exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 11

Speedup: 1.0×

• 49.9% 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

real(8) function code(x)
    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

real(8) function code(x)
    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, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{-x\_m}\\ \frac{--1}{\mathsf{E}\left(\right) \cdot {\left(t\_0 \cdot t\_0\right)}^{\left(0.5 \cdot x\_m\right)}} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m))))
   (/ (- -1.0) (* (E) (pow (* t_0 t_0) (* 0.5 x_m))))))

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-flip N/A

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

   5. pow-flip N/A

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

   6. exp-to-pow N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-log-exp N/A

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

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

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

   10. distribute-lft-neg-in N/A

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

   11. rem-log-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. neg-log N/A

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

   14. exp-to-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. pow-neg N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow-1 N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lift-neg.f64 N/A

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

   18. lower-exp.f64 N/A

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

   19. lower-neg.f64 100.0

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

8. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 100.0

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-flip N/A

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

   5. pow-flip N/A

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

   6. exp-to-pow N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-log-exp N/A

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

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

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

   10. distribute-lft-neg-in N/A

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

   11. rem-log-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. neg-log N/A

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

   14. exp-to-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. pow-neg N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow-1 N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lift-neg.f64 N/A

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

   18. lower-exp.f64 N/A

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

   19. lower-neg.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{--1}{{\left(e^{-x}\right)}^{x} \cdot \mathsf{E}\left(\right)} \]
10. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-flip N/A

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

   5. pow-flip N/A

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

   6. exp-to-pow N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-log-exp N/A

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

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

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

   10. distribute-lft-neg-in N/A

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

   11. rem-log-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. neg-log N/A

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

   14. exp-to-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lift-exp.f64 N/A

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

   4. rec-exp N/A

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

   5. lift-neg.f64 N/A

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

   6. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 0.0002:\\ \;\;\;\;\frac{1}{\frac{\mathsf{E}\left(\right)}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (* x_m x_m) 0.0002)
   (/ 1.0 (/ (E) (fma x_m x_m 1.0)))
   (exp (* x_m x_m))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-4

   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. exp-neg N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. exp-1-e N/A

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

      13. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. lift-exp.f64 N/A

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

      8. rem-log-exp N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. rem-log-exp N/A

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

      12. lift-exp.f64 N/A

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

      13. neg-log N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-neg.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 99.7

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

   9. Applied rewrites 99.7%

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

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 99.7

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

   11. Applied rewrites 99.7%

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

   if 2.0000000000000001e-4 < (*.f64 x x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.3

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

   8. Applied rewrites 99.3%

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

Alternative 6: 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. Step-by-step derivation

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 7: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1 - \left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)}{1 - x\_m \cdot x\_m}}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{\mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (* x_m x_m) 5e+293)
   (/ (/ (- 1.0 (* (* x_m x_m) (* x_m x_m))) (- 1.0 (* x_m x_m))) (E))
   (/ (* x_m x_m) (E))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000033e293

   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. exp-neg N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. exp-1-e N/A

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

      13. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. lift-exp.f64 N/A

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

      8. rem-log-exp N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. rem-log-exp N/A

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

      12. lift-exp.f64 N/A

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

      13. neg-log N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-neg.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 71.2

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

   9. Applied rewrites 71.2%

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

   11. Applied rewrites 84.1%

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

   if 5.00000000000000033e293 < (*.f64 x 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. exp-neg N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. exp-1-e N/A

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

      13. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. lift-exp.f64 N/A

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

      8. rem-log-exp N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. rem-log-exp N/A

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

      12. lift-exp.f64 N/A

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

      13. neg-log N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-neg.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 100.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - x \cdot x}}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{E}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.0% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-flip N/A

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

   5. pow-flip N/A

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

   6. exp-to-pow N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-log-exp N/A

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

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

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

   10. distribute-lft-neg-in N/A

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

   11. rem-log-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. neg-log N/A

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

   14. exp-to-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 78.0

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

9. Applied rewrites 78.0%

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

    1. lift-/.f64 N/A

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

    2. clear-num N/A

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

    3. lower-/.f64 N/A

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

    4. lower-/.f64 78.0

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

11. Applied rewrites 78.0%

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

Alternative 9: 75.8% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 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. exp-neg N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. exp-1-e N/A

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

      13. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.2%

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

   if 1 < (*.f64 x x)

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. exp-1-e N/A

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

      13. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-flip N/A

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

      5. pow-flip N/A

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

      6. exp-to-pow N/A

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

      7. lift-exp.f64 N/A

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

      8. rem-log-exp N/A

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

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

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

      10. distribute-lft-neg-in N/A

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

      11. rem-log-exp N/A

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

      12. lift-exp.f64 N/A

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

      13. neg-log N/A

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

      14. exp-to-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-neg.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 53.0

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

   9. Applied rewrites 53.0%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 53.0%

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

Alternative 10: 76.0% accurate, 6.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

   \[e^{-\left(1 - x \cdot x\right)} \]
2. Add Preprocessing
3. 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. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 78.0

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

7. Applied rewrites 78.0%

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

Alternative 11: 50.8% accurate, 9.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 54.6%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))