exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 1.0×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

\[\begin{array}{l} \\ {\left(e^{-2}\right)}^{\left(\mathsf{fma}\left(x, x, -1\right) \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (exp -2.0) (* (fma x x -1.0) -0.5)))

C

double code(double x) {
	return pow(exp(-2.0), (fma(x, x, -1.0) * -0.5));
}

Julia

function code(x)
	return exp(-2.0) ^ Float64(fma(x, x, -1.0) * -0.5)
end

Wolfram

code[x_] := N[Power[N[Exp[-2.0], $MachinePrecision], N[(N[(x * x + -1.0), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. prod-exp N/A

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

   5. metadata-eval N/A

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

   6. lower-exp.f64 100.0

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

   \[\leadsto {\left(e^{-2}\right)}^{\left(\mathsf{fma}\left(x, x, -1\right) \cdot -0.5\right)} \]
8. 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, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. prod-exp N/A

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

   5. metadata-eval N/A

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

   6. lower-exp.f64 100.0

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{{\left(e^{-2}\right)}^{\left(-0.5 \cdot \mathsf{fma}\left(x, x, -1\right)\right)}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow-unpow N/A

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

   4. lift-fma.f64 N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. pow-exp N/A

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

   14. metadata-eval N/A

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

   15. exp-1-e N/A

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

   16. lower-E.f64 N/A

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

   17. lift-exp.f64 N/A

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

   18. pow-exp N/A

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

   19. metadata-eval N/A

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

   20. exp-1-e N/A

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

   21. lower-E.f64 N/A

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

   22. div-inv N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

8. Applied rewrites 99.9%

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

Alternative 4: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.005) (/ (fma x x 1.0) (E)) (exp (* x x))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.0050000000000000001

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. prod-exp N/A

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

      5. metadata-eval N/A

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

      6. lower-exp.f64 100.0

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{\left(e^{-2}\right)}^{\left(-0.5 \cdot \mathsf{fma}\left(x, x, -1\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow-unpow N/A

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

      4. lift-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. sqr-pow N/A

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

      9. pow-prod-down N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. pow-exp N/A

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

      14. metadata-eval N/A

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

      15. exp-1-e N/A

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

      16. lower-E.f64 N/A

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

      17. lift-exp.f64 N/A

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

      18. pow-exp N/A

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

      19. metadata-eval N/A

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

      20. exp-1-e N/A

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

      21. lower-E.f64 N/A

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

      22. div-inv N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. rem-log-exp N/A

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

       4. metadata-eval N/A

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

       5. log-E N/A

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

       6. log-E N/A

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

       7. prod-exp N/A

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

       8. log-E N/A

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

       9. e-exp-1 N/A

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

       10. log-E N/A

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

       11. e-exp-1 N/A

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

       12. unpow2 N/A

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

       13. associate-*r/ N/A

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

       14. associate-/l* N/A

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

       15. associate-*r* N/A

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

   11. Applied rewrites 99.0%

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

   if 0.0050000000000000001 < (*.f64 x x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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 6: 75.6% 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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. prod-exp N/A

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

   5. metadata-eval N/A

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

   6. lower-exp.f64 100.0

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{{\left(e^{-2}\right)}^{\left(-0.5 \cdot \mathsf{fma}\left(x, x, -1\right)\right)}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow-unpow N/A

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

   4. lift-fma.f64 N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. pow-exp N/A

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

   14. metadata-eval N/A

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

   15. exp-1-e N/A

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

   16. lower-E.f64 N/A

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

   17. lift-exp.f64 N/A

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

   18. pow-exp N/A

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

   19. metadata-eval N/A

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

   20. exp-1-e N/A

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

   21. lower-E.f64 N/A

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

   22. div-inv N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval N/A

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. rem-log-exp N/A

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

    4. metadata-eval N/A

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

    5. log-E N/A

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

    6. log-E N/A

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

    7. prod-exp N/A

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

    8. log-E N/A

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

    9. e-exp-1 N/A

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

    10. log-E N/A

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

    11. e-exp-1 N/A

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

    12. unpow2 N/A

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

    13. associate-*r/ N/A

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

    14. associate-/l* N/A

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

    15. associate-*r* N/A

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

11. Applied rewrites 75.4%

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

Reproduce

herbie shell --seed 2024254 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))