Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.0% → 99.9%

Time: 3.3s

Alternatives: 7

Speedup: 5.5×

• 60.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ t_1 := 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\\ \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + t\_1\right) - 1 \leq \infty:\\ \;\;\;\;\left(t\_0 \cdot t\_0 + t\_1\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, \left(a \cdot a\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a)))
        (t_1 (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))))
   (if (<= (- (+ (pow (+ (* a a) (* b b)) 2.0) t_1) 1.0) INFINITY)
     (- (+ (* t_0 t_0) t_1) 1.0)
     (- (fma t_0 (* a a) (* (* a a) 4.0)) 1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double t_1 = 4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a)));
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + t_1) - 1.0) <= ((double) INFINITY)) {
		tmp = ((t_0 * t_0) + t_1) - 1.0;
	} else {
		tmp = fma(t_0, (a * a), ((a * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	t_1 = Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + t_1) - 1.0) <= Inf)
		tmp = Float64(Float64(Float64(t_0 * t_0) + t_1) - 1.0);
	else
		tmp = Float64(fma(t_0, Float64(a * a), Float64(Float64(a * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision] - 1.0), $MachinePrecision], Infinity], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(t$95$0 * N[(a * a), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. pow2 N/A

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

      16. +-commutative N/A

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

      17. pow2 N/A

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

      18. lower-fma.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 99.8

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

   3. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

   3. Applied rewrites 5.1%

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

   4. Taylor expanded in b around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 36.3

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in a around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(3 + a\right)\\ t_1 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + t\_0\right)\right) - 1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, \mathsf{fma}\left(a \cdot a, 1 - a, t\_0\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, a \cdot a, \left(a \cdot a\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) (+ 3.0 a))) (t_1 (fma b b (* a a))))
   (if (<=
        (-
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) t_0)))
         1.0)
        INFINITY)
     (- (fma t_1 t_1 (* (fma (* a a) (- 1.0 a) t_0) 4.0)) 1.0)
     (- (fma t_1 (* a a) (* (* a a) 4.0)) 1.0))))

C

double code(double a, double b) {
	double t_0 = (b * b) * (3.0 + a);
	double t_1 = fma(b, b, (a * a));
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + t_0))) - 1.0) <= ((double) INFINITY)) {
		tmp = fma(t_1, t_1, (fma((a * a), (1.0 - a), t_0) * 4.0)) - 1.0;
	} else {
		tmp = fma(t_1, (a * a), ((a * a) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * Float64(3.0 + a))
	t_1 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + t_0))) - 1.0) <= Inf)
		tmp = Float64(fma(t_1, t_1, Float64(fma(Float64(a * a), Float64(1.0 - a), t_0) * 4.0)) - 1.0);
	else
		tmp = Float64(fma(t_1, Float64(a * a), Float64(Float64(a * a) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], Infinity], N[(N[(t$95$1 * t$95$1 + N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision] + t$95$0), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(t$95$1 * N[(a * a), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

   3. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

   3. Applied rewrites 5.1%

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

   4. Taylor expanded in b around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 36.3

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in a around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 3: 97.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), a \cdot a, \left(a \cdot a\right) \cdot 4\right) - 1\\ \mathbf{if}\;a \leq -0.88:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1650000:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (fma (fma b b (* a a)) (* a a) (* (* a a) 4.0)) 1.0)))
   (if (<= a -0.88)
     t_0
     (if (<= a 1650000.0) (- (* (* (fma b b 12.0) b) b) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = fma(fma(b, b, (a * a)), (a * a), ((a * a) * 4.0)) - 1.0;
	double tmp;
	if (a <= -0.88) {
		tmp = t_0;
	} else if (a <= 1650000.0) {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(fma(fma(b, b, Float64(a * a)), Float64(a * a), Float64(Float64(a * a) * 4.0)) - 1.0)
	tmp = 0.0
	if (a <= -0.88)
		tmp = t_0;
	elseif (a <= 1650000.0)
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -0.88], t$95$0, If[LessEqual[a, 1650000.0], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.880000000000000004 or 1.65e6 < a

   1. Initial program 48.2%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

   3. Applied rewrites 50.8%

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

   4. Taylor expanded in b around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 66.9

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

   6. Applied rewrites 66.9%

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

   7. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in a around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 96.2

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

   12. Applied rewrites 96.2%

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

   if -0.880000000000000004 < a < 1.65e6

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 98.7

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

   8. Applied rewrites 98.7%

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

Alternative 4: 98.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

3. Applied rewrites 75.3%

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

4. Taylor expanded in b around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 82.8

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

6. Applied rewrites 82.8%

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

7. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 98.5

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

9. Applied rewrites 98.5%

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

Alternative 5: 93.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -180000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{+42}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -180000000.0)
     t_0
     (if (<= a 1.05e+42) (- (* (* (fma b b 12.0) b) b) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -180000000.0) {
		tmp = t_0;
	} else if (a <= 1.05e+42) {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -180000000.0)
		tmp = t_0;
	elseif (a <= 1.05e+42)
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -180000000.0], t$95$0, If[LessEqual[a, 1.05e+42], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e8 or 1.04999999999999998e42 < a

   1. Initial program 44.3%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 92.0

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

   4. Applied rewrites 92.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 92.0

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

   6. Applied rewrites 92.0%

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

   if -1.8e8 < a < 1.04999999999999998e42

   1. Initial program 99.6%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 95.1

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

   6. Applied rewrites 95.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 95.2

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

   8. Applied rewrites 95.2%

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

Alternative 6: 82.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -26000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 64000:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -26000000.0)
     t_0
     (if (<= a 64000.0) (- (* b (* b 12.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -26000000.0) {
		tmp = t_0;
	} else if (a <= 64000.0) {
		tmp = (b * (b * 12.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * a)
    if (a <= (-26000000.0d0)) then
        tmp = t_0
    else if (a <= 64000.0d0) then
        tmp = (b * (b * 12.0d0)) - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -26000000.0) {
		tmp = t_0;
	} else if (a <= 64000.0) {
		tmp = (b * (b * 12.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if a <= -26000000.0:
		tmp = t_0
	elif a <= 64000.0:
		tmp = (b * (b * 12.0)) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -26000000.0)
		tmp = t_0;
	elseif (a <= 64000.0)
		tmp = Float64(Float64(b * Float64(b * 12.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * a);
	tmp = 0.0;
	if (a <= -26000000.0)
		tmp = t_0;
	elseif (a <= 64000.0)
		tmp = (b * (b * 12.0)) - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -26000000.0], t$95$0, If[LessEqual[a, 64000.0], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6e7 or 64000 < a

   1. Initial program 47.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 89.1

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

   6. Applied rewrites 89.1%

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

   if -2.6e7 < a < 64000

   1. Initial program 99.7%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in b around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {b}^{2} \cdot 12 - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot 12 - 1 \]

      3. pow2 N/A

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

      4. lift-*.f64 75.4

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

   7. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.4

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

   9. Applied rewrites 75.4%

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

Alternative 7: 51.1% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(b \cdot 12\right) - 1 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lower-pow.f64 69.2

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in b around 0

   \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto {b}^{2} \cdot 12 - 1 \]

   2. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot 12 - 1 \]

   3. pow2 N/A

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

   4. lift-*.f64 51.1

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

7. Applied rewrites 51.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 51.1

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

9. Applied rewrites 51.1%

   \[\leadsto b \cdot \left(b \cdot \color{blue}{12}\right) - 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025107 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))