exp2 (problem 3.3.7)

Percentage Accurate: 76.5% → 99.3%

Time: 6.5s

Precision: binary64

Cost: 26436

• Unsound rule application detected in e-graph. Results may not be sound.

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 0.0) (* x x) 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 <= 0.0) {
		tmp = x * x;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (exp(x) - 2.0d0) + exp(-x)
    if (t_0 <= 0.0d0) then
        tmp = x * x
    else
        tmp = t_0
    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 t_0 = (Math.exp(x) - 2.0) + Math.exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = x * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = (math.exp(x) - 2.0) + math.exp(-x)
	tmp = 0
	if t_0 <= 0.0:
		tmp = x * x
	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 <= 0.0)
		tmp = Float64(x * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = (exp(x) - 2.0) + exp(-x);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = x * x;
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(x * x), $MachinePrecision], t$95$0]]

Target

Original	76.5%
Target	100.0%
Herbie	99.3%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 54.4%

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

   2. Simplified 54.4%

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

      [Start] 54.4	\[ \left(e^{x} - 2\right) + e^{-x} \]
      associate-+l- [=>] 54.4	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
      sub-neg [=>] 54.4	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
      sub-neg [=>] 54.4	\[ e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]
      +-commutative [=>] 54.4	\[ e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]
      distribute-neg-in [=>] 54.4	\[ e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]
      remove-double-neg [=>] 54.4	\[ e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]
      metadata-eval [=>] 54.4	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2}} \]

   4. Simplified 100.0%

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

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

   if 0.0 < (+.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;\mathsf{expm1}\left(-x\right)\\ \mathbf{elif}\;x \leq 0.00018:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;\mathsf{expm1}\left(-x\right)\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + \left(\left(1 + 0.08333333333333333 \cdot {x}^{4}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;\mathsf{expm1}\left(-x\right)\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4
Accuracy	94.3%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 5
Accuracy	80.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.16666666666666666\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6
Accuracy	88.8%
Cost	832

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

Alternative 7
Accuracy	76.8%
Cost	192

\[x \cdot x \]

Alternative 8
Accuracy	27.2%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023161 
(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))))