exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 11

Speedup: 1.0×

• 49.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(E * N[Power[N[Exp[(-x$95$m)], $MachinePrecision], x$95$m], $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. pow-exp N/A

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

   4. sqr-neg-rev N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. exp-prod N/A

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

   8. mul-1-neg N/A

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

   9. lower-pow.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-exp.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-E.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. associate-/r/ N/A

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

   9. pow-neg N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-E.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. lift-neg.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m)
	return exp(Float64(1.0 - x_m)) ^ fma(-1.0, x_m, -1.0)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[Exp[N[(1.0 - x$95$m), $MachinePrecision]], $MachinePrecision], N[(-1.0 * x$95$m + -1.0), $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-/.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. metadata-eval N/A

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

   13. sqr-neg-rev N/A

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

   14. mul-1-neg N/A

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

   15. mul-1-neg N/A

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

   16. difference-of-squares-rev N/A

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

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

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

   18. exp-prod N/A

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

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

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

6. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.exp((x_m - -1.0)), (x_m - 1.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.exp((x_m - -1.0)), (x_m - 1.0))

Julia

x_m = abs(x)
function code(x_m)
	return exp(Float64(x_m - -1.0)) ^ Float64(x_m - 1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = exp((x_m - -1.0)) ^ (x_m - 1.0);
end

Wolfram

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

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

6. Applied rewrites 100.0%

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

Alternative 4: 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 5: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_E), (x_m - 1.0));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.E, (x_m - 1.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.e, (x_m - 1.0))

Julia

x_m = abs(x)
function code(x_m)
	return exp(1) ^ Float64(x_m - 1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.71828182845904523536 ^ (x_m - 1.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[E, N[(x$95$m - 1.0), $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%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. exp-1-e N/A

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

   2. lift-E.f64 98.7

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

9. Applied rewrites 98.7%

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

Alternative 6: 92.1% accurate, 2.0× speedup

Math

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

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))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(E / N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 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. pow-exp N/A

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

   4. sqr-neg-rev N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. exp-prod N/A

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

   8. mul-1-neg N/A

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

   9. lower-pow.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-exp.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

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. pow-exp 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-neg-rev N/A

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

   3. pow-exp N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 92.1

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

9. Applied rewrites 92.1%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-fma.f64 N/A

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

    7. associate-*r* N/A

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

    8. lower-fma.f64 N/A

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

11. Applied rewrites 92.1%

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

Alternative 7: 91.9% accurate, 2.0× speedup

Math

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

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))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(E / N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$95$m * x$95$m), $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. pow-exp N/A

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

   4. sqr-neg-rev N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. exp-prod N/A

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

   8. mul-1-neg N/A

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

   9. lower-pow.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-exp.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

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. pow-exp 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-neg-rev N/A

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

   3. pow-exp N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 92.1

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

9. Applied rewrites 92.1%

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

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 91.9

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

12. Applied rewrites 91.9%

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

Alternative 8: 88.1% accurate, 2.5× speedup

Math

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

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))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(E / N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(x$95$m * x$95$m), $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. pow-exp N/A

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

   4. sqr-neg-rev N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. exp-prod N/A

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

   8. mul-1-neg N/A

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

   9. lower-pow.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-exp.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

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. pow-exp N/A

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

   2. sqr-neg-rev N/A

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

   3. pow-exp N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 88.1

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

9. Applied rewrites 88.1%

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

Alternative 9: 75.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{e}\\ \end{array} \end{array} \]

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = (x_m * x_m) / ((double) M_E);
	}
	return tmp;
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = 1.0 / math.e
	else:
		tmp = (x_m * x_m) / math.e
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = (x_m * x_m) / 2.71828182845904523536;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(1.0 / E), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   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. pow-exp N/A

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

      4. sqr-neg-rev N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. exp-prod N/A

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

      8. mul-1-neg N/A

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

      9. lower-pow.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-exp.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. associate-/r/ N/A

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

      2. pow-exp N/A

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

      3. sqr-neg-rev N/A

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

      4. pow-exp N/A

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

      5. associate-/r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-E.f64 52.1

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

   9. Applied rewrites 52.1%

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

   10. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 52.1

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

   12. Applied rewrites 52.1%

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

Alternative 10: 75.9% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m + 1.0), $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. Taylor expanded in x around 0

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

   1. pow-exp N/A

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

   2. pow2 N/A

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

   3. e-exp-1 N/A

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

   4. div-exp N/A

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

   5. pow2 N/A

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

   6. exp-neg N/A

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

   7. div-add-rev N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-E.f64 75.9

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

7. Applied rewrites 75.9%

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

Alternative 11: 51.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{e} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 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 2025091 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))