Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.9% → 99.9%

Time: 8.9s

Alternatives: 14

Speedup: 5.7×

• 61.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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 0.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 84.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 22000:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right), \left(\mathsf{fma}\left(b \cdot b, 2, 4\right) \cdot a\right) \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 22000.0)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (-
    (fma (* b b) (fma b b (fma -12.0 a 4.0)) (* (* (fma (* b b) 2.0 4.0) a) a))
    1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 22000.0) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma((b * b), fma(b, b, fma(-12.0, a, 4.0)), ((fma((b * b), 2.0, 4.0) * a) * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 22000.0)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(fma(Float64(b * b), fma(b, b, fma(-12.0, a, 4.0)), Float64(Float64(fma(Float64(b * b), 2.0, 4.0) * a) * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 22000.0], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 22000

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. distribute-lft-in N/A

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

       9. associate-+r+ N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

   11. Applied rewrites 78.9%

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

   if 22000 < b

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 98.6%

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

Alternative 3: 98.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \lor \neg \left(a \leq 0.00029\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -1.4) (not (<= a 0.00029)))
   (- (* (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) a) a) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -1.4) || !(a <= 0.00029)) {
		tmp = ((fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -1.4) || !(a <= 0.00029))
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -1.4], N[Not[LessEqual[a, 0.00029]], $MachinePrecision]], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3999999999999999 or 2.9e-4 < a

   1. Initial program 61.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.7%

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

   if -1.3999999999999999 < a < 2.9e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 58.7%

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

   10. Applied rewrites 52.8%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \lor \neg \left(a \leq 0.00029\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right)\\ \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;\left(t\_0 \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 0.00029:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0))))
   (if (<= a -1.4)
     (- (* (* t_0 a) a) 1.0)
     (if (<= a 0.00029)
       (- (* (* (fma b b 4.0) b) b) 1.0)
       (- (* t_0 (* a a)) 1.0)))))

C

double code(double a, double b) {
	double t_0 = fma((4.0 + a), a, fma((b * b), 2.0, 4.0));
	double tmp;
	if (a <= -1.4) {
		tmp = ((t_0 * a) * a) - 1.0;
	} else if (a <= 0.00029) {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	} else {
		tmp = (t_0 * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0))
	tmp = 0.0
	if (a <= -1.4)
		tmp = Float64(Float64(Float64(t_0 * a) * a) - 1.0);
	elseif (a <= 0.00029)
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	else
		tmp = Float64(Float64(t_0 * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.4], N[(N[(N[(t$95$0 * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 0.00029], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(t$95$0 * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.3999999999999999

   1. Initial program 47.0%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.2%

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

   if -1.3999999999999999 < a < 2.9e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 58.7%

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

   10. Applied rewrites 52.8%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.9%

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

   if 2.9e-4 < a

   1. Initial program 71.5%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.4%

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

   10. Applied rewrites 97.4%

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

Alternative 5: 94.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 6600\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -17.0) (not (<= a 6600.0)))
   (* (* a a) (* a a))
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -17.0) || !(a <= 6600.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -17.0) || !(a <= 6600.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -17.0], N[Not[LessEqual[a, 6600.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -17 or 6600 < a

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 87.4

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.3%

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

   if -17 < a < 6600

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 57.4%

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

   10. Applied rewrites 51.6%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 99.2%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 6600\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 6600\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -17.0) (not (<= a 6600.0)))
   (* (* a a) (* a a))
   (- (* (* b b) (fma b b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -17.0) || !(a <= 6600.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -17.0) || !(a <= 6600.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -17.0], N[Not[LessEqual[a, 6600.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -17 or 6600 < a

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 87.4

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.3%

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

   if -17 < a < 6600

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 6600\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 54000:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 54000.0)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (- (* (* (fma b b (fma -12.0 a 4.0)) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 54000.0) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, fma(-12.0, a, 4.0)) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 54000.0)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, fma(-12.0, a, 4.0)) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 54000.0], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 54000

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. distribute-lft-in N/A

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

       9. associate-+r+ N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

   11. Applied rewrites 78.9%

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

   if 54000 < b

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 94.3

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

   5. Applied rewrites 94.3%

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

   7. Applied rewrites 94.3%

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

Alternative 8: 82.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 54000:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(-12, a, 4\right)\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 54000.0)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (- (* (* b b) (fma b b (fma -12.0 a 4.0))) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 54000.0) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((b * b) * fma(b, b, fma(-12.0, a, 4.0))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 54000.0)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, fma(-12.0, a, 4.0))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 54000.0], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + N[(-12.0 * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 54000

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. distribute-lft-in N/A

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

       9. associate-+r+ N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

   11. Applied rewrites 78.9%

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

   if 54000 < b

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 94.3

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

   5. Applied rewrites 94.3%

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

Alternative 9: 93.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 6600\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -17.0) (not (<= a 6600.0)))
   (* (* a a) (* a a))
   (- (* (* (* b b) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -17.0) || !(a <= 6600.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (((b * b) * b) * b) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-17.0d0)) .or. (.not. (a <= 6600.0d0))) then
        tmp = (a * a) * (a * a)
    else
        tmp = (((b * b) * b) * b) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -17.0) || !(a <= 6600.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (((b * b) * b) * b) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -17.0) or not (a <= 6600.0):
		tmp = (a * a) * (a * a)
	else:
		tmp = (((b * b) * b) * b) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -17.0) || !(a <= 6600.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(Float64(b * b) * b) * b) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -17.0) || ~((a <= 6600.0)))
		tmp = (a * a) * (a * a);
	else
		tmp = (((b * b) * b) * b) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -17.0], N[Not[LessEqual[a, 6600.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -17 or 6600 < a

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 87.4

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.3%

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

   if -17 < a < 6600

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 57.4%

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

   10. Applied rewrites 51.6%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 98.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 6600\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 54000:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 54000.0)
   (- (* (* (fma (+ 4.0 a) a 4.0) a) a) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 54000.0) {
		tmp = ((fma((4.0 + a), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 54000.0)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 54000.0], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 54000

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. distribute-lft-in N/A

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

       9. associate-+r+ N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

   11. Applied rewrites 78.9%

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

   if 54000 < b

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 32.4%

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

   10. Applied rewrites 28.4%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 90.4%

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

Alternative 11: 82.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 750\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -17.0) (not (<= a 750.0)))
   (* (* a a) (* a a))
   (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -17.0) || !(a <= 750.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-17.0d0)) .or. (.not. (a <= 750.0d0))) then
        tmp = (a * a) * (a * a)
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a <= -17.0) || !(a <= 750.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -17.0) or not (a <= 750.0):
		tmp = (a * a) * (a * a)
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -17.0) || !(a <= 750.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a <= -17.0) || ~((a <= 750.0)))
		tmp = (a * a) * (a * a);
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -17.0], N[Not[LessEqual[a, 750.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -17 or 750 < a

   1. Initial program 60.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in a around inf

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

      1. lower-pow.f64 87.4

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

   8. Applied rewrites 87.4%

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

   10. Applied rewrites 87.3%

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

   if -17 < a < 750

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 74.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -17 \lor \neg \left(a \leq 750\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 54000:\\ \;\;\;\;\left(\left(\left(a - -4\right) \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 54000.0)
   (- (* (* (* (- a -4.0) a) a) a) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 54000.0) {
		tmp = ((((a - -4.0) * a) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 54000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a - -4.0) * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 54000.0], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 54000

   1. Initial program 85.9%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. pow-sqr N/A

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

       8. distribute-lft-in N/A

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

       9. associate-+r+ N/A

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

       10. *-commutative N/A

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

       11. unpow2 N/A

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

       12. distribute-lft-in N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

   11. Applied rewrites 78.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 77.3%

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

   if 54000 < b

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 32.4%

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

   10. Applied rewrites 28.4%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 90.4%

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

Alternative 13: 59.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e+151) (- (* (* a a) 4.0) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e+151) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.1d+151) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.1e+151) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.1e+151:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e+151)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.1e+151)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e+151], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000003e151

   1. Initial program 84.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+r+ N/A

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

      4. associate-+r+ N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.7%

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

   if 1.10000000000000003e151 < b

   1. Initial program 57.5%

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

   3. Taylor expanded in a around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

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

      9. cube-unmult N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. unpow2 N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. cube-unmult N/A

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

      16. metadata-eval N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. pow-sqr N/A

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

      21. metadata-eval N/A

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

      22. lower-fma.f64 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f64 N/A

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

      25. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

Alternative 14: 50.8% accurate, 11.4× 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 80.3%

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

3. Taylor expanded in a around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. pow-plus N/A

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

   8. metadata-eval N/A

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

   9. cube-unmult N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. cube-unmult N/A

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

   16. metadata-eval N/A

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

   17. pow-plus N/A

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

   18. associate-*r* N/A

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

   19. unpow2 N/A

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

   20. pow-sqr N/A

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

   21. metadata-eval N/A

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

   22. lower-fma.f64 N/A

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

   23. unpow2 N/A

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

   24. lower-*.f64 N/A

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

   25. lower-pow.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 50.6%

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

Reproduce

herbie shell --seed 2024351 
(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))