Hyperbolic sine

?

Percentage Accurate: 54.6% → 99.7%
Time: 6.2s
Precision: binary64
Cost: 39433

?

\[\frac{e^{x} - e^{-x}}{2} \]
↓
\[\begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.0005\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \end{array} \]
(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0005)))
     (/ t_0 2.0)
     (/ (+ (* x 2.0) (* 0.3333333333333333 (* x (* x x)))) 2.0))))
double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}
↓
double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0005)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	}
	return tmp;
}

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}
↓
public static double code(double x) {
	double t_0 = Math.exp(x) - Math.exp(-x);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0005)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	}
	return tmp;
}

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0
↓
def code(x):
	t_0 = math.exp(x) - math.exp(-x)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0005):
		tmp = t_0 / 2.0
	else:
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0
	return tmp

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end
↓
function code(x)
	t_0 = Float64(exp(x) - exp(Float64(-x)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.0005))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * Float64(x * Float64(x * x)))) / 2.0);
	end
	return tmp
end

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end
↓
function tmp_2 = code(x)
	t_0 = exp(x) - exp(-x);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0005)))
		tmp = t_0 / 2.0;
	else
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	end
	tmp_2 = tmp;
end

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0005]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

\frac{e^{x} - e^{-x}}{2}
↓
\begin{array}{l}
t_0 := e^{x} - e^{-x}\\
\mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.0005\right):\\
\;\;\;\;\frac{t_0}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\


\end{array}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 9 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < -inf.0 or 5.0000000000000001e-4 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

    1. Initial program 100.0%

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

    if -inf.0 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 5.0000000000000001e-4

    1. Initial program 9.4%

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

    2. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]

    3. Applied egg-rr100.0%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]
      Step-by-step derivation

      [Start]100.0

      \[ \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2} \]

      unpow3 [=>]100.0

      \[ \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification100.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -\infty \lor \neg \left(e^{x} - e^{-x} \leq 0.0005\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy91.8%
Cost8972
\[\begin{array}{l} t_0 := x \cdot \left(x \cdot 0.3333333333333333\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := \frac{0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ t_3 := \frac{x \cdot \frac{t_1 - 4}{t_0 - 2}}{2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+77}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot \frac{{t_0}^{3} + 8}{t_1 + \left(4 - 2 \cdot t_0\right)}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+101}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 2
Accuracy87.8%
Cost1865
\[\begin{array}{l} t_0 := x \cdot \left(x \cdot 0.3333333333333333\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+154} \lor \neg \left(x \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{t_0 \cdot t_0 - 4}{t_0 - 2}}{2}\\ \end{array} \]
Alternative 3
Accuracy83.5%
Cost841
\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \lor \neg \left(x \leq 2.45\right):\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \]
Alternative 4
Accuracy83.8%
Cost832
\[\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2} \]
Alternative 5
Accuracy83.7%
Cost704
\[\frac{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}{2} \]
Alternative 6
Accuracy51.9%
Cost320
\[\frac{x \cdot 2}{2} \]
Alternative 7
Accuracy2.8%
Cost64
\[-1 \]
Alternative 8
Accuracy3.5%
Cost64
\[0 \]

Reproduce?

herbie shell --seed 2023160 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))