exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 10.8s

Alternatives: 14

Speedup: 1.0×

• 50.6% 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

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(e^{x\_m + 1}\right)}^{\left(x\_m + -1\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (pow (exp (+ x_m 1.0)) (+ x_m -1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(exp((x_m + 1.0)), (x_m + -1.0));
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.exp((x_m + 1.0)), (x_m + -1.0))

Julia

x_m = abs(x)
function code(x_m)
	return exp(Float64(x_m + 1.0)) ^ Float64(x_m + -1.0)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[Exp[N[(x$95$m + 1.0), $MachinePrecision]], $MachinePrecision], N[(x$95$m + -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. 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. difference-of-sqr--1 N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 91.5% accurate, 0.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (exp (+ -1.0 (* x_m x_m))) 0.5)
   (/ 1.0 (/ E (fma x_m (fma x_m (* (* x_m x_m) 0.5) x_m) 1.0)))
   (* 0.16666666666666666 (* x_m (/ (* x_m (* x_m (* x_m (* x_m x_m)))) E)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (exp((-1.0 + (x_m * x_m))) <= 0.5) {
		tmp = 1.0 / (((double) M_E) / fma(x_m, fma(x_m, ((x_m * x_m) * 0.5), x_m), 1.0));
	} else {
		tmp = 0.16666666666666666 * (x_m * ((x_m * (x_m * (x_m * (x_m * x_m)))) / ((double) M_E)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (exp(Float64(-1.0 + Float64(x_m * x_m))) <= 0.5)
		tmp = Float64(1.0 / Float64(exp(1) / fma(x_m, fma(x_m, Float64(Float64(x_m * x_m) * 0.5), x_m), 1.0)));
	else
		tmp = Float64(0.16666666666666666 * Float64(x_m * Float64(Float64(x_m * Float64(x_m * Float64(x_m * Float64(x_m * x_m)))) / exp(1))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.5], N[(1.0 / N[(E / N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x$95$m * N[(N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]), $MachinePrecision]), $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

      \[\leadsto \color{blue}{e^{-1} + {x}^{2} \cdot \left(e^{-1} + \frac{1}{2} \cdot \left({x}^{2} \cdot e^{-1}\right)\right)} \]
   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 99.7%

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

   7. Applied rewrites 99.7%

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

   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

   4. Applied rewrites 49.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot x}{e}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), \frac{1}{e}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 33.6%

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot x}{e}, \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 0.027777777777777776, -0.25\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.16666666666666666, -0.5\right)}}, 1\right), \frac{1}{e}\right) \]

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 83.6%

       \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x}{e} \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))