exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 4

Speedup: 1.0×

• 49.2% 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, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := -0.5 + x\_m \cdot 0.5\\ {\left(e^{x\_m \cdot 2}\right)}^{t\_0} \cdot {\left(e^{2}\right)}^{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ -0.5 (* x_m 0.5))))
   (* (pow (exp (* x_m 2.0)) t_0) (pow (exp 2.0) t_0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = -0.5 + (x_m * 0.5);
	return pow(exp((x_m * 2.0)), t_0) * pow(exp(2.0), t_0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    t_0 = (-0.5d0) + (x_m * 0.5d0)
    code = (exp((x_m * 2.0d0)) ** t_0) * (exp(2.0d0) ** t_0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = -0.5 + (x_m * 0.5);
	return Math.pow(Math.exp((x_m * 2.0)), t_0) * Math.pow(Math.exp(2.0), t_0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = -0.5 + (x_m * 0.5)
	return math.pow(math.exp((x_m * 2.0)), t_0) * math.pow(math.exp(2.0), t_0)

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(-0.5 + Float64(x_m * 0.5))
	return Float64((exp(Float64(x_m * 2.0)) ^ t_0) * (exp(2.0) ^ t_0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	t_0 = -0.5 + (x_m * 0.5);
	tmp = (exp((x_m * 2.0)) ^ t_0) * (exp(2.0) ^ t_0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(-0.5 + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[Exp[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision], t$95$0], $MachinePrecision] * N[Power[N[Exp[2.0], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. exp-prod 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

   1. sqr-pow 100.0%

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

   2. pow-prod-down 100.0%

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

   3. exp-sum 100.0%

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

   4. exp-sum 99.9%

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

   5. swap-sqr 100.0%

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

   6. unpow-prod-down 74.2%

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

   7. div-inv 74.2%

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

   8. +-commutative 74.2%

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

   9. metadata-eval 74.2%

      \[\leadsto {\left(e^{x} \cdot e^{x}\right)}^{\left(\left(-1 + x\right) \cdot \color{blue}{0.5}\right)} \cdot {\left(e^{1} \cdot e^{1}\right)}^{\left(\frac{x + -1}{2}\right)} \]

   10. exp-1-e 74.2%

       \[\leadsto {\left(e^{x} \cdot e^{x}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(\color{blue}{e} \cdot e^{1}\right)}^{\left(\frac{x + -1}{2}\right)} \]

   11. exp-1-e 74.2%

       \[\leadsto {\left(e^{x} \cdot e^{x}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e \cdot \color{blue}{e}\right)}^{\left(\frac{x + -1}{2}\right)} \]

   12. div-inv 74.2%

       \[\leadsto {\left(e^{x} \cdot e^{x}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e \cdot e\right)}^{\color{blue}{\left(\left(x + -1\right) \cdot \frac{1}{2}\right)}} \]

   13. +-commutative 74.2%

       \[\leadsto {\left(e^{x} \cdot e^{x}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e \cdot e\right)}^{\left(\color{blue}{\left(-1 + x\right)} \cdot \frac{1}{2}\right)} \]

   14. metadata-eval 74.2%

       \[\leadsto {\left(e^{x} \cdot e^{x}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e \cdot e\right)}^{\left(\left(-1 + x\right) \cdot \color{blue}{0.5}\right)} \]

8. Applied egg-rr 74.2%

   \[\leadsto \color{blue}{{\left(e^{x} \cdot e^{x}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e \cdot e\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)}} \]
9. Step-by-step derivation

   1. exp-lft-sqr 74.2%

      \[\leadsto {\color{blue}{\left(e^{x \cdot 2}\right)}}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \cdot {\left(e \cdot e\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   2. *-commutative 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\color{blue}{\left(0.5 \cdot \left(-1 + x\right)\right)}} \cdot {\left(e \cdot e\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   3. distribute-rgt-in 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\color{blue}{\left(-1 \cdot 0.5 + x \cdot 0.5\right)}} \cdot {\left(e \cdot e\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   4. metadata-eval 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(\color{blue}{-0.5} + x \cdot 0.5\right)} \cdot {\left(e \cdot e\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   5. exp-1-e 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\left(\color{blue}{e^{1}} \cdot e\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   6. exp-1-e 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\left(e^{1} \cdot \color{blue}{e^{1}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   7. prod-exp 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\color{blue}{\left(e^{1 + 1}\right)}}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   8. metadata-eval 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\left(e^{\color{blue}{2}}\right)}^{\left(\left(-1 + x\right) \cdot 0.5\right)} \]

   9. *-commutative 74.2%

      \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\left(e^{2}\right)}^{\color{blue}{\left(0.5 \cdot \left(-1 + x\right)\right)}} \]

   10. distribute-rgt-in 74.2%

       \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\left(e^{2}\right)}^{\color{blue}{\left(-1 \cdot 0.5 + x \cdot 0.5\right)}} \]

   11. metadata-eval 74.2%

       \[\leadsto {\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\left(e^{2}\right)}^{\left(\color{blue}{-0.5} + x \cdot 0.5\right)} \]

10. Simplified 74.2%

    \[\leadsto \color{blue}{{\left(e^{x \cdot 2}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot {\left(e^{2}\right)}^{\left(-0.5 + x \cdot 0.5\right)}} \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return exp(((x_m * 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 * x_m) + (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(N[(x$95$m * x$95$m), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x + -1}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[E, N[(x$95$m + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. exp-prod 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 72.6%

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

   1. exp-1-e 72.6%

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

9. Simplified 72.6%

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

Alternative 4: 51.6% accurate, 35.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{e} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 1.0 E))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / ((double) M_E);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 / Math.E;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 / math.e

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / exp(1))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 / 2.71828182845904523536;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / E), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. neg-sub0 100.0%

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

   2. sqr-neg 100.0%

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

   3. associate--r- 100.0%

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

   4. metadata-eval 100.0%

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

   5. +-commutative 100.0%

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

   6. sqr-neg 100.0%

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

3. Simplified 100.0%

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

   1. difference-of-sqr--1 100.0%

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

   2. exp-prod 100.0%

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

   3. sub-neg 100.0%

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

   4. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 72.6%

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

   1. exp-1-e 72.6%

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

9. Simplified 72.6%

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

10. Taylor expanded in x around 0 50.6%

    \[\leadsto \color{blue}{\frac{1}{e}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))