exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 17

Speedup: 1.0×

• 49.6% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return Float64((exp(x_m) ^ Float64(x_m + -1.0)) * (exp(1) ^ Float64(x_m + -1.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (exp(x_m) ^ (x_m + -1.0)) * (2.71828182845904523536 ^ (x_m + -1.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[Exp[x$95$m], $MachinePrecision], N[(x$95$m + -1.0), $MachinePrecision]], $MachinePrecision] * N[Power[E, N[(x$95$m + -1.0), $MachinePrecision]], $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. 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. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. exp-sum N/A

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

   5. e-exp-1 N/A

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

   6. *-un-lft-identity N/A

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

   7. lift-E.f64 N/A

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

   8. associate-*r* N/A

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

   9. unpow-prod-down N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-exp.f64 N/A

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

   14. lower-pow.f64 100.0

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

8. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 100.0

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

10. Applied rewrites 100.0%

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

Alternative 2: 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 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. 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

Reproduce

herbie shell --seed 2024229 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))