exp2 (problem 3.3.7)

Percentage Accurate: 76.6% → 99.9%

Time: 5.6s

Precision: binary64

Cost: 26436

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 1e-7) (fma x x (* 0.08333333333333333 (pow x 4.0))) t_0)))

C

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

↓

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 1e-7)
		tmp = fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-7], N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Target

Original	76.6%
Target	100.0%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 9.9999999999999995e-8

   1. Initial program 55.0%

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

   2. Simplified 55.1%

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

      [Start] 55.0%	\[ \left(e^{x} - 2\right) + e^{-x} \]
      associate-+l- [=>] 55.1%	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
      sub-neg [=>] 55.1%	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
      sub-neg [=>] 55.1%	\[ e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]
      +-commutative [=>] 55.1%	\[ e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]
      distribute-neg-in [=>] 55.1%	\[ e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]
      remove-double-neg [=>] 55.1%	\[ e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]
      metadata-eval [=>] 55.1%	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]
      Step-by-step derivation

      [Start] 100.0%	\[ x \cdot x + 0.08333333333333333 \cdot {x}^{4} \]
      fma-def [=>] 100.0%	\[ \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]

   if 9.9999999999999995e-8 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	26436

\[\begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	26436

\[\begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 10^{-7}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0056 \lor \neg \left(x \leq 0.0058\right):\\ \;\;\;\;2 \cdot \cosh x - 2\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -0.000205 \lor \neg \left(x \leq 0.0002\right):\\ \;\;\;\;2 \cdot \cosh x - 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5
Accuracy	93.7%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 6
Accuracy	87.9%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 7
Accuracy	76.1%
Cost	192

\[x \cdot x \]

Alternative 8
Accuracy	4.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(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))))