exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 10

Speedup: 1.0×

• 50.0% 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, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{2 \cdot x\_m}\right)}^{\left(0.5 \cdot x\_m\right)}}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (/ (E) (pow (exp (* 2.0 x_m)) (* 0.5 x_m)))))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

      \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

   8. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{x}\right)}^{x}}}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. exp-lft-sqr-rev N/A

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

   5. *-commutative N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-*.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 100.0

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{2 \cdot x}\right)}^{\left(0.5 \cdot \color{blue}{x}\right)}}} \]

11. Applied rewrites 100.0%

    \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{2 \cdot x}\right)}^{\color{blue}{\left(0.5 \cdot x\right)}}}} \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{{\left(e^{x\_m}\right)}^{x\_m}}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (pow (exp x_m) x_m) (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

      \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

   8. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{x}\right)}^{x}}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-E.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-exp N/A

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

   7. pow2 N/A

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

   8. e-exp-1 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{1}}}{e^{{x}^{2}}}} \]

   9. div-exp N/A

      \[\leadsto \frac{1}{\color{blue}{e^{1 - {x}^{2}}}} \]

   10. pow2 N/A

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

   11. exp-neg N/A

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

   12. sqr-neg-rev N/A

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

   13. fp-cancel-sign-sub N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. pow2 N/A

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

   16. distribute-neg-in N/A

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

   17. metadata-eval N/A

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

   18. mul-1-neg N/A

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

   19. distribute-lft-neg-out N/A

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

   20. metadata-eval N/A

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

   21. *-lft-identity N/A

       \[\leadsto e^{-1 + \color{blue}{{x}^{2}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x}}{\mathsf{E}\left(\right)}} \]
7. Add Preprocessing

Alternative 3: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;e^{-1 + x\_m \cdot x\_m} \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{\mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5) (/ 1.0 (E)) (/ (* x_m x_m) (E))))

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x)))) < 0.5

   1. Initial program 100.0%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

      8. exp-diff N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

      10. exp-1-e N/A

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

      11. lower-E.f64 N/A

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

      12. pow2 N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lift-E.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x))))

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   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 0

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

      1. exp-neg N/A

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

      2. pow2 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      3. div-exp N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      4. e-exp-1 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      5. pow2 N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      6. pow-exp N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

      7. metadata-eval N/A

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

      8. rec-exp N/A

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

      9. e-exp-1 N/A

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

      10. metadata-eval N/A

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

      11. rec-exp N/A

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

      12. e-exp-1 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. mul-1-neg N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. *-rgt-identity N/A

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

      20. mul-1-neg N/A

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

          \[\leadsto \frac{x \cdot x}{\mathsf{E}\left(\right)} \]

       2. lift-*.f64 51.5

          \[\leadsto \frac{x \cdot x}{\mathsf{E}\left(\right)} \]

   11. Applied rewrites 51.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-1 + x \cdot x} \leq 0.5:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{E}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.16666666666666666, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot x\_m, x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.1)
   (/
    1.0
    (/
     (E)
     (fma
      (* (fma (fma (* x_m x_m) 0.16666666666666666 0.5) (* x_m x_m) 1.0) x_m)
      x_m
      1.0)))
   (exp (* x_m x_m))))

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 99.9%

      \[e^{-\left(1 - x \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

         \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

      8. exp-diff N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

      10. exp-1-e N/A

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

      11. lower-E.f64 N/A

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

      12. pow2 N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{x}\right)}^{x}}}} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-exp.f64 N/A

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

      9. lower-/.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}}} \]
   8. Step-by-step derivation

      1. unpow-prod-down N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + \color{blue}{1}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 1\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 1\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 1\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      11. pow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

      13. pow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

      15. pow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

      16. lift-*.f64 94.7

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

   9. Applied rewrites 94.7%

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

       2. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + \color{blue}{1}}} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + 1}} \]

       4. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 1}} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 1}} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\left(\frac{1}{6} \cdot \left(x \cdot x\right) + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 1}} \]

       7. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\left(\left(\frac{1}{6} \cdot \left(x \cdot x\right) + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x\right) \cdot x + 1}} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\left(\left(\frac{1}{6} \cdot \left(x \cdot x\right) + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x, \color{blue}{x}, 1\right)}} \]

   11. Applied rewrites 94.7%

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), x \cdot x, 1\right) \cdot x, \color{blue}{x}, 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (exp (fma x_m x_m -1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(x$95$m * x$95$m + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto e^{{x}^{2} - 1 \cdot \color{blue}{1}} \]

   2. fp-cancel-sub-sign-inv N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

      \[\leadsto e^{x \cdot x + -1 \cdot 1} \]

   5. metadata-eval N/A

      \[\leadsto e^{x \cdot x + -1} \]

   6. lower-fma.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 6: 92.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.16666666666666666, 0.5\right), x\_m \cdot x\_m, 1\right) \cdot x\_m, x\_m, 1\right)}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/
  1.0
  (/
   (E)
   (fma
    (* (fma (fma (* x_m x_m) 0.16666666666666666 0.5) (* x_m x_m) 1.0) x_m)
    x_m
    1.0))))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

      \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

   8. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{x}\right)}^{x}}}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}}} \]
8. Step-by-step derivation

   1. unpow-prod-down N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   2. sqr-pow N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + \color{blue}{1}}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 1\right)}} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 1\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 1\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   13. pow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

   15. pow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

   16. lift-*.f64 90.6

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

9. Applied rewrites 90.6%

   \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

    2. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + \color{blue}{1}}} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + 1}} \]

    4. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 1}} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 1}} \]

    6. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\left(\frac{1}{6} \cdot \left(x \cdot x\right) + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 1}} \]

    7. associate-*r* N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(\left(\left(\frac{1}{6} \cdot \left(x \cdot x\right) + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x\right) \cdot x + 1}} \]

    8. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\left(\left(\frac{1}{6} \cdot \left(x \cdot x\right) + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x, \color{blue}{x}, 1\right)}} \]

11. Applied rewrites 90.6%

    \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), x \cdot x, 1\right) \cdot x, \color{blue}{x}, 1\right)}} \]
12. Add Preprocessing

Alternative 7: 92.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666, x\_m \cdot x\_m, 1\right), x\_m \cdot x\_m, 1\right)}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/
  1.0
  (/
   (E)
   (fma
    (fma (* (* x_m x_m) 0.16666666666666666) (* x_m x_m) 1.0)
    (* x_m x_m)
    1.0))))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

      \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

   8. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{x}\right)}^{x}}}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}}} \]
8. Step-by-step derivation

   1. unpow-prod-down N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   2. sqr-pow N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{1} + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + \color{blue}{1}}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1}} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 1\right)}} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 1\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 1\right)}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right), {x}^{2}, 1\right)}} \]

   13. pow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), {x}^{2}, 1\right)}} \]

   15. pow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

   16. lift-*.f64 90.6

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 1\right)}} \]

9. Applied rewrites 90.6%

   \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}}} \]

10. Taylor expanded in x around inf

    \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {x}^{2}, x \cdot x, 1\right), x \cdot x, 1\right)}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{6}, x \cdot x, 1\right), x \cdot x, 1\right)}} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{6}, x \cdot x, 1\right), x \cdot x, 1\right)}} \]

    3. pow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{6}, x \cdot x, 1\right), x \cdot x, 1\right)}} \]

    4. lift-*.f64 90.3

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.16666666666666666, x \cdot x, 1\right), x \cdot x, 1\right)}} \]

12. Applied rewrites 90.3%

    \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.16666666666666666, x \cdot x, 1\right), x \cdot x, 1\right)}} \]
13. Add Preprocessing

Alternative 8: 88.2% accurate, 2.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (/ (E) (fma (fma (* x_m x_m) 0.5 1.0) (* x_m x_m) 1.0))))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

      \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

   8. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{E}\left(\right)}{{\left(e^{x}\right)}^{x}}}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. unpow-prod-down N/A

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

   2. sqr-pow N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right), {x}^{2}, 1\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right), {x}^{2}, 1\right)}} \]

   11. pow2 N/A

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

   12. lift-*.f64 86.1

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

9. Applied rewrites 86.1%

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

Alternative 9: 75.9% accurate, 6.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (fma x_m x_m 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

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

5. Applied rewrites 48.5%

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

6. Taylor expanded in x around 0

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

   1. exp-neg N/A

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

   2. pow2 N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   3. div-exp N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   4. e-exp-1 N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   5. pow2 N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   6. pow-exp N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{-1} \]

   7. metadata-eval N/A

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

   8. rec-exp N/A

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

   9. e-exp-1 N/A

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

   10. metadata-eval N/A

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

   11. rec-exp N/A

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

   12. e-exp-1 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. mul-1-neg N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. distribute-rgt-neg-out N/A

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

   19. *-rgt-identity N/A

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

   20. mul-1-neg N/A

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

8. Applied rewrites 74.1%

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

Alternative 10: 51.0% accurate, 9.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 1.0 (E)))

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. pow2 N/A

      \[\leadsto \frac{1}{e^{1 - \color{blue}{{x}^{2}}}} \]

   8. exp-diff N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{1}}{e^{{x}^{2}}}}} \]

   10. exp-1-e N/A

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

   11. lower-E.f64 N/A

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lift-E.f64 48.5

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

7. Applied rewrites 48.5%

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

Reproduce

herbie shell --seed 2025054 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))