Hyperbolic sine

Percentage Accurate: 54.5% → 98.9%

Time: 4.0s

Alternatives: 5

Speedup: 1.9×

• 49.6% 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: 98.9% 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 53.1%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 54.0%

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

   1. associate-/l* 53.5%

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

   2. associate-/r/ 53.7%

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

   3. metadata-eval 53.7%

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

   4. *-un-lft-identity 53.7%

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

   5. log1p-expm1-u 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right) \]
7. Add Preprocessing

Alternative 2: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.0)
   (/ (+ (* 0.3333333333333333 (pow x 3.0)) (* x 2.0)) 2.0)
   (/ (* 0.016666666666666666 (pow x 5.0)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 5.0) {
		tmp = ((0.3333333333333333 * pow(x, 3.0)) + (x * 2.0)) / 2.0;
	} else {
		tmp = (0.016666666666666666 * pow(x, 5.0)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.0d0) then
        tmp = ((0.3333333333333333d0 * (x ** 3.0d0)) + (x * 2.0d0)) / 2.0d0
    else
        tmp = (0.016666666666666666d0 * (x ** 5.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.0) {
		tmp = ((0.3333333333333333 * Math.pow(x, 3.0)) + (x * 2.0)) / 2.0;
	} else {
		tmp = (0.016666666666666666 * Math.pow(x, 5.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.0:
		tmp = ((0.3333333333333333 * math.pow(x, 3.0)) + (x * 2.0)) / 2.0
	else:
		tmp = (0.016666666666666666 * math.pow(x, 5.0)) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.0)
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(x * 2.0)) / 2.0);
	else
		tmp = Float64(Float64(0.016666666666666666 * (x ^ 5.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.0)
		tmp = ((0.3333333333333333 * (x ^ 3.0)) + (x * 2.0)) / 2.0;
	else
		tmp = (0.016666666666666666 * (x ^ 5.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.0], N[(N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5

   1. Initial program 38.4%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 90.5%

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

   if 5 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 84.4%

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

   4. Taylor expanded in x around inf 84.4%

      \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot {x}^{5}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3) x (/ (* 0.016666666666666666 (pow x 5.0)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = x;
	} else {
		tmp = (0.016666666666666666 * pow(x, 5.0)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.3d0) then
        tmp = x
    else
        tmp = (0.016666666666666666d0 * (x ** 5.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = x;
	} else {
		tmp = (0.016666666666666666 * Math.pow(x, 5.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.3:
		tmp = x
	else:
		tmp = (0.016666666666666666 * math.pow(x, 5.0)) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.3)
		tmp = x;
	else
		tmp = Float64(Float64(0.016666666666666666 * (x ^ 5.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.3)
		tmp = x;
	else
		tmp = (0.016666666666666666 * (x ^ 5.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.3], x, N[(N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 38.4%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 69.3%

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

   4. Taylor expanded in x around 0 68.9%

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

   if 3.2999999999999998 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 84.4%

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

   4. Taylor expanded in x around inf 84.4%

      \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot {x}^{5}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 53.1%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 54.0%

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

4. Final simplification 54.0%

   \[\leadsto \frac{x \cdot 2}{2} \]
5. Add Preprocessing

Alternative 5: 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 53.1%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 54.0%

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

4. Taylor expanded in x around 0 53.7%

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

5. Final simplification 53.7%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))