exp neg sub

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 8

Speedup: 1.0×

• 50.0% 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

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(e^{x\_m \cdot 2 - -1}\right)}^{\left(-0.5 + x\_m \cdot 0.5\right)} \cdot e^{0.5 \cdot \left(x\_m + -1\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow (exp (- (* x_m 2.0) -1.0)) (+ -0.5 (* x_m 0.5)))
  (exp (* 0.5 (+ x_m -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(exp(((x_m * 2.0) - -1.0)), (-0.5 + (x_m * 0.5))) * exp((0.5 * (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 * 2.0d0) - (-1.0d0))) ** ((-0.5d0) + (x_m * 0.5d0))) * exp((0.5d0 * (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 * 2.0) - -1.0)), (-0.5 + (x_m * 0.5))) * Math.exp((0.5 * (x_m + -1.0)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.exp(((x_m * 2.0) - -1.0)), (-0.5 + (x_m * 0.5))) * math.exp((0.5 * (x_m + -1.0)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((exp(Float64(Float64(x_m * 2.0) - -1.0)) ^ Float64(-0.5 + Float64(x_m * 0.5))) * exp(Float64(0.5 * Float64(x_m + -1.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (exp(((x_m * 2.0) - -1.0)) ^ (-0.5 + (x_m * 0.5))) * exp((0.5 * (x_m + -1.0)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[Exp[N[(N[(x$95$m * 2.0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(-0.5 + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(0.5 * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $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 99.9%

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

   2. exp-prod 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(e^{x + 1}\right)}^{\left(x + -1\right)}} \]
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 99.9%

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

   3. exp-sum 99.9%

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

   4. associate-*r* 99.9%

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

   5. unpow-prod-down 79.7%

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

   6. div-inv 79.7%

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

   7. +-commutative 79.7%

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

   8. metadata-eval 79.7%

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

   9. div-inv 79.7%

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

   10. metadata-eval 79.7%

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

   11. pow-unpow 79.7%

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

   12. exp-prod 79.7%

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

   13. *-un-lft-identity 79.7%

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

   14. +-commutative 79.7%

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

8. Applied egg-rr 79.7%

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

   1. metadata-eval 79.7%

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

   2. sub-neg 79.7%

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

   3. div-exp 79.6%

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

   4. associate-*l/ 79.6%

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

   5. prod-exp 79.6%

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

   6. count-2 79.6%

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

   7. div-exp 79.7%

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

   8. *-commutative 79.7%

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

   9. *-commutative 79.7%

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

   10. distribute-rgt-in 79.7%

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

   11. metadata-eval 79.7%

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

   12. unpow1/2 79.7%

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

   13. +-commutative 79.7%

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

10. Simplified 79.7%

    \[\leadsto \color{blue}{{\left(e^{x \cdot 2 - -1}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot \sqrt{e^{x + -1}}} \]
11. Step-by-step derivation

    1. pow1/2 79.7%

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

    2. pow-exp 79.7%

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

12. Applied egg-rr 79.7%

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

13. Final simplification 79.7%

    \[\leadsto {\left(e^{x \cdot 2 - -1}\right)}^{\left(-0.5 + x \cdot 0.5\right)} \cdot e^{0.5 \cdot \left(x + -1\right)} \]
14. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.98:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;e^{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 0.98) (/ 1.0 E) (exp x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.98) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = exp(x_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.98) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = Math.exp(x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.98:
		tmp = 1.0 / math.e
	else:
		tmp = math.exp(x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.98)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = exp(x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.98)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = exp(x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.98], N[(1.0 / E), $MachinePrecision], N[Exp[x$95$m], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.97999999999999998

   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 99.9%

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

      2. exp-prod 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 68.3%

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

      1. exp-1-e 68.3%

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

   9. Simplified 68.3%

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

   10. Taylor expanded in x around 0 70.8%

       \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 0.97999999999999998 < x

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate--r- 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

      1. difference-of-sqr--1 99.9%

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

      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 98.7%

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

      1. exp-1-e 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in x around inf 98.7%

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

       1. *-un-lft-identity 98.7%

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

       2. e-exp-1 98.7%

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

       3. pow-exp 98.7%

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

       4. *-un-lft-identity 98.7%

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

   12. Applied egg-rr 98.7%

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

       1. *-lft-identity 98.7%

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

   14. Simplified 98.7%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Exp[N[(-1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $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. Final simplification 100.0%

   \[\leadsto e^{-1 + x \cdot x} \]
6. Add Preprocessing

Alternative 4: 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 99.9%

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

   2. exp-prod 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 77.0%

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

   1. exp-1-e 77.0%

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

9. Simplified 77.0%

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

Alternative 5: 75.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.95:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + x\_m \cdot \left(x\_m \cdot 0.5 + 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.95) (/ 1.0 E) (+ 1.0 (* x_m (+ (* x_m 0.5) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = 1.0 / ((double) M_E);
	} else {
		tmp = 1.0 + (x_m * ((x_m * 0.5) + 1.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = 1.0 / Math.E;
	} else {
		tmp = 1.0 + (x_m * ((x_m * 0.5) + 1.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.95:
		tmp = 1.0 / math.e
	else:
		tmp = 1.0 + (x_m * ((x_m * 0.5) + 1.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.95)
		tmp = Float64(1.0 / exp(1));
	else
		tmp = Float64(1.0 + Float64(x_m * Float64(Float64(x_m * 0.5) + 1.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.95)
		tmp = 1.0 / 2.71828182845904523536;
	else
		tmp = 1.0 + (x_m * ((x_m * 0.5) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.95], N[(1.0 / E), $MachinePrecision], N[(1.0 + N[(x$95$m * N[(N[(x$95$m * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   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 99.9%

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

      2. exp-prod 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 68.3%

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

      1. exp-1-e 68.3%

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

   9. Simplified 68.3%

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

   10. Taylor expanded in x around 0 70.8%

       \[\leadsto \color{blue}{\frac{1}{e}} \]

   if 0.94999999999999996 < x

   1. Initial program 99.9%

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

      1. neg-sub0 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate--r- 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. sqr-neg 99.9%

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

   3. Simplified 99.9%

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

      1. difference-of-sqr--1 99.9%

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

      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 98.7%

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

      1. exp-1-e 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in x around inf 98.7%

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

   11. Taylor expanded in x around 0 44.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(\log e + 0.5 \cdot \left(x \cdot {\log e}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. log-E 44.4%

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

       2. associate-*r* 44.4%

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

       3. log-E 44.4%

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

       4. metadata-eval 44.4%

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

       5. associate-*r* 44.4%

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

   13. Simplified 44.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(1 + 0.5 \cdot \left(x \cdot 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;\frac{1}{e}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.5 + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.8% 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 99.9%

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

   2. exp-prod 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 77.0%

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

   1. exp-1-e 77.0%

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

9. Simplified 77.0%

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

10. Taylor expanded in x around 0 51.5%

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

Alternative 7: 11.6% accurate, 35.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m + 1.0), $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 99.9%

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

   2. exp-prod 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 77.0%

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

   1. exp-1-e 77.0%

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

9. Simplified 77.0%

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

10. Taylor expanded in x around inf 37.6%

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

11. Taylor expanded in x around 0 10.6%

    \[\leadsto \color{blue}{1 + x \cdot \log e} \]
12. Step-by-step derivation

    1. +-commutative 10.6%

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

    2. log-E 10.6%

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

    3. *-rgt-identity 10.6%

       \[\leadsto \color{blue}{x} + 1 \]

13. Simplified 10.6%

    \[\leadsto \color{blue}{x + 1} \]
14. Add Preprocessing

Alternative 8: 10.4% accurate, 106.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

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 99.9%

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

   2. exp-prod 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 77.0%

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

   1. exp-1-e 77.0%

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

9. Simplified 77.0%

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

10. Taylor expanded in x around inf 37.6%

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

11. Taylor expanded in x around 0 10.6%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x)
  :name "exp neg sub"
  :precision binary64
  (exp (- (- 1.0 (* x x)))))