Hyperbolic tangent

Average Error: 58.2 → 0.8

Time: 58.1s

Precision: binary64

Cost: 13504

Math

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

↓

\[\frac{\mathsf{expm1}\left(x \cdot -2\right)}{-1 - e^{\left(-6 \cdot x\right) \cdot 0.3333333333333333}} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ (expm1 (* x -2.0)) (- -1.0 (exp (* (* -6.0 x) 0.3333333333333333)))))

C

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

↓

double code(double x) {
	return expm1((x * -2.0)) / (-1.0 - exp(((-6.0 * x) * 0.3333333333333333)));
}

Java

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

↓

public static double code(double x) {
	return Math.expm1((x * -2.0)) / (-1.0 - Math.exp(((-6.0 * x) * 0.3333333333333333)));
}

Python

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

↓

def code(x):
	return math.expm1((x * -2.0)) / (-1.0 - math.exp(((-6.0 * x) * 0.3333333333333333)))

Julia

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

↓

function code(x)
	return Float64(expm1(Float64(x * -2.0)) / Float64(-1.0 - exp(Float64(Float64(-6.0 * x) * 0.3333333333333333))))
end

Wolfram

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

↓

code[x_] := N[(N[(Exp[N[(x * -2.0), $MachinePrecision]] - 1), $MachinePrecision] / N[(-1.0 - N[Exp[N[(N[(-6.0 * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.2

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

2. Simplified 0.8

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(x \cdot -2\right)}{-1 - {\left(e^{x}\right)}^{-2}}} \]
   Proof

3. Applied egg-rr 0.8

   \[\leadsto \frac{\mathsf{expm1}\left(x \cdot -2\right)}{-1 - \color{blue}{e^{\left(x \cdot \left(-4 + -2\right)\right) \cdot 0.3333333333333333}}} \]

4. Applied egg-rr 0.8

   \[\leadsto \frac{\mathsf{expm1}\left(x \cdot -2\right)}{-1 - e^{\color{blue}{\left(-6 \cdot x\right) \cdot 0.3333333333333333}}} \]

Alternatives

Alternative 1
Error	0.8
Cost	13376

\[\frac{\mathsf{expm1}\left(x \cdot -2\right)}{-1 - e^{-2 \cdot x}} \]

Alternative 2
Error	2.3
Cost	6784

\[-0.3333333333333333 \cdot {x}^{3} + x \]

Alternative 3
Error	2.3
Cost	704

\[\left(4 - \left(3 + \left(0.3333333333333333 \cdot x\right) \cdot x\right)\right) \cdot x \]

Alternative 4
Error	2.3
Cost	576

\[\left(1 + \left(-0.3333333333333333 \cdot x\right) \cdot x\right) \cdot x \]

Alternative 5
Error	2.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))