Average Error: 57.9 → 0.0
Time: 3.2s
Precision: binary64
Cost: 6720
\[\frac{e^{x} - e^{-x}}{2} \]
↓
\[\frac{2 \cdot \sinh x}{2} \]
(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))
↓
(FPCore (x) :precision binary64 (/ (* 2.0 (sinh x)) 2.0))
double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}
↓
double code(double x) {
	return (2.0 * sinh(x)) / 2.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 * sinh(x)) / 2.0d0
end function

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.sinh(x)) / 2.0;
}

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0
↓
def code(x):
	return (2.0 * 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 * sinh(x)) / 2.0)
end

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end
↓
function tmp = code(x)
	tmp = (2.0 * 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[Sinh[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

\frac{e^{x} - e^{-x}}{2}
↓
\frac{2 \cdot \sinh x}{2}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  2. Applied egg-rr57.9

    \[\leadsto \frac{\color{blue}{e^{x} + \left(-e^{-x}\right)}}{2} \]

  3. Simplified0.0

    \[\leadsto \frac{\color{blue}{2 \cdot \sinh x}}{2} \]
    Proof
    (*.f64 2 (sinh.f64 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> *-commutative_binary64 (*.f64 (sinh.f64 x) 2)): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite=> sinh-def_binary64 (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) 2)) 2): 247 points increase in error, 9 points decrease in error
    (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
    (Rewrite=> associate-/l*_binary64 (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (/.f64 2 2))): 0 points increase in error, 0 points decrease in error
    (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (Rewrite=> metadata-eval 1)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> /-rgt-identity_binary64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= unsub-neg_binary64 (+.f64 (exp.f64 x) (neg.f64 (exp.f64 (neg.f64 x))))): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022334 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))