exp2 (problem 3.3.7)

Average Error: 29.3 → 0.6

Time: 10.8s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (fma x x (* (pow x 4.0) 0.08333333333333333)))

C

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

↓

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

Julia

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

↓

function code(x)
	return fma(x, x, Float64((x ^ 4.0) * 0.08333333333333333))
end

Wolfram

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

↓

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

Target

Original	29.3
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 29.3

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

2. Applied egg-rr 30.1

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + -2\right)\right)} + e^{-x} \]

3. Taylor expanded in x around 0 0.6

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

4. Simplified 0.6

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{4} \cdot 0.08333333333333333\right)} \]
   Proof
   (fma.f64 x x (*.f64 (pow.f64 x 4) 1/12)): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite<= *-commutative_binary64 (*.f64 1/12 (pow.f64 x 4)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (*.f64 1/12 (pow.f64 x 4)))): 2 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (*.f64 1/12 (pow.f64 x 4))): 1 points increase in error, 0 points decrease in error

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	6912

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

Alternative 2
Error	1.0
Cost	192

\[x \cdot x \]

Alternative 3
Error	60.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022311 
(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))))