exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 10

Speedup: 1.0×

• 49.6% 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, 0.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* (pow (exp x) (- x)) E)))

C

double code(double x) {
	return 1.0 / (pow(exp(x), -x) * ((double) M_E));
}

Java

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

Python

def code(x):
	return 1.0 / (math.pow(math.exp(x), -x) * math.e)

Julia

function code(x)
	return Float64(1.0 / Float64((exp(x) ^ Float64(-x)) * exp(1)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((exp(x) ^ -x) * 2.71828182845904523536);
end

Wolfram

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

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

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-E.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. frac-2neg N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. sqr-pow N/A

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

   12. associate-*r/ N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. frac-2neg N/A

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

   17. pow-flip N/A

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

   18. lower-pow.f64 N/A

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

   19. lift-exp.f64 N/A

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

   20. lower-neg.f64 N/A

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

   21. lift-E.f64 100.0

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

8. Applied rewrites 100.0%

   \[\leadsto \frac{1}{\color{blue}{{\left(e^{x}\right)}^{\left(-x\right)} \cdot e}} \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (pow (exp x) x) E))

C

double code(double x) {
	return pow(exp(x), x) / ((double) M_E);
}

Java

public static double code(double x) {
	return Math.pow(Math.exp(x), x) / Math.E;
}

Python

def code(x):
	return math.pow(math.exp(x), x) / math.e

Julia

function code(x)
	return Float64((exp(x) ^ x) / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) ^ x) / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / E), $MachinePrecision]

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

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   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%

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

Alternative 3: 76.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (exp(-(1.0 - (x * x))) <= 0.5) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = (x * x) / ((double) M_E);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.exp(-(1.0 - (x * x))) <= 0.5) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = (x * x) / Math.E;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(-(1.0 - (x * x))) <= 0.5:
		tmp = 1.0 / math.e
	else:
		tmp = (x * x) / math.e
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (exp(Float64(-Float64(1.0 - Float64(x * x)))) <= 0.5)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(Float64(x * x) / exp(1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (exp(-(1.0 - (x * x))) <= 0.5)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = (x * x) / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Exp[(-N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], 0.5], N[(1.0 / E), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]]

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

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

      12. pow2 N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 99.0

         \[\leadsto \frac{1}{e} \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{1}{\color{blue}{e}} \]

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x 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 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 53.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{{x}^{2}}{e} \]
   10. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \frac{x \cdot x}{e} \]

       2. lift-*.f64 53.1

          \[\leadsto \frac{x \cdot x}{e} \]

   11. Applied rewrites 53.1%

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

Alternative 4: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{e}}\\ \mathbf{if}\;x \leq 1.46:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \frac{x \cdot x}{e}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt E))))
   (if (<= x 1.46) (fma t_0 t_0 (/ (* x x) E)) (exp (* x x)))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_E));
	double tmp;
	if (x <= 1.46) {
		tmp = fma(t_0, t_0, ((x * x) / ((double) M_E)));
	} else {
		tmp = exp((x * x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(exp(1)))
	tmp = 0.0
	if (x <= 1.46)
		tmp = fma(t_0, t_0, Float64(Float64(x * x) / exp(1)));
	else
		tmp = exp(Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[E], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.46], N[(t$95$0 * t$95$0 + N[(N[(x * x), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision], N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.46

   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 68.1%

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

   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 84.7%

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

      1. lift-fma.f64 N/A

         \[\leadsto \frac{x \cdot x + 1}{e} \]

      2. lift-E.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. div-add-rev N/A

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

      7. metadata-eval N/A

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

      8. add-sqr-sqrt N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-E.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-E.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-E.f64 84.7

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

   10. Applied rewrites 84.7%

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

   if 1.46 < 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 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

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

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: 92.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{e}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, 1\right), x \cdot x, 1\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   E
   (fma (fma (fma 0.16666666666666666 (* x x) 0.5) (* x x) 1.0) (* x x) 1.0))))

C

double code(double x) {
	return 1.0 / (((double) M_E) / fma(fma(fma(0.16666666666666666, (x * x), 0.5), (x * x), 1.0), (x * x), 1.0));
}

Julia

function code(x)
	return Float64(1.0 / Float64(exp(1) / fma(fma(fma(0.16666666666666666, Float64(x * x), 0.5), Float64(x * x), 1.0), Float64(x * x), 1.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(E / N[(N[(N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. sqr-pow N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 92.0

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

9. Applied rewrites 92.0%

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

Alternative 7: 88.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / (((double) M_E) / fma(fma(0.5, (x * x), 1.0), (x * x), 1.0));
}

Julia

function code(x)
	return Float64(1.0 / Float64(exp(1) / fma(fma(0.5, Float64(x * x), 1.0), Float64(x * x), 1.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(E / N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lower-/.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. sqr-pow N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 88.2

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

9. Applied rewrites 88.2%

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

Alternative 8: 76.5% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fma (* (- x) x) E (- E)) (* E (- E))))

C

double code(double x) {
	return fma((-x * x), ((double) M_E), -((double) M_E)) / (((double) M_E) * -((double) M_E));
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(-x) * x), exp(1), Float64(-exp(1))) / Float64(exp(1) * Float64(-exp(1))))
end

Wolfram

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

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

5. Applied rewrites 51.3%

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

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 76.5%

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

   1. lift-E.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. pow2 N/A

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

   5. div-add N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

10. Applied rewrites 76.5%

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

Alternative 9: 76.5% accurate, 6.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fma(x, x, 1.0) / ((double) M_E);
}

Julia

function code(x)
	return Float64(fma(x, x, 1.0) / exp(1))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.3%

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

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 76.5%

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

Alternative 10: 51.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e} \end{array} \]

FPCore

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

C

double code(double x) {
	return 1.0 / ((double) M_E);
}

Java

public static double code(double x) {
	return 1.0 / Math.E;
}

Python

def code(x):
	return 1.0 / math.e

Julia

function code(x)
	return Float64(1.0 / exp(1))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / 2.71828182845904523536;
end

Wolfram

code[x_] := N[(1.0 / E), $MachinePrecision]

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

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

   12. pow2 N/A

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

   13. exp-prod N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-exp.f64 100.0

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

4. Applied rewrites 100.0%

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

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 51.3

      \[\leadsto \frac{1}{e} \]

7. Applied rewrites 51.3%

   \[\leadsto \frac{1}{\color{blue}{e}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))