exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 8

Speedup: 1.0×

• 49.1% 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

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

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 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{E}\left(\right)}, \left(x \cdot x\right) \cdot 0.5, \left(-1 - x \cdot x\right) \cdot \frac{-1}{\mathsf{E}\left(\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2.0)
   (fma (/ (* x x) (E)) (* (* x x) 0.5) (* (- -1.0 (* x x)) (/ -1.0 (E))))
   (exp (* x x))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.5, \mathsf{fma}\left(x, x, 1\right)\right) \cdot e^{-1}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.3%

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

   if 2 < (*.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-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. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{E}\left(\right)}, \left(x \cdot x\right) \cdot 0.5, \left(-1 - x \cdot x\right) \cdot \frac{-1}{\mathsf{E}\left(\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 87.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.5, \mathsf{fma}\left(x, x, 1\right)\right) \cdot e^{-1}} \]
6. Step-by-step derivation

7. Applied rewrites 86.7%

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

9. Applied rewrites 87.0%

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

10. Final simplification 87.0%

    \[\leadsto \mathsf{fma}\left(\left(\left(-x\right) \cdot x\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right), \frac{-1}{\mathsf{E}\left(\right)}, \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right) \]
11. Add Preprocessing

Alternative 4: 88.1% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 87.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.5, \mathsf{fma}\left(x, x, 1\right)\right) \cdot e^{-1}} \]
6. Step-by-step derivation

7. Applied rewrites 86.7%

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

9. Applied rewrites 86.7%

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

10. Final simplification 86.7%

    \[\leadsto \mathsf{fma}\left(x, \frac{x}{\mathsf{E}\left(\right)} \cdot \left(0.5 \cdot \left(x \cdot x\right)\right), \frac{\mathsf{fma}\left(x, x, 1\right)}{\mathsf{E}\left(\right)}\right) \]
11. Add Preprocessing

Alternative 5: 87.7% accurate, 2.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. distribute-lft-in N/A

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

   6. *-rgt-identity N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 87.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 0.5, \mathsf{fma}\left(x, x, 1\right)\right) \cdot e^{-1}} \]
6. Step-by-step derivation

7. Applied rewrites 86.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 85.4%

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

11. Final simplification 85.4%

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

Alternative 6: 76.5% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 5e-5) (/ 1.0 (E)) (* (/ x (E)) x)))

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000024e-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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. e-exp-1 N/A

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

      2. rec-exp N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. e-exp-1 N/A

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

      7. rec-exp N/A

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

      8. metadata-eval N/A

         \[\leadsto e^{-1} + {x}^{2} \cdot e^{\color{blue}{-1}} \]

      9. distribute-rgt1-in N/A

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

      10. metadata-eval N/A

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

      11. rec-exp N/A

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

      12. e-exp-1 N/A

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

      13. associate-*r/ N/A

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

      14. distribute-lft1-in N/A

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

      15. rgt-mult-inverse N/A

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

      16. distribute-lft-in N/A

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

      17. distribute-lft-in N/A

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

      18. rgt-mult-inverse N/A

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

      19. *-rgt-identity N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.5%

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

   if 5.00000000000000024e-5 < (*.f64 x x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot e^{-1}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-exp.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{e^{-1}} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.7%

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

   10. Applied rewrites 48.7%

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

4. Final simplification 74.0%

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

Alternative 7: 76.8% 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

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

   1. e-exp-1 N/A

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

   2. rec-exp N/A

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. associate-*r/ N/A

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

   6. e-exp-1 N/A

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

   7. rec-exp N/A

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

   8. metadata-eval N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{\color{blue}{-1}} \]

   9. distribute-rgt1-in N/A

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

   10. metadata-eval N/A

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

   11. rec-exp N/A

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

   12. e-exp-1 N/A

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

   13. associate-*r/ N/A

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

   14. distribute-lft1-in N/A

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

   15. rgt-mult-inverse N/A

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

   16. distribute-lft-in N/A

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

   17. distribute-lft-in N/A

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

   18. rgt-mult-inverse N/A

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

   19. *-rgt-identity N/A

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

7. Applied rewrites 74.9%

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

Alternative 8: 51.7% 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

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

   1. e-exp-1 N/A

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

   2. rec-exp N/A

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. associate-*r/ N/A

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

   6. e-exp-1 N/A

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

   7. rec-exp N/A

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

   8. metadata-eval N/A

      \[\leadsto e^{-1} + {x}^{2} \cdot e^{\color{blue}{-1}} \]

   9. distribute-rgt1-in N/A

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

   10. metadata-eval N/A

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

   11. rec-exp N/A

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

   12. e-exp-1 N/A

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

   13. associate-*r/ N/A

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

   14. distribute-lft1-in N/A

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

   15. rgt-mult-inverse N/A

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

   16. distribute-lft-in N/A

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

   17. distribute-lft-in N/A

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

   18. rgt-mult-inverse N/A

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

   19. *-rgt-identity N/A

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

7. Applied rewrites 74.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 52.2%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))