exp2 (problem 3.3.7)

Average Error: 29.8 → 0.7

Time: 10.8s

Precision: binary64

Cost: 13632

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (* 0.002777777777777778 (pow x 6.0))
  (+ (* 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 (0.002777777777777778 * pow(x, 6.0)) + ((x * x) + (pow(x, 4.0) * 0.08333333333333333));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.002777777777777778d0 * (x ** 6.0d0)) + ((x * x) + ((x ** 4.0d0) * 0.08333333333333333d0))
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

↓

public static double code(double x) {
	return (0.002777777777777778 * Math.pow(x, 6.0)) + ((x * x) + (Math.pow(x, 4.0) * 0.08333333333333333));
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

↓

def code(x):
	return (0.002777777777777778 * math.pow(x, 6.0)) + ((x * x) + (math.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 Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(x * x) + Float64((x ^ 4.0) * 0.08333333333333333)))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

↓

function tmp = code(x)
	tmp = (0.002777777777777778 * (x ^ 6.0)) + ((x * x) + ((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[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	29.8
Target	0.0
Herbie	0.7

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

Derivation

1. Initial program 29.8

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

2. Simplified 29.8

   \[\leadsto \color{blue}{-2 + \left(e^{x} + e^{-x}\right)} \]
   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)))): 6 points increase in error, 3 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 0.7

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

4. Applied egg-rr 0.7

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

5. Applied egg-rr 0.7

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

6. Final simplification 0.7

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

Alternatives

Alternative 1
Error	0.8
Cost	6912

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

Alternative 2
Error	1.2
Cost	192

\[x \cdot x \]

Alternative 3
Error	60.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022317 
(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))))