Average Error: 58.2 → 0.5

Time: 4.1s

Precision: binary64

Cost: 19520

\[\frac{e^{x} - e^{-x}}{2} \]

↓

\[\frac{2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sinh x\right)\right)}{2} \]

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

↓

(FPCore (x) :precision binary64 (/ (* 2.0 (log1p (expm1 (sinh x)))) 2.0))

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

↓

double code(double x) {
	return (2.0 * log1p(expm1(sinh(x)))) / 2.0;
}

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

↓

public static double code(double x) {
	return (2.0 * Math.log1p(Math.expm1(Math.sinh(x)))) / 2.0;
}

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

↓

def code(x):
	return (2.0 * math.log1p(math.expm1(math.sinh(x)))) / 2.0

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

↓

function code(x)
	return Float64(Float64(2.0 * log1p(expm1(sinh(x)))) / 2.0)
end

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

↓

code[x_] := N[(N[(2.0 * N[Log[1 + N[(Exp[N[Sinh[x], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

\frac{e^{x} - e^{-x}}{2}

↓

\frac{2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sinh x\right)\right)}{2}

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Initial program 58.2
\[\frac{e^{x} - e^{-x}}{2} \]
Applied egg-rr0.0
\[\leadsto \frac{\color{blue}{2 \cdot \sinh x}}{2} \]
Applied egg-rr0.5
\[\leadsto \frac{2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sinh x\right)\right)}}{2} \]
Final simplification0.5
\[\leadsto \frac{2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sinh x\right)\right)}{2} \]

Alternatives

Alternative 1
Error	0.0
Cost	6720

\[\frac{2 \cdot \sinh x}{2} \]

Alternative 2
Error	0.8
Cost	1088

\[\frac{x \cdot \left(x \cdot \left(\left(1 + x \cdot 0.3333333333333333\right) + -1\right)\right) + 2 \cdot x}{2} \]

Alternative 3
Error	0.8
Cost	832

\[\frac{2 \cdot x + x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2} \]

Alternative 4
Error	1.1
Cost	320

\[\frac{2 \cdot x}{2} \]

Reproduce

herbie shell --seed 2022298 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))