* a a) (* a a))
   (if (<= a 47.0)
     (- (fma (* b b) (* b b) (* (* b b) 4.0)) 1.0)
     (* (* (* (* a a) a) a) (+ 1.0 (/ 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.9e+78) {
		tmp = (a * a) * (a * a);
	} else if (a <= 47.0) {
		tmp = fma((b * b), (b * b), ((b * b) * 4.0)) - 1.0;
	} else {
		tmp = (((a * a) * a) * a) * (1.0 + (4.0 / a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.9e+78)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 47.0)
		tmp = Float64(fma(Float64(b * b), Float64(b * b), Float64(Float64(b * b) * 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) * Float64(1.0 + Float64(4.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.9e+78], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 47.0], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.9e78

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

   if -5.9e78 < a < 47

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

   if 47 < a

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. unpow3 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 46.0

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

   6. Applied rewrites 46.0%

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

Alternative 8: 93.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+78}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 47:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) \cdot \left(1 + \frac{4}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e+78)
   (* (* a a) (* a a))
   (if (<= a 47.0)
     (- (* (* b b) (fma b b 4.0)) 1.0)
     (* (* (* (* a a) a) a) (+ 1.0 (/ 4.0 a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.9e+78) {
		tmp = (a * a) * (a * a);
	} else if (a <= 47.0) {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	} else {
		tmp = (((a * a) * a) * a) * (1.0 + (4.0 / a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.9e+78)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 47.0)
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) * Float64(1.0 + Float64(4.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.9e+78], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 47.0], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * N[(1.0 + N[(4.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.9e78

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

   if -5.9e78 < a < 47

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 69.3

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

   6. Applied rewrites 69.3%

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

   if 47 < a

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. unpow3 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 46.0

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

   6. Applied rewrites 46.0%

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

Alternative 9: 93.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+78}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 47:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{4}{a} + 1\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e+78)
   (* (* a a) (* a a))
   (if (<= a 47.0)
     (- (* (* b b) (fma b b 4.0)) 1.0)
     (* (* (+ (/ 4.0 a) 1.0) (* (* a a) a)) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.9e+78) {
		tmp = (a * a) * (a * a);
	} else if (a <= 47.0) {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	} else {
		tmp = (((4.0 / a) + 1.0) * ((a * a) * a)) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.9e+78)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 47.0)
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(4.0 / a) + 1.0) * Float64(Float64(a * a) * a)) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.9e+78], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 47.0], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(4.0 / a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.9e78

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

   if -5.9e78 < a < 47

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 69.3

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

   6. Applied rewrites 69.3%

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

   if 47 < a

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. lower-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 46.0

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

   9. Applied rewrites 46.0%

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

   11. Applied rewrites 46.0%

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

Alternative 10: 93.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.9 \cdot 10^{+78}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 47:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(4 + a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.9e+78)
   (* (* a a) (* a a))
   (if (<= a 47.0)
     (- (* (* b b) (fma b b 4.0)) 1.0)
     (* (+ 4.0 a) (* (* a a) a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.9e+78) {
		tmp = (a * a) * (a * a);
	} else if (a <= 47.0) {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	} else {
		tmp = (4.0 + a) * ((a * a) * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.9e+78)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 47.0)
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	else
		tmp = Float64(Float64(4.0 + a) * Float64(Float64(a * a) * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.9e+78], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 47.0], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 + a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.9e78

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

   if -5.9e78 < a < 47

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 69.3

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

   6. Applied rewrites 69.3%

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

   if 47 < a

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. lower-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 46.0

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

   9. Applied rewrites 46.0%

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

   10. Taylor expanded in a around 0

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

       1. pow3 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 46.0

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

   12. Applied rewrites 46.0%

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

Alternative 11: 81.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -4500000:\\ \;\;\;\;t\_0 \cdot a\\ \mathbf{elif}\;a \leq 10.5:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(4 + a\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) a)))
   (if (<= a -4500000.0)
     (* t_0 a)
     (if (<= a 10.5) (- (* (* b b) 4.0) 1.0) (* (+ 4.0 a) t_0)))))

C

double code(double a, double b) {
	double t_0 = (a * a) * a;
	double tmp;
	if (a <= -4500000.0) {
		tmp = t_0 * a;
	} else if (a <= 10.5) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = (4.0 + a) * t_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) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * a
    if (a <= (-4500000.0d0)) then
        tmp = t_0 * a
    else if (a <= 10.5d0) then
        tmp = ((b * b) * 4.0d0) - 1.0d0
    else
        tmp = (4.0d0 + a) * t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * a;
	double tmp;
	if (a <= -4500000.0) {
		tmp = t_0 * a;
	} else if (a <= 10.5) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = (4.0 + a) * t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * a
	tmp = 0
	if a <= -4500000.0:
		tmp = t_0 * a
	elif a <= 10.5:
		tmp = ((b * b) * 4.0) - 1.0
	else:
		tmp = (4.0 + a) * t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * a)
	tmp = 0.0
	if (a <= -4500000.0)
		tmp = Float64(t_0 * a);
	elseif (a <= 10.5)
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(4.0 + a) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * a;
	tmp = 0.0;
	if (a <= -4500000.0)
		tmp = t_0 * a;
	elseif (a <= 10.5)
		tmp = ((b * b) * 4.0) - 1.0;
	else
		tmp = (4.0 + a) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -4500000.0], N[(t$95$0 * a), $MachinePrecision], If[LessEqual[a, 10.5], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 + a), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.5e6

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. unpow3 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 45.6

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

   6. Applied rewrites 45.6%

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

   if -4.5e6 < a < 10.5

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   if 10.5 < a

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. lower-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 46.0

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

   9. Applied rewrites 46.0%

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

   10. Taylor expanded in a around 0

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

       1. pow3 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 46.0

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

   12. Applied rewrites 46.0%

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

Alternative 12: 81.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -4500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 10.6:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* (* a a) a) a)))
   (if (<= a -4500000.0) t_0 (if (<= a 10.6) (- (* (* b b) 4.0) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * a) * a;
	double tmp;
	if (a <= -4500000.0) {
		tmp = t_0;
	} else if (a <= 10.6) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = ((a * a) * a) * a
    if (a <= (-4500000.0d0)) then
        tmp = t_0
    else if (a <= 10.6d0) then
        tmp = ((b * b) * 4.0d0) - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = ((a * a) * a) * a;
	double tmp;
	if (a <= -4500000.0) {
		tmp = t_0;
	} else if (a <= 10.6) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((a * a) * a) * a
	tmp = 0
	if a <= -4500000.0:
		tmp = t_0
	elif a <= 10.6:
		tmp = ((b * b) * 4.0) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * a) * a)
	tmp = 0.0
	if (a <= -4500000.0)
		tmp = t_0;
	elseif (a <= 10.6)
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((a * a) * a) * a;
	tmp = 0.0;
	if (a <= -4500000.0)
		tmp = t_0;
	elseif (a <= 10.6)
		tmp = ((b * b) * 4.0) - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -4500000.0], t$95$0, If[LessEqual[a, 10.6], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e6 or 10.5999999999999996 < a

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. unpow3 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 45.6

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

   6. Applied rewrites 45.6%

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

   if -4.5e6 < a < 10.5999999999999996

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

Alternative 13: 81.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -4500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 10.6:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -4500000.0) t_0 (if (<= a 10.6) (- (* (* b b) 4.0) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -4500000.0) {
		tmp = t_0;
	} else if (a <= 10.6) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * a)
    if (a <= (-4500000.0d0)) then
        tmp = t_0
    else if (a <= 10.6d0) then
        tmp = ((b * b) * 4.0d0) - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -4500000.0) {
		tmp = t_0;
	} else if (a <= 10.6) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if a <= -4500000.0:
		tmp = t_0
	elif a <= 10.6:
		tmp = ((b * b) * 4.0) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -4500000.0)
		tmp = t_0;
	elseif (a <= 10.6)
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * a);
	tmp = 0.0;
	if (a <= -4500000.0)
		tmp = t_0;
	elseif (a <= 10.6)
		tmp = ((b * b) * 4.0) - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4500000.0], t$95$0, If[LessEqual[a, 10.6], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e6 or 10.5999999999999996 < a

   1. Initial program 74.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

   if -4.5e6 < a < 10.5999999999999996

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

Alternative 14: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.2:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b b)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.2)
   (- (* (* b b) 4.0) 1.0)
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.2) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b * b) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.2d0)) then
        tmp = ((b * b) * 4.0d0) - 1.0d0
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.2) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.2:
		tmp = ((b * b) * 4.0) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.2)
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.2)
		tmp = ((b * b) * 4.0) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.2], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.20000000000000001

   1. Initial program 74.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 51.0

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

   11. Applied rewrites 51.0%

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

   if -0.20000000000000001 < (-.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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 74.0%

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   7. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.9

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

   9. Applied rewrites 45.9%

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

Alternative 15: 51.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return ((b * b) * 4.0) - 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) * 4.0d0) - 1.0d0
end function

Java

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

Python

def code(a, b):
	return ((b * b) * 4.0) - 1.0

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 74.0%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(4 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left({b}^{4} + \color{blue}{4 \cdot {b}^{2}}\right) - 1 \]

   2. metadata-eval N/A

      \[\leadsto \left({b}^{\left(2 + 2\right)} + 4 \cdot {b}^{2}\right) - 1 \]

   3. pow-prod-up N/A

      \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{4} \cdot {b}^{2}\right) - 1 \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left({b}^{2}, \color{blue}{{b}^{2}}, 4 \cdot {b}^{2}\right) - 1 \]

   5. pow2 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot b, {\color{blue}{b}}^{2}, 4 \cdot {b}^{2}\right) - 1 \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot b, {\color{blue}{b}}^{2}, 4 \cdot {b}^{2}\right) - 1 \]

   7. pow2 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot \color{blue}{b}, 4 \cdot {b}^{2}\right) - 1 \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot \color{blue}{b}, 4 \cdot {b}^{2}\right) - 1 \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, {b}^{2} \cdot 4\right) - 1 \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, {b}^{2} \cdot 4\right) - 1 \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   12. lift-*.f64 69.3

       \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

4. Applied rewrites 69.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 4\right)} - 1 \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b} \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot b, b \cdot \color{blue}{b}, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(b \cdot b\right) \cdot 4}\right) - 1 \]

   4. pow2 N/A

      \[\leadsto \left({b}^{2} \cdot \left(b \cdot b\right) + \left(\color{blue}{b} \cdot b\right) \cdot 4\right) - 1 \]

   5. pow2 N/A

      \[\leadsto \left({b}^{2} \cdot {b}^{2} + \left(b \cdot \color{blue}{b}\right) \cdot 4\right) - 1 \]

   6. pow2 N/A

      \[\leadsto \left({b}^{2} \cdot \left(b \cdot b\right) + \left(b \cdot \color{blue}{b}\right) \cdot 4\right) - 1 \]

   7. associate-*r* N/A

      \[\leadsto \left(\left({b}^{2} \cdot b\right) \cdot b + \color{blue}{\left(b \cdot b\right)} \cdot 4\right) - 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, \color{blue}{b}, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   10. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   11. lift-*.f64 69.4

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

6. Applied rewrites 69.4%

   \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, \color{blue}{b}, \left(b \cdot b\right) \cdot 4\right) - 1 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   3. pow3 N/A

      \[\leadsto \mathsf{fma}\left({b}^{3}, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

   4. lower-pow.f64 69.4

      \[\leadsto \mathsf{fma}\left({b}^{3}, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

8. Applied rewrites 69.4%

   \[\leadsto \mathsf{fma}\left({b}^{3}, b, \left(b \cdot b\right) \cdot 4\right) - 1 \]

9. Taylor expanded in b around 0

   \[\leadsto 4 \cdot \color{blue}{{b}^{2}} - 1 \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto {b}^{2} \cdot 4 - 1 \]

    2. pow2 N/A

       \[\leadsto \left(b \cdot b\right) \cdot 4 - 1 \]

    3. lift-*.f64 N/A

       \[\leadsto \left(b \cdot b\right) \cdot 4 - 1 \]

    4. lift-*.f64 51.0

       \[\leadsto \left(b \cdot b\right) \cdot 4 - 1 \]

11. Applied rewrites 51.0%

    \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{4} - 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025128 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))