exp2 (problem 3.3.7)

Average Error: 46.11% → 0.94%

Time: 8.9s

Precision: binary64

Cost: 7296

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (* x x)
  (* (pow x 4.0) (+ (* 0.002777777777777778 (* x x)) 0.08333333333333333))))

C

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

↓

double code(double x) {
	return (x * x) + (pow(x, 4.0) * ((0.002777777777777778 * (x * x)) + 0.08333333333333333));
}

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
    code = (x * x) + ((x ** 4.0d0) * ((0.002777777777777778d0 * (x * x)) + 0.08333333333333333d0))
end function

Java

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

↓

public static double code(double x) {
	return (x * x) + (Math.pow(x, 4.0) * ((0.002777777777777778 * (x * x)) + 0.08333333333333333));
}

Python

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

↓

def code(x):
	return (x * x) + (math.pow(x, 4.0) * ((0.002777777777777778 * (x * x)) + 0.08333333333333333))

Julia

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

↓

function code(x)
	return Float64(Float64(x * x) + Float64((x ^ 4.0) * Float64(Float64(0.002777777777777778 * Float64(x * x)) + 0.08333333333333333)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (x * x) + ((x ^ 4.0) * ((0.002777777777777778 * (x * x)) + 0.08333333333333333));
end

Wolfram

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

↓

code[x_] := N[(N[(x * x), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	46.11%
Target	0.07%
Herbie	0.94%

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

Derivation

1. Initial program 46.11

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

2. Simplified 46.13

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

   [Start] 46.11	\[ \left(e^{x} - 2\right) + e^{-x} \]
   associate-+l- [=>] 46.13	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
   sub-neg [=>] 46.13	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
   neg-sub0 [=>] 46.13	\[ e^{x} + \color{blue}{\left(0 - \left(2 - e^{-x}\right)\right)} \]
   associate--r- [=>] 46.13	\[ e^{x} + \color{blue}{\left(\left(0 - 2\right) + e^{-x}\right)} \]
   metadata-eval [=>] 46.13	\[ e^{x} + \left(\color{blue}{-2} + e^{-x}\right) \]
   metadata-eval [<=] 46.13	\[ e^{x} + \left(\color{blue}{\left(-2\right)} + e^{-x}\right) \]
   +-commutative [=>] 46.13	\[ e^{x} + \color{blue}{\left(e^{-x} + \left(-2\right)\right)} \]
   metadata-eval [=>] 46.13	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

3. Taylor expanded in x around 0 0.94

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

4. Simplified 0.94

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   Proof

   [Start] 0.94	\[ 0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right) \]
   fma-def [=>] 0.94	\[ \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   unpow2 [=>] 0.94	\[ \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

5. Applied egg-rr 0.94

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

6. Applied egg-rr 0.94

   \[\leadsto \color{blue}{{x}^{4} \cdot \left(0.002777777777777778 \cdot \left(x \cdot x\right) + 0.08333333333333333\right)} + x \cdot x \]

7. Final simplification 0.94

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

Alternatives

Alternative 1
Error	1.11%
Cost	6912

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

Alternative 2
Error	1.56%
Cost	192

\[x \cdot x \]

Alternative 3
Error	94.06%
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023089 
(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))))