exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 6

Speedup: 1.0×

• 50.2% 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, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate--r- N/A

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

   7. lift-*.f64 N/A

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

   8. neg-sub0 N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. pow-exp N/A

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

   12. lift-*.f64 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. distribute-neg-in N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-fma.f64 N/A

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

   19. neg-mul-1 N/A

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

   20. pow-exp N/A

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

   21. lift-exp.f64 N/A

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

   22. pow-unpow N/A

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

   23. lift-exp.f64 N/A

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

   24. metadata-eval N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.0001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, x, -0.5\right) \cdot \mathsf{E}\left(\right), x \cdot x, \mathsf{E}\left(\right)\right), x \cdot x, \mathsf{E}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.0001)
   (/
    1.0
    (fma
     (- (fma (* (fma (* 0.16666666666666666 x) x -0.5) (E)) (* x x) (E)))
     (* x x)
     (E)))
   (exp (* x x))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000005e-4

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. exp-neg N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-exp.f64 N/A

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

      12. exp-1-e N/A

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

      13. lower-E.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. 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 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

   12. Applied rewrites 100.0%

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

   if 1.00000000000000005e-4 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. 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. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.0001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, x, -0.5\right) \cdot \mathsf{E}\left(\right), x \cdot x, \mathsf{E}\left(\right)\right), x \cdot x, \mathsf{E}\left(\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 4: 75.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -5:\\ \;\;\;\;\frac{x \cdot x}{\mathsf{E}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{E}\left(\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -5.0) (/ (* x x) (E)) (/ 1.0 (E))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -5

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. 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 56.2

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

   7. Applied rewrites 56.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 56.2%

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

   if -5 < (-.f64 #s(literal 1 binary64) (*.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. Taylor expanded in x around 0

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

   7. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{E}\left(\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.8% 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 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. exp-neg N/A

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

   4. lift--.f64 N/A

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

   5. exp-diff N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 78.5

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

7. Applied rewrites 78.5%

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

Alternative 6: 50.6% 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 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 52.2%

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

Reproduce

herbie shell --seed 2024272 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))