Hyperbolic sine

Average Accuracy: 52.9% → 99.7%

Time: 5.1s

Precision: binary64

Cost: 39433

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• 48.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{-7}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) + x \cdot 2}{2}\\ \end{array} \]

FPCore

(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 1e-7)))
     (/ t_0 2.0)
     (/ (+ (* x (* x (* x 0.3333333333333333))) (* x 2.0)) 2.0))))

C

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 <= 1e-7)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	}
	return tmp;
}

Java

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 <= 1e-7)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	}
	return tmp;
}

Python

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 <= 1e-7):
		tmp = t_0 / 2.0
	else:
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0
	return tmp

Julia

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 <= 1e-7))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(x * Float64(x * 0.3333333333333333))) + Float64(x * 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

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 <= 1e-7)))
		tmp = t_0 / 2.0;
	else
		tmp = ((x * (x * (x * 0.3333333333333333))) + (x * 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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, 1e-7]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(x * N[(x * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < -inf.0 or 9.9999999999999995e-8 < (-.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))) < 9.9999999999999995e-8

   1. Initial program 6.8%

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

   2. Taylor expanded in x around 0 100.0%

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

   3. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2} \]
      unpow3 [=>] 100.0	\[ \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]
      associate-*r* [=>] 100.0	\[ \frac{2 \cdot x + \color{blue}{\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x}}{2} \]
      distribute-rgt-out [=>] 100.0	\[ \frac{\color{blue}{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}}{2} \]
      *-commutative [<=] 100.0	\[ \frac{x \cdot \left(2 + \color{blue}{\left(x \cdot x\right) \cdot 0.3333333333333333}\right)}{2} \]
      +-commutative [<=] 100.0	\[ \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)}}{2} \]
      associate-*l* [=>] 100.0	\[ \frac{x \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right)}{2} \]
      fma-def [=>] 100.0	\[ \frac{x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}}{2} \]

   4. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \frac{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}{2} \]
      fma-udef [=>] 100.0	\[ \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)}}{2} \]
      distribute-rgt-in [=>] 100.0	\[ \frac{\color{blue}{\left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) \cdot x + 2 \cdot x}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -\infty \lor \neg \left(e^{x} - e^{-x} \leq 10^{-7}\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) + x \cdot 2}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.2%
Cost	1865

\[\begin{array}{l} t_0 := x \cdot \left(x \cdot 0.3333333333333333\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+155} \lor \neg \left(x \leq 10^{+103}\right):\\ \;\;\;\;\frac{x}{\frac{6}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{t_0 \cdot t_0 - 4}{t_0 - 2}}{2}\\ \end{array} \]

Alternative 2
Accuracy	84.3%
Cost	832

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

Alternative 3
Accuracy	84.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2.45 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\frac{x}{\frac{6}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \]

Alternative 4
Accuracy	84.3%
Cost	704

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

Alternative 5
Accuracy	53.5%
Cost	320

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

Alternative 6
Accuracy	2.9%
Cost	64

\[-1 \]

Alternative 7
Accuracy	3.5%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023158 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))