Hyperbolic sine

Percentage Accurate: 54.5% → 99.1%

Time: 3.5s

Alternatives: 6

Speedup: 1.9×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (expm1 x)))

C

double code(double x) {
	return log1p(expm1(x));
}

Java

public static double code(double x) {
	return Math.log1p(Math.expm1(x));
}

Python

def code(x):
	return math.log1p(math.expm1(x))

Julia

function code(x)
	return log1p(expm1(x))
end

Wolfram

code[x_] := N[Log[1 + N[(Exp[x] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around 0 54.4%

   \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]
3. Step-by-step derivation

   1. associate-/l* 53.9%

      \[\leadsto \color{blue}{\frac{2}{\frac{2}{x}}} \]

   2. associate-/r/ 54.1%

      \[\leadsto \color{blue}{\frac{2}{2} \cdot x} \]

   3. metadata-eval 54.1%

      \[\leadsto \color{blue}{1} \cdot x \]

   4. *-un-lft-identity 54.1%

      \[\leadsto \color{blue}{x} \]

   5. log1p-expm1-u 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)} \]

5. Final simplification 99.4%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right) \]

Alternative 2: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 4.8\right):\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 4.8)))
   (/ (* 0.016666666666666666 (pow x 5.0)) 2.0)
   (+ x (* 0.16666666666666666 (pow x 3.0)))))

C

double code(double x) {
	double tmp;
	if ((x <= -5.0) || !(x <= 4.8)) {
		tmp = (0.016666666666666666 * pow(x, 5.0)) / 2.0;
	} else {
		tmp = x + (0.16666666666666666 * pow(x, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.0d0)) .or. (.not. (x <= 4.8d0))) then
        tmp = (0.016666666666666666d0 * (x ** 5.0d0)) / 2.0d0
    else
        tmp = x + (0.16666666666666666d0 * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -5.0) || !(x <= 4.8)) {
		tmp = (0.016666666666666666 * Math.pow(x, 5.0)) / 2.0;
	} else {
		tmp = x + (0.16666666666666666 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5.0) or not (x <= 4.8):
		tmp = (0.016666666666666666 * math.pow(x, 5.0)) / 2.0
	else:
		tmp = x + (0.16666666666666666 * math.pow(x, 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 4.8))
		tmp = Float64(Float64(0.016666666666666666 * (x ^ 5.0)) / 2.0);
	else
		tmp = Float64(x + Float64(0.16666666666666666 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -5.0) || ~((x <= 4.8)))
		tmp = (0.016666666666666666 * (x ^ 5.0)) / 2.0;
	else
		tmp = x + (0.16666666666666666 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5.0], N[Not[LessEqual[x, 4.8]], $MachinePrecision]], N[(N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x + N[(0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 4.79999999999999982 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 84.6%

      \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + 2 \cdot x\right)}}{2} \]

   3. Taylor expanded in x around inf 84.6%

      \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot {x}^{5}}}{2} \]

   if -5 < x < 4.79999999999999982

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{x + 0.16666666666666666 \cdot {x}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 4.8\right):\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot {x}^{3}\\ \end{array} \]

Alternative 3: 83.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 2.4\right):\\ \;\;\;\;0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.5) (not (<= x 2.4))) (* 0.16666666666666666 (pow x 3.0)) x))

C

double code(double x) {
	double tmp;
	if ((x <= -2.5) || !(x <= 2.4)) {
		tmp = 0.16666666666666666 * pow(x, 3.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.5d0)) .or. (.not. (x <= 2.4d0))) then
        tmp = 0.16666666666666666d0 * (x ** 3.0d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.5) || !(x <= 2.4)) {
		tmp = 0.16666666666666666 * Math.pow(x, 3.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.5) or not (x <= 2.4):
		tmp = 0.16666666666666666 * math.pow(x, 3.0)
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.5) || !(x <= 2.4))
		tmp = Float64(0.16666666666666666 * (x ^ 3.0));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.5) || ~((x <= 2.4)))
		tmp = 0.16666666666666666 * (x ^ 3.0);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.5], N[Not[LessEqual[x, 2.4]], $MachinePrecision]], N[(0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 2.39999999999999991 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.7%

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

   3. Taylor expanded in x around inf 74.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{3}} \]

   if -2.5 < x < 2.39999999999999991

   1. Initial program 8.9%

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

   2. Taylor expanded in x around 0 98.9%

      \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]

   3. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 2.4\right):\\ \;\;\;\;0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x + 0.16666666666666666 \cdot {x}^{3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (* 0.16666666666666666 (pow x 3.0))))

C

double code(double x) {
	return x + (0.16666666666666666 * pow(x, 3.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (0.16666666666666666d0 * (x ** 3.0d0))
end function

Java

public static double code(double x) {
	return x + (0.16666666666666666 * Math.pow(x, 3.0));
}

Python

def code(x):
	return x + (0.16666666666666666 * math.pow(x, 3.0))

Julia

function code(x)
	return Float64(x + Float64(0.16666666666666666 * (x ^ 3.0)))
end

MATLAB

function tmp = code(x)
	tmp = x + (0.16666666666666666 * (x ^ 3.0));
end

Wolfram

code[x_] := N[(x + N[(0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around 0 87.6%

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

3. Taylor expanded in x around 0 87.6%

   \[\leadsto \color{blue}{x + 0.16666666666666666 \cdot {x}^{3}} \]

4. Final simplification 87.6%

   \[\leadsto x + 0.16666666666666666 \cdot {x}^{3} \]

Alternative 5: 52.0% accurate, 41.2× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 2.0) 2.0))

C

double code(double x) {
	return (x * 2.0) / 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return (x * 2.0) / 2.0;
}

Python

def code(x):
	return (x * 2.0) / 2.0

Julia

function code(x)
	return Float64(Float64(x * 2.0) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / 2.0;
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around 0 54.4%

   \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]

3. Final simplification 54.4%

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

Alternative 6: 52.0% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around 0 54.4%

   \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]

3. Taylor expanded in x around 0 54.1%

   \[\leadsto \color{blue}{x} \]

4. Final simplification 54.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))