exp2 (problem 3.3.7)

Average Error: 29.8 → 0.6

Time: 1.0min

Precision: binary64

Cost: 26624

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return (0.002777777777777778 * pow(x, 6.0)) + (fma(x, x, (pow(x, 4.0) * 0.08333333333333333)) + (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 Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(fma(x, x, Float64((x ^ 4.0) * 0.08333333333333333)) + 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[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x + N[(N[Power[x, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	29.8
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 29.8

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

2. Taylor expanded in x around 0 0.6

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

3. Applied egg-rr 0.6

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

Alternatives

Alternative 1
Error	0.7
Cost	19904

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

Alternative 2
Error	0.8
Cost	13184

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

Alternative 3
Error	0.9
Cost	6912

\[\left(0.08333333333333333 \cdot {x}^{3} + x\right) \cdot x \]

Alternative 4
Error	1.1
Cost	192

\[x \cdot x \]

Alternative 5
Error	60.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023033 
(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))))