exp2 (problem 3.3.7)

Average Error: 29.4 → 0.6

Time: 7.1s

Precision: binary64

Cost: 19904

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (fma x x (* 0.08333333333333333 (pow x 4.0)))
  (* 0.002777777777777778 (pow x 6.0))))

C

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

↓

double code(double x) {
	return fma(x, x, (0.08333333333333333 * pow(x, 4.0))) + (0.002777777777777778 * pow(x, 6.0));
}

Julia

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

↓

function code(x)
	return Float64(fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0))) + Float64(0.002777777777777778 * (x ^ 6.0)))
end

Wolfram

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

↓

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

Target

Original	29.4
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 29.4

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

2. Simplified 29.5

   \[\leadsto \color{blue}{-2 + \left(e^{x} + e^{-x}\right)} \]
   Proof
   (+.f64 -2 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= metadata-eval (neg.f64 2)) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (neg.f64 2) (exp.f64 x)) (exp.f64 (neg.f64 x)))): 1 points increase in error, 11 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 x) (neg.f64 2))) (exp.f64 (neg.f64 x))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (exp.f64 x) 2)) (exp.f64 (neg.f64 x))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x around 0 0.6

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

4. Simplified 0.6

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\right)} \]
   Proof
   (fma.f64 1/360 (pow.f64 x 6) (fma.f64 x x (*.f64 1/12 (pow.f64 x 4)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 1/360 (pow.f64 x 6) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (*.f64 1/12 (pow.f64 x 4))))): 3 points increase in error, 0 points decrease in error
   (fma.f64 1/360 (pow.f64 x 6) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (*.f64 1/12 (pow.f64 x 4)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/360 (pow.f64 x 6)) (+.f64 (pow.f64 x 2) (*.f64 1/12 (pow.f64 x 4))))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 0.6

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

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	13632

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

Alternative 2
Error	0.8
Cost	13184

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

Alternative 3
Error	0.8
Cost	6912

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

Alternative 4
Error	1.1
Cost	192

\[x \cdot x \]

Alternative 5
Error	60.2
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2022328 
(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))))