exp2 (problem 3.3.7)

Average Error: 30.1 → 0.5

Time: 9.2s

Precision: binary64

Cost: 39168

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (fma
  0.002777777777777778
  (pow x 6.0)
  (fma
   x
   x
   (fma 0.08333333333333333 (pow x 4.0) (* 4.96031746031746e-5 (pow x 8.0))))))

C

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

↓

double code(double x) {
	return fma(0.002777777777777778, pow(x, 6.0), fma(x, x, fma(0.08333333333333333, pow(x, 4.0), (4.96031746031746e-5 * pow(x, 8.0)))));
}

Julia

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

↓

function code(x)
	return fma(0.002777777777777778, (x ^ 6.0), fma(x, x, fma(0.08333333333333333, (x ^ 4.0), Float64(4.96031746031746e-5 * (x ^ 8.0)))))
end

Wolfram

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

↓

code[x_] := N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision] + N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	30.1
Target	0.0
Herbie	0.5

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

Derivation

1. Initial program 30.1

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

2. Simplified 30.1

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

   [Start] 30.1	\[ \left(e^{x} - 2\right) + e^{-x} \]
   associate-+l- [=>] 30.1	\[ \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]
   sub-neg [=>] 30.1	\[ \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]
   neg-sub0 [=>] 30.1	\[ e^{x} + \color{blue}{\left(0 - \left(2 - e^{-x}\right)\right)} \]
   associate--r- [=>] 30.1	\[ e^{x} + \color{blue}{\left(\left(0 - 2\right) + e^{-x}\right)} \]
   metadata-eval [=>] 30.1	\[ e^{x} + \left(\color{blue}{-2} + e^{-x}\right) \]
   metadata-eval [<=] 30.1	\[ e^{x} + \left(\color{blue}{\left(-2\right)} + e^{-x}\right) \]
   +-commutative [=>] 30.1	\[ e^{x} + \color{blue}{\left(e^{-x} + \left(-2\right)\right)} \]
   metadata-eval [=>] 30.1	\[ e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

3. Taylor expanded in x around 0 0.5

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

4. Simplified 0.5

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, \mathsf{fma}\left(0.08333333333333333, {x}^{4}, 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)\right)} \]
   Proof

   [Start] 0.5	\[ 0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + \left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right) \]
   fma-def [=>] 0.5	\[ \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + \left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)} \]
   unpow2 [=>] 0.5	\[ \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + \left(0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right) \]
   fma-def [=>] 0.5	\[ \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4} + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)}\right) \]
   fma-def [=>] 0.5	\[ \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left(0.08333333333333333, {x}^{4}, 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)}\right)\right) \]

5. Final simplification 0.5

   \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, \mathsf{fma}\left(0.08333333333333333, {x}^{4}, 4.96031746031746 \cdot 10^{-5} \cdot {x}^{8}\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.7
Cost	13184

\[\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right) \]

Alternative 2
Error	0.7
Cost	6912

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

Alternative 3
Error	1.1
Cost	192

\[x \cdot x \]

Alternative 4
Error	60.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023017 
(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))))