exp2 (problem 3.3.7)

Average Error: 29.4 → 0.6

Time: 9.2s

Precision: binary64

Cost: 26176

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	29.4
Target	0.1
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 29.4

   \[\leadsto \color{blue}{-2 + \left(e^{x} + e^{-x}\right)} \]
   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, 4 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

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(0.002777777777777778, {x}^{6}, 0.08333333333333333 \cdot {x}^{4}\right)\right)} \]
   Proof
   (fma.f64 x x (fma.f64 1/360 (pow.f64 x 6) (*.f64 1/12 (pow.f64 x 4)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/360 (pow.f64 x 6)) (*.f64 1/12 (pow.f64 x 4))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 x x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/12 (pow.f64 x 4)) (*.f64 1/360 (pow.f64 x 6))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) (+.f64 (*.f64 1/12 (pow.f64 x 4)) (*.f64 1/360 (pow.f64 x 6))))): 2 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (+.f64 (*.f64 1/12 (pow.f64 x 4)) (*.f64 1/360 (pow.f64 x 6)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (pow.f64 x 2) (*.f64 1/12 (pow.f64 x 4))) (*.f64 1/360 (pow.f64 x 6)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_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. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	19840

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

Alternative 2
Error	0.8
Cost	13248

\[{\left(\mathsf{hypot}\left(x, \left(x \cdot x\right) \cdot \sqrt{0.08333333333333333}\right)\right)}^{2} \]

Alternative 3
Error	0.8
Cost	13184

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

Alternative 4
Error	0.8
Cost	6912

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

Alternative 5
Error	1.1
Cost	192

\[x \cdot x \]

Reproduce

herbie shell --seed 2022289 
(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))))