exp2 (problem 3.3.7)

Percentage Accurate: 76.5% → 99.7%

Time: 4.3s

Precision: binary64

Cost: 19972

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\left(e^{x} - 2\right) + e^{-x} \]

↓

\[\begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

↓

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 2e-13)
   (+ (* x x) (* 0.08333333333333333 (* (* x x) (* x x))))
   (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

↓

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 2e-13) {
		tmp = (x * x) + (0.08333333333333333 * ((x * x) * (x * x)));
	} else {
		tmp = (2.0 * cosh(x)) - 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((exp(x) - 2.0d0) + exp(-x)) <= 2d-13) then
        tmp = (x * x) + (0.08333333333333333d0 * ((x * x) * (x * x)))
    else
        tmp = (2.0d0 * cosh(x)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

↓

public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 2e-13) {
		tmp = (x * x) + (0.08333333333333333 * ((x * x) * (x * x)));
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

↓

def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 2e-13:
		tmp = (x * x) + (0.08333333333333333 * ((x * x) * (x * x)))
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

↓

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 2e-13)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (((exp(x) - 2.0) + exp(-x)) <= 2e-13)
		tmp = (x * x) + (0.08333333333333333 * ((x * x) * (x * x)));
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2e-13], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Target

Original	76.5%
Target	100.0%
Herbie	99.7%

\[4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-13

   1. Initial program 55.6%

      \[\left(e^{x} - 2\right) + e^{-x} \]

   2. Simplified 55.6%

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

      [Start] 55.6%	\[ \left(e^{x} - 2\right) + e^{-x} \]
      associate-+l- [=>] 55.6%	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
      sub-neg [=>] 55.6%	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
      sub-neg [=>] 55.6%	\[ e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]
      +-commutative [=>] 55.6%	\[ e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]
      distribute-neg-in [=>] 55.6%	\[ e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]
      remove-double-neg [=>] 55.6%	\[ e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]
      metadata-eval [=>] 55.6%	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + 0.08333333333333333 \cdot {x}^{4}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ {x}^{2} + 0.08333333333333333 \cdot {x}^{4} \]
      unpow2 [=>] 100.0%	\[ \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   5. Applied egg-rr 100.0%

      \[\leadsto x \cdot x + 0.08333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
      Step-by-step derivation

      [Start] 100.0%	\[ x \cdot x + 0.08333333333333333 \cdot {x}^{4} \]
      metadata-eval [<=] 100.0%	\[ x \cdot x + 0.08333333333333333 \cdot {x}^{\color{blue}{\left(2 + 2\right)}} \]
      pow-prod-up [<=] 100.0%	\[ x \cdot x + 0.08333333333333333 \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \]
      pow-prod-down [=>] 100.0%	\[ x \cdot x + 0.08333333333333333 \cdot \color{blue}{{\left(x \cdot x\right)}^{2}} \]
      pow2 [<=] 100.0%	\[ x \cdot x + 0.08333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   if 2.0000000000000001e-13 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]

   2. Simplified 100.0%

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

      [Start] 100.0%	\[ \left(e^{x} - 2\right) + e^{-x} \]
      associate-+l- [=>] 100.0%	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
      sub-neg [=>] 100.0%	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
      sub-neg [=>] 100.0%	\[ e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]
      +-commutative [=>] 100.0%	\[ e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]
      distribute-neg-in [=>] 100.0%	\[ e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]
      remove-double-neg [=>] 100.0%	\[ e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]
      metadata-eval [=>] 100.0%	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]
      Step-by-step derivation

      [Start] 100.0%	\[ e^{x} + \left(e^{-x} + -2\right) \]
      associate-+r+ [=>] 100.0%	\[ \color{blue}{\left(e^{x} + e^{-x}\right) + -2} \]
      cosh-undef [=>] 100.0%	\[ \color{blue}{2 \cdot \cosh x} + -2 \]
      fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(2, \cosh x, -2\right)} \]
      metadata-eval [<=] 100.0%	\[ \mathsf{fma}\left(2, \cosh x, \color{blue}{-2}\right) \]
      fma-neg [<=] 100.0%	\[ \color{blue}{2 \cdot \cosh x - 2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	19972

\[\begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]

Alternative 2
Accuracy	93.7%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3
Accuracy	87.9%
Cost	832

\[x \cdot x + 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

Alternative 4
Accuracy	76.1%
Cost	192

\[x \cdot x \]

Alternative 5
Accuracy	4.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))