exp2 (problem 3.3.7)

Average Error: 29.9 → 0.3

Time: 8.8s

Precision: binary64

Cost: 26436

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 1e-10)
     (pow x 2.0)
     (+ (exp x) (- t_0 2.0)))))

C

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

↓

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 1e-10) {
		tmp = pow(x, 2.0);
	} else {
		tmp = exp(x) + (t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (((exp(x) - 2.0d0) + t_0) <= 1d-10) then
        tmp = x ** 2.0d0
    else
        tmp = exp(x) + (t_0 - 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 t_0 = Math.exp(-x);
	double tmp;
	if (((Math.exp(x) - 2.0) + t_0) <= 1e-10) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.exp(x) + (t_0 - 2.0);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if ((math.exp(x) - 2.0) + t_0) <= 1e-10:
		tmp = math.pow(x, 2.0)
	else:
		tmp = math.exp(x) + (t_0 - 2.0)
	return tmp

Julia

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

↓

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 1e-10)
		tmp = x ^ 2.0;
	else
		tmp = Float64(exp(x) + Float64(t_0 - 2.0));
	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);
	tmp = 0.0;
	if (((exp(x) - 2.0) + t_0) <= 1e-10)
		tmp = x ^ 2.0;
	else
		tmp = exp(x) + (t_0 - 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_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + t$95$0), $MachinePrecision], 1e-10], N[Power[x, 2.0], $MachinePrecision], N[(N[Exp[x], $MachinePrecision] + N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	29.9
Target	0.0
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 1.00000000000000004e-10

   1. Initial program 30.4

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

   2. Simplified 30.4

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

      [Start] 30.4	\[ \left(e^{x} - 2\right) + e^{-x} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 30.4	\[ \color{blue}{e^{-x} + \left(e^{x} - 2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 30.5	\[ \color{blue}{\left(e^{x} + e^{-x}\right) - 2} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 30.5	\[ \color{blue}{\left(e^{-x} + e^{x}\right)} - 2 \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 30.4	\[ \color{blue}{e^{x} + \left(e^{-x} - 2\right)} \]

   3. Taylor expanded in x around 0 0.1

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

   if 1.00000000000000004e-10 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 10.5

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

   2. Simplified 10.6

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

      [Start] 10.5	\[ \left(e^{x} - 2\right) + e^{-x} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 10.5	\[ \color{blue}{e^{-x} + \left(e^{x} - 2\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 11.1	\[ \color{blue}{\left(e^{x} + e^{-x}\right) - 2} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 11.1	\[ \color{blue}{\left(e^{-x} + e^{x}\right)} - 2 \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 10.6	\[ \color{blue}{e^{x} + \left(e^{-x} - 2\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.5
Cost	26688

\[\left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right) + \left({x}^{2} + 0.002777777777777778 \cdot {x}^{6}\right) \]

Alternative 2
Error	0.6
Cost	19968

\[\left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right) + 0.002777777777777778 \cdot {x}^{6} \]

Alternative 3
Error	0.7
Cost	13248

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

Alternative 4
Error	1.1
Cost	6528

\[{x}^{2} \]

Alternative 5
Error	60.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023090 
(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))))