exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 15

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

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

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: 87.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (exp((-1.0 + (x * x))) <= 0.5) {
		tmp = fma(x, (x * ((double) M_E)), ((double) M_E)) / (((double) M_E) * ((double) M_E));
	} else {
		tmp = (0.5 * (x * (x * (x * x)))) / ((double) M_E);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(Float64(-1.0 + Float64(x * x))) <= 0.5)
		tmp = Float64(fma(x, Float64(x * exp(1)), exp(1)) / Float64(exp(1) * exp(1)));
	else
		tmp = Float64(Float64(0.5 * Float64(x * Float64(x * Float64(x * x)))) / exp(1));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(N[(x * N[(x * E), $MachinePrecision] + E), $MachinePrecision] / N[(E * E), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.f64 x x)))) < 0.5

   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. rec-exp N/A

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

      6. lower-/.f64 N/A

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

      7. exp-1-e N/A

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

      8. lower-E.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

      \[\leadsto \frac{x \cdot x}{e} - \color{blue}{\frac{-1}{e}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.7%

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

   if 0.5 < (exp.f64 (neg.f64 (-.f64 #s(literal 1 binary64) (*.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

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

      1. *-lft-identity N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.8%

      \[\leadsto x \cdot \color{blue}{\frac{0.5 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{e}} \]
   9. Step-by-step derivation

   10. Applied rewrites 75.8%

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.5}{e}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

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

Reproduce

herbie shell --seed 2024226 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))