exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 7.6s

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

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, 0.5× speedup

Math

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

FPCore

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

Derivation

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. 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 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

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. Add Preprocessing

Alternative 3: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lift-exp.f64 N/A

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

   11. *-rgt-identity N/A

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

   12. pow-exp N/A

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

   13. lift-*.f64 N/A

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

   14. lift-E.f64 N/A

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

   15. e-exp-1 N/A

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

   16. div-exp N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-sqr-1 N/A

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

   19. lift-+.f64 N/A

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

   20. lift--.f64 N/A

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

   21. exp-prod N/A

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

   22. lower-pow.f64 N/A

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

8. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\sqrt{\mathsf{E}\left(\right)}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)}}{\mathsf{E}\left(\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow (sqrt (E)) (* 2.0 (* x x))) (E)))

Derivation

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. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. *-lft-identity N/A

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

   5. exp-prod N/A

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

   6. e-exp-1 N/A

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

   7. add-sqr-sqrt N/A

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

   8. pow2 N/A

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

   9. pow-pow N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-E.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt{\mathsf{E}\left(\right)}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)}}}{\mathsf{E}\left(\right)} \]
7. Add Preprocessing

Alternative 5: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

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. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[e^{-\left(1 - x \cdot x\right)} \]
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 99.9

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

4. Applied rewrites 99.9%

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

Alternative 7: 74.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow (E) (- x 1.0)))

Derivation

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. 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. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lift-exp.f64 N/A

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

   11. *-rgt-identity N/A

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

   12. pow-exp N/A

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

   13. lift-*.f64 N/A

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

   14. lift-E.f64 N/A

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

   15. e-exp-1 N/A

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

   16. div-exp N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-sqr-1 N/A

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

   19. lift-+.f64 N/A

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

   20. lift--.f64 N/A

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

   21. exp-prod N/A

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

   22. lower-pow.f64 N/A

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

    1. exp-1-e N/A

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

    2. lower-E.f64 72.7

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

11. Applied rewrites 72.7%

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

Alternative 8: 87.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+304}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\mathsf{fma}\left(x, x, -1\right)}}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1e+304)
   (/ (/ (* (fma x x -1.0) (fma x x 1.0)) (fma x x -1.0)) (E))
   (/ (* x x) (E))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999994e303

   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 64.3

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

   9. Applied rewrites 64.3%

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

   11. Applied rewrites 82.4%

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

   if 9.9999999999999994e303 < (*.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. Add Preprocessing

Alternative 9: 75.4% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1.0) (/ 1.0 (E)) (/ (* x x) (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 98.7%

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

   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 45.0

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

   9. Applied rewrites 45.0%

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

   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 45.0%

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

Alternative 10: 75.7% accurate, 6.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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. 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 71.9

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

7. Applied rewrites 71.9%

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

Alternative 11: 51.3% accurate, 9.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 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. Taylor expanded in x around 0

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

7. Applied rewrites 50.6%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))