sqrtexp (problem 3.4.4)

Average Error: 40.8 → 0.1

Time: 3.3s

Precision: binary64

Cost: 19520

Math

\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

↓

\[\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))

↓

(FPCore (x) :precision binary64 (sqrt (/ (expm1 (+ x x)) (expm1 x))))

C

double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}

↓

double code(double x) {
	return sqrt((expm1((x + x)) / expm1(x)));
}

Java

public static double code(double x) {
	return Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
}

↓

public static double code(double x) {
	return Math.sqrt((Math.expm1((x + x)) / Math.expm1(x)));
}

Python

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))

↓

def code(x):
	return math.sqrt((math.expm1((x + x)) / math.expm1(x)))

Julia

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end

↓

function code(x)
	return sqrt(Float64(expm1(Float64(x + x)) / expm1(x)))
end

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Sqrt[N[(N[(Exp[N[(x + x), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 40.8

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]
   Proof
   (sqrt.f64 (/.f64 (expm1.f64 (+.f64 x x)) (expm1.f64 x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (/.f64 (expm1.f64 (Rewrite=> count-2_binary64 (*.f64 2 x))) (expm1.f64 x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 2 x)) 1)) (expm1.f64 x))): 166 points increase in error, 0 points decrease in error
   (sqrt.f64 (/.f64 (-.f64 (exp.f64 (*.f64 2 x)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 x) 1)))): 159 points increase in error, 6 points decrease in error

3. Final simplification 0.1

   \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}} \]

Alternatives

Alternative 1
Error	0.0
Cost	13056

\[\mathsf{hypot}\left(1, e^{x \cdot 0.5}\right) \]

Alternative 2
Error	0.0
Cost	12992

\[\sqrt{1 + e^{x}} \]

Reproduce

herbie shell --seed 2022334 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))