Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.1% → 98.9%

Time: 4.0s

Alternatives: 7

Speedup: 5.5×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\mathsf{TRUE}\left(\right)\]

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: 73.1% 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: 98.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

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

4. Applied rewrites 77.6%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 98.9

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

7. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow2 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 98.9

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

9. Applied rewrites 98.9%

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

Alternative 2: 88.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;b \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, a \cdot a, \left(a \cdot a\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, b \cdot b, \left(b \cdot b\right) \cdot 12\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<= b 8.8e-8)
     (- (fma t_0 (* a a) (* (* a a) 4.0)) 1.0)
     (- (fma t_0 (* b b) (* (* b b) 12.0)) 1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (b <= 8.8e-8) {
		tmp = fma(t_0, (a * a), ((a * a) * 4.0)) - 1.0;
	} else {
		tmp = fma(t_0, (b * b), ((b * b) * 12.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (b <= 8.8e-8)
		tmp = Float64(fma(t_0, Float64(a * a), Float64(Float64(a * a) * 4.0)) - 1.0);
	else
		tmp = Float64(fma(t_0, Float64(b * b), Float64(Float64(b * b) * 12.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 8.8e-8], N[(N[(t$95$0 * N[(a * a), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(t$95$0 * N[(b * b), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.7999999999999994e-8

   1. Initial program 77.2%

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

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      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 83.6

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

   7. Applied rewrites 83.6%

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

   8. Taylor expanded in a around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 68.7

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

   10. Applied rewrites 68.7%

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

   11. Taylor expanded in a around 0

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

       1. pow2 N/A

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

       2. lift-*.f64 82.9

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

   13. Applied rewrites 82.9%

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

   if 8.7999999999999994e-8 < b

   1. Initial program 69.8%

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

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 96.0

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

   10. Applied rewrites 96.0%

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

Alternative 3: 86.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), a \cdot a, \left(b \cdot b\right) \cdot 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.027)
   (- (fma (fma b b (* a a)) (* a a) (* (* b b) 12.0)) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.027) {
		tmp = fma(fma(b, b, (a * a)), (a * a), ((b * b) * 12.0)) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(fma(fma(b, b, Float64(a * a)), Float64(a * a), Float64(Float64(b * b) * 12.0)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.027], N[(N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 77.3%

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

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in a around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 83.8

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

   10. Applied rewrites 83.8%

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

   if 0.0269999999999999997 < b

   1. Initial program 69.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. 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 94.5

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

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right)} - 1 \]
   6. 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 94.4

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

   7. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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 94.5

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

   9. Applied rewrites 94.5%

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

Alternative 4: 80.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -600000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-73}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{+35}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -600000.0)
     t_0
     (if (<= a 3.7e-73)
       (- (* b (* b 12.0)) 1.0)
       (if (<= a 8.8e+35) (* (* b b) (* b b)) t_0)))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -600000.0) {
		tmp = t_0;
	} else if (a <= 3.7e-73) {
		tmp = (b * (b * 12.0)) - 1.0;
	} else if (a <= 8.8e+35) {
		tmp = (b * b) * (b * b);
	} 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 <= (-600000.0d0)) then
        tmp = t_0
    else if (a <= 3.7d-73) then
        tmp = (b * (b * 12.0d0)) - 1.0d0
    else if (a <= 8.8d+35) then
        tmp = (b * b) * (b * b)
    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 <= -600000.0) {
		tmp = t_0;
	} else if (a <= 3.7e-73) {
		tmp = (b * (b * 12.0)) - 1.0;
	} else if (a <= 8.8e+35) {
		tmp = (b * b) * (b * b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if a <= -600000.0:
		tmp = t_0
	elif a <= 3.7e-73:
		tmp = (b * (b * 12.0)) - 1.0
	elif a <= 8.8e+35:
		tmp = (b * b) * (b * b)
	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 <= -600000.0)
		tmp = t_0;
	elseif (a <= 3.7e-73)
		tmp = Float64(Float64(b * Float64(b * 12.0)) - 1.0);
	elseif (a <= 8.8e+35)
		tmp = Float64(Float64(b * b) * Float64(b * b));
	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 <= -600000.0)
		tmp = t_0;
	elseif (a <= 3.7e-73)
		tmp = (b * (b * 12.0)) - 1.0;
	elseif (a <= 8.8e+35)
		tmp = (b * b) * (b * b);
	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, -600000.0], t$95$0, If[LessEqual[a, 3.7e-73], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 8.8e+35], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6e5 or 8.7999999999999994e35 < a

   1. Initial program 47.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.3

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

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{{a}^{4}} \]
   6. 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 91.2

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

   7. Applied rewrites 91.2%

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

   if -6e5 < a < 3.7000000000000001e-73

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. 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 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. 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 73.7

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

   8. Applied rewrites 73.7%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
   9. 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 73.7

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

   10. Applied rewrites 73.7%

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

   if 3.7000000000000001e-73 < a < 8.7999999999999994e35

   1. Initial program 99.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.9

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

   7. Applied rewrites 68.9%

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

Alternative 5: 93.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -1.85 \cdot 10^{+34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{+35}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -1.85e+34)
     t_0
     (if (<= a 8.8e+35) (- (* (* (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 <= -1.85e+34) {
		tmp = t_0;
	} else if (a <= 8.8e+35) {
		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 <= -1.85e+34)
		tmp = t_0;
	elseif (a <= 8.8e+35)
		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, -1.85e+34], t$95$0, If[LessEqual[a, 8.8e+35], 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 < -1.85000000000000004e34 or 8.7999999999999994e35 < a

   1. Initial program 43.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 96.8

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

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{{a}^{4}} \]
   6. 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 96.7

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

   7. Applied rewrites 96.7%

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

   if -1.85000000000000004e34 < a < 8.7999999999999994e35

   1. Initial program 97.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. 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 94.2

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

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, 12, {b}^{4}\right)} - 1 \]
   6. 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 94.1

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

   7. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      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 94.2

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

   9. Applied rewrites 94.2%

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

Alternative 6: 81.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -600000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+35}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -600000.0) t_0 (if (<= a 4e+35) (- (* 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 <= -600000.0) {
		tmp = t_0;
	} else if (a <= 4e+35) {
		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 <= (-600000.0d0)) then
        tmp = t_0
    else if (a <= 4d+35) 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 <= -600000.0) {
		tmp = t_0;
	} else if (a <= 4e+35) {
		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 <= -600000.0:
		tmp = t_0
	elif a <= 4e+35:
		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 <= -600000.0)
		tmp = t_0;
	elseif (a <= 4e+35)
		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 <= -600000.0)
		tmp = t_0;
	elseif (a <= 4e+35)
		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, -600000.0], t$95$0, If[LessEqual[a, 4e+35], 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 < -6e5 or 3.9999999999999999e35 < a

   1. Initial program 47.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 91.3

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

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{{a}^{4}} \]
   6. 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 91.2

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

   7. Applied rewrites 91.2%

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

   if -6e5 < a < 3.9999999999999999e35

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   4. 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 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. 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 69.2

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

   8. Applied rewrites 69.2%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
   9. 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 69.2

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

   10. Applied rewrites 69.2%

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

Alternative 7: 50.8% 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 75.2%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
4. 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 71.6

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

5. Applied rewrites 71.6%

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

6. Taylor expanded in b around 0

   \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
7. 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 48.0

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

8. Applied rewrites 48.0%

   \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
9. 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 48.0

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

10. Applied rewrites 48.0%

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

Reproduce

herbie shell --seed 2025050 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  :pre (TRUE)
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))