Hyperbolic sine

Percentage Accurate: 54.9% → 99.7%

Time: 3.2s

Alternatives: 6

Speedup: 18.7×

• 50.1% 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: 54.9% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -5000000000000 \lor \neg \left(t_0 \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 -5000000000000.0) (not (<= t_0 5e-8)))
     (/ t_0 2.0)
     (/ (+ (* x 2.0) (* 0.3333333333333333 (pow x 3.0))) 2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -5000000000000.0) || !(t_0 <= 5e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + (0.3333333333333333 * pow(x, 3.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - exp(-x)
    if ((t_0 <= (-5000000000000.0d0)) .or. (.not. (t_0 <= 5d-8))) then
        tmp = t_0 / 2.0d0
    else
        tmp = ((x * 2.0d0) + (0.3333333333333333d0 * (x ** 3.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - Math.exp(-x);
	double tmp;
	if ((t_0 <= -5000000000000.0) || !(t_0 <= 5e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + (0.3333333333333333 * Math.pow(x, 3.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - math.exp(-x)
	tmp = 0
	if (t_0 <= -5000000000000.0) or not (t_0 <= 5e-8):
		tmp = t_0 / 2.0
	else:
		tmp = ((x * 2.0) + (0.3333333333333333 * math.pow(x, 3.0))) / 2.0
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - exp(Float64(-x)))
	tmp = 0.0
	if ((t_0 <= -5000000000000.0) || !(t_0 <= 5e-8))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * (x ^ 3.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - exp(-x);
	tmp = 0.0;
	if ((t_0 <= -5000000000000.0) || ~((t_0 <= 5e-8)))
		tmp = t_0 / 2.0;
	else
		tmp = ((x * 2.0) + (0.3333333333333333 * (x ^ 3.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5000000000000.0], N[Not[LessEqual[t$95$0, 5e-8]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < -5e12 or 4.9999999999999998e-8 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   if -5e12 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 4.9999999999999998e-8

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -5000000000000 \lor \neg \left(e^{x} - e^{-x} \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2}\\ \end{array} \]

Alternative 2: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* x 2.0) (* 0.3333333333333333 (pow x 3.0))) 2.0))

C

double code(double x) {
	return ((x * 2.0) + (0.3333333333333333 * pow(x, 3.0))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * 2.0d0) + (0.3333333333333333d0 * (x ** 3.0d0))) / 2.0d0
end function

Java

public static double code(double x) {
	return ((x * 2.0) + (0.3333333333333333 * Math.pow(x, 3.0))) / 2.0;
}

Python

def code(x):
	return ((x * 2.0) + (0.3333333333333333 * math.pow(x, 3.0))) / 2.0

Julia

function code(x)
	return Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * (x ^ 3.0))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * 2.0) + (0.3333333333333333 * (x ^ 3.0))) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in x around 0 86.7%

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

3. Final simplification 86.7%

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

Alternative 3: 83.8% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x (+ 2.0 (* 0.3333333333333333 (* x x)))) 2.0))

C

double code(double x) {
	return (x * (2.0 + (0.3333333333333333 * (x * x)))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (2.0d0 + (0.3333333333333333d0 * (x * x)))) / 2.0d0
end function

Java

public static double code(double x) {
	return (x * (2.0 + (0.3333333333333333 * (x * x)))) / 2.0;
}

Python

def code(x):
	return (x * (2.0 + (0.3333333333333333 * (x * x)))) / 2.0

Julia

function code(x)
	return Float64(Float64(x * Float64(2.0 + Float64(0.3333333333333333 * Float64(x * x)))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x * (2.0 + (0.3333333333333333 * (x * x)))) / 2.0;
end

Wolfram

code[x_] := N[(N[(x * N[(2.0 + N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in x around 0 86.7%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
3. Step-by-step derivation

   1. unpow3 86.7%

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

   2. associate-*r* 86.4%

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

   3. distribute-rgt-out 86.4%

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

   4. *-commutative 86.4%

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

   5. +-commutative 86.4%

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

   6. associate-*l* 86.4%

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

   7. fma-def 86.4%

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

4. Simplified 86.4%

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

   1. fma-udef 86.4%

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

   2. associate-*r* 86.4%

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

   3. *-commutative 86.4%

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

6. Applied egg-rr 86.4%

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

7. Final simplification 86.4%

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

Alternative 4: 51.6% 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 54.4%

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

2. Taylor expanded in x around 0 52.8%

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

3. Final simplification 52.8%

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

Alternative 5: 2.9% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -1.0d0
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 54.4%

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

3. Applied egg-rr 2.9%

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

4. Final simplification 2.9%

   \[\leadsto -1 \]

Alternative 6: 3.4% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 54.4%

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

3. Applied egg-rr 3.5%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

4. Final simplification 3.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))