exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 6

Speedup: 1.0×

• 50.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = exp(-(1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\left(1 - x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (- 1.0 (* x x)))))

C

double code(double x) {
	return exp(-(1.0 - (x * x)));
}

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(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = exp(-(1.0d0 - (x * x)))
end function

Java

public static double code(double x) {
	return Math.exp(-(1.0 - (x * x)));
}

Python

def code(x):
	return math.exp(-(1.0 - (x * x)))

Julia

function code(x)
	return exp(Float64(-Float64(1.0 - Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = exp(-(1.0 - (x * x)));
end

Wolfram

code[x_] := N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(x, x, -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (fma x x -1.0)))

C

double code(double x) {
	return exp(fma(x, x, -1.0));
}

Julia

function code(x)
	return exp(fma(x, x, -1.0))
end

Wolfram

code[x_] := N[Exp[N[(x * x + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, x, 1\right), x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (* (/ 1.0 (E)) (fma (fma (* 0.5 x) x 1.0) (* x x) 1.0))
   (exp (* x x))))

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

         \[\leadsto e^{-1} + \color{blue}{\left({x}^{2} \cdot e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(e^{-1} + {x}^{2} \cdot e^{-1}\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot e^{-1} + e^{-1}\right)} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot e^{-1}\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \cdot e^{-1}}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right) \cdot e^{-1}} \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

      10. lower-exp.f64 N/A

          \[\leadsto \color{blue}{e^{-1}} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{-1} \cdot \left(\color{blue}{x \cdot x} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto e^{-1} \cdot \color{blue}{\mathsf{fma}\left(x, x, {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{e^{-1} \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.7%

      \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\color{blue}{x}, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 91.7%

      \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot 0.5}, \mathsf{fma}\left(x, x, 1\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 91.7%

       \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, x, 1\right), \color{blue}{x \cdot x}, 1\right) \]

   if 1.75 < x

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto e^{\color{blue}{-1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 3.1%

      \[\leadsto e^{\color{blue}{-1}} \]

   6. Taylor expanded in x around inf

      \[\leadsto e^{\color{blue}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

      2. lower-*.f64 100.0

         \[\leadsto e^{\color{blue}{x \cdot x}} \]

   8. Applied rewrites 100.0%

      \[\leadsto e^{\color{blue}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, x, 1\right), x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ 1.0 (E)) (fma (fma (* 0.5 x) x 1.0) (* x x) 1.0)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

      \[\leadsto e^{-1} + \color{blue}{\left({x}^{2} \cdot e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(e^{-1} + {x}^{2} \cdot e^{-1}\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot e^{-1} + e^{-1}\right)} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot e^{-1}\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \cdot e^{-1}}\right) \]

   7. distribute-rgt1-in N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right) \cdot e^{-1}} \]

   8. distribute-rgt-out N/A

      \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

   10. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{-1}} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

   11. unpow2 N/A

       \[\leadsto e^{-1} \cdot \left(\color{blue}{x \cdot x} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto e^{-1} \cdot \color{blue}{\mathsf{fma}\left(x, x, {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)} \]

5. Applied rewrites 89.9%

   \[\leadsto \color{blue}{e^{-1} \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 89.9%

   \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\color{blue}{x}, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 89.9%

   \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right) \cdot 0.5}, \mathsf{fma}\left(x, x, 1\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 89.9%

    \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, x, 1\right), \color{blue}{x \cdot x}, 1\right) \]
12. Add Preprocessing

Alternative 4: 62.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) (/ 1.0 (E)) (/ (* x x) (E))))

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

         \[\leadsto e^{-1} + \color{blue}{\left({x}^{2} \cdot e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(e^{-1} + {x}^{2} \cdot e^{-1}\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot e^{-1} + e^{-1}\right)} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot e^{-1}\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \cdot e^{-1}}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right) \cdot e^{-1}} \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

      10. lower-exp.f64 N/A

          \[\leadsto \color{blue}{e^{-1}} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{-1} \cdot \left(\color{blue}{x \cdot x} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto e^{-1} \cdot \color{blue}{\mathsf{fma}\left(x, x, {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)} \]

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{e^{-1} \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.1%

      \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\color{blue}{x}, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 70.3%

       \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot 1 \]
   11. Step-by-step derivation

   12. Applied rewrites 70.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]

   if 1 < x

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

         \[\leadsto e^{-1} + \color{blue}{\left({x}^{2} \cdot e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(e^{-1} + {x}^{2} \cdot e^{-1}\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot e^{-1} + e^{-1}\right)} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot e^{-1}\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \cdot e^{-1}}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto {x}^{2} \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right) \cdot e^{-1}} \]

      8. distribute-rgt-out N/A

         \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

      10. lower-exp.f64 N/A

          \[\leadsto \color{blue}{e^{-1}} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

      11. unpow2 N/A

          \[\leadsto e^{-1} \cdot \left(\color{blue}{x \cdot x} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto e^{-1} \cdot \color{blue}{\mathsf{fma}\left(x, x, {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)} \]

   5. Applied rewrites 84.3%

      \[\leadsto \color{blue}{e^{-1} \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.3%

      \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\color{blue}{x}, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   9. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(1 + {x}^{2}\right) \cdot e^{\color{blue}{\mathsf{neg}\left(1\right)}} \]

      4. rec-exp N/A

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

      5. e-exp-1 N/A

         \[\leadsto \left(1 + {x}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(1 + {x}^{2}\right) \cdot 1}{\mathsf{E}\left(\right)}} \]

      7. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + {x}^{2}}{\mathsf{E}\left(\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

      12. lower-E.f64 52.5

          \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

   10. Applied rewrites 52.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{{x}^{2}}{\mathsf{E}\left(\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 52.5%

       \[\leadsto \frac{x \cdot x}{\mathsf{E}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x x 1.0) (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

      \[\leadsto e^{-1} + \color{blue}{\left({x}^{2} \cdot e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(e^{-1} + {x}^{2} \cdot e^{-1}\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot e^{-1} + e^{-1}\right)} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot e^{-1}\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \cdot e^{-1}}\right) \]

   7. distribute-rgt1-in N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right) \cdot e^{-1}} \]

   8. distribute-rgt-out N/A

      \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

   10. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{-1}} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

   11. unpow2 N/A

       \[\leadsto e^{-1} \cdot \left(\color{blue}{x \cdot x} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto e^{-1} \cdot \color{blue}{\mathsf{fma}\left(x, x, {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)} \]

5. Applied rewrites 89.9%

   \[\leadsto \color{blue}{e^{-1} \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 89.9%

   \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\color{blue}{x}, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
9. Step-by-step derivation

   1. distribute-rgt1-in N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(1 + {x}^{2}\right) \cdot e^{\color{blue}{\mathsf{neg}\left(1\right)}} \]

   4. rec-exp N/A

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

   5. e-exp-1 N/A

      \[\leadsto \left(1 + {x}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\left(1 + {x}^{2}\right) \cdot 1}{\mathsf{E}\left(\right)}} \]

   7. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{1 + {x}^{2}}}{\mathsf{E}\left(\right)} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + {x}^{2}}{\mathsf{E}\left(\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\color{blue}{{x}^{2} + 1}}{\mathsf{E}\left(\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\color{blue}{x \cdot x} + 1}{\mathsf{E}\left(\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\mathsf{E}\left(\right)} \]

   12. lower-E.f64 78.0

       \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\mathsf{E}\left(\right)}} \]

10. Applied rewrites 78.0%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}} \]
11. Add Preprocessing

Alternative 6: 50.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

      \[\leadsto e^{-1} + \color{blue}{\left({x}^{2} \cdot e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

   2. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(e^{-1} + {x}^{2} \cdot e^{-1}\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot e^{-1} + e^{-1}\right)} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \left(\frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)\right)} \]

   5. associate-*r* N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + {x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot e^{-1}\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \left(e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) \cdot e^{-1}}\right) \]

   7. distribute-rgt1-in N/A

      \[\leadsto {x}^{2} \cdot e^{-1} + \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right) \cdot e^{-1}} \]

   8. distribute-rgt-out N/A

      \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{e^{-1} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right)} \]

   10. lower-exp.f64 N/A

       \[\leadsto \color{blue}{e^{-1}} \cdot \left({x}^{2} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

   11. unpow2 N/A

       \[\leadsto e^{-1} \cdot \left(\color{blue}{x \cdot x} + \left({x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto e^{-1} \cdot \color{blue}{\mathsf{fma}\left(x, x, {x}^{2} \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + 1\right)} \]

5. Applied rewrites 89.9%

   \[\leadsto \color{blue}{e^{-1} \cdot \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 89.9%

   \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot \mathsf{fma}\left(\color{blue}{x}, x, \mathsf{fma}\left({x}^{4}, 0.5, 1\right)\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 51.7%

    \[\leadsto \frac{1}{\mathsf{E}\left(\right)} \cdot 1 \]
11. Step-by-step derivation

12. Applied rewrites 51.7%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{E}\left(\right)}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025003 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))