Average Error: 57.9 → 0.0
Time: 5.4s
Precision: binary64
Cost: 6464
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
↓
\[\tanh x \]
(FPCore (x)
 :precision binary64
 (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))
↓
(FPCore (x) :precision binary64 (tanh x))
double code(double x) {
	return (exp(x) - exp(-x)) / (exp(x) + exp(-x));
}
↓
double code(double x) {
	return tanh(x);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / (exp(x) + exp(-x))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = tanh(x)
end function

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.tanh(x);
}

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

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

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / (exp(x) + exp(-x));
end
↓
function tmp = code(x)
	tmp = tanh(x);
end

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[Tanh[x], $MachinePrecision]

\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\tanh x

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  2. Applied egg-rr58.0

    \[\leadsto \color{blue}{\frac{e^{x}}{2 \cdot \cosh x} - \frac{e^{-x}}{2 \cdot \cosh x}} \]

  3. Simplified0.0

    \[\leadsto \color{blue}{\tanh x} \]
    Proof
    (tanh.f64 x): 0 points increase in error, 0 points decrease in error
    (Rewrite=> tanh-def-a_binary64 (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))): 256 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) 1))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (/.f64 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (Rewrite<= metadata-eval (/.f64 2 2)))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) 2) 2))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) 2) 2))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (*.f64 (Rewrite<= cosh-def_binary64 (cosh.f64 x)) 2)): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (Rewrite<= *-commutative_binary64 (*.f64 2 (cosh.f64 x)))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (exp.f64 x) (*.f64 2 (cosh.f64 x))) (/.f64 (exp.f64 (neg.f64 x)) (*.f64 2 (cosh.f64 x))))): 5 points increase in error, 5 points decrease in error

  4. Final simplification0.0

    \[\leadsto \tanh x \]

Reproduce

herbie shell --seed 2022334 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))