Hyperbolic sine

Percentage Accurate: 54.2% → 99.9%

Time: 2.8s

Alternatives: 7

Speedup: 1.9×

• 49.3% 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: 54.2% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.01\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.01)))
     (/ t_0 2.0)
     (/
      (fma
       2.0
       x
       (+
        (* 0.3333333333333333 (pow x 3.0))
        (* 0.016666666666666666 (pow x 5.0))))
      2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.01)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = fma(2.0, x, ((0.3333333333333333 * pow(x, 3.0)) + (0.016666666666666666 * pow(x, 5.0)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(exp(x) - exp(Float64(-x)))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 0.01))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(fma(2.0, x, Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(0.016666666666666666 * (x ^ 5.0)))) / 2.0);
	end
	return 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, -1.0], N[Not[LessEqual[t$95$0, 0.01]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(2.0 * x + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < -1 or 0.0100000000000000002 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   if -1 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.0100000000000000002

   1. Initial program 8.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + 2 \cdot x\right)}}{2} \]
   3. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.3333333333333333, {x}^{3}, 0.016666666666666666 \cdot {x}^{5}\right)\right)}}{2} \]
   5. Step-by-step derivation

      1. fma-udef 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(2, x, \color{blue}{0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}}\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -1 \lor \neg \left(e^{x} - e^{-x} \leq 0.01\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 0.01\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 -1.0) (not (<= t_0 0.01)))
     (/ t_0 2.0)
     (/
      (+
       (* 0.016666666666666666 (pow x 5.0))
       (+ (* 0.3333333333333333 (pow x 3.0)) (* x 2.0)))
      2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 0.01)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((0.016666666666666666 * pow(x, 5.0)) + ((0.3333333333333333 * pow(x, 3.0)) + (x * 2.0))) / 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 <= (-1.0d0)) .or. (.not. (t_0 <= 0.01d0))) then
        tmp = t_0 / 2.0d0
    else
        tmp = ((0.016666666666666666d0 * (x ** 5.0d0)) + ((0.3333333333333333d0 * (x ** 3.0d0)) + (x * 2.0d0))) / 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 <= -1.0) || !(t_0 <= 0.01)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((0.016666666666666666 * Math.pow(x, 5.0)) + ((0.3333333333333333 * Math.pow(x, 3.0)) + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - math.exp(-x)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 0.01):
		tmp = t_0 / 2.0
	else:
		tmp = ((0.016666666666666666 * math.pow(x, 5.0)) + ((0.3333333333333333 * math.pow(x, 3.0)) + (x * 2.0))) / 2.0
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - exp(Float64(-x)))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 0.01))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(0.016666666666666666 * (x ^ 5.0)) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(x * 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - exp(-x);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 0.01)))
		tmp = t_0 / 2.0;
	else
		tmp = ((0.016666666666666666 * (x ^ 5.0)) + ((0.3333333333333333 * (x ^ 3.0)) + (x * 2.0))) / 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, -1.0], N[Not[LessEqual[t$95$0, 0.01]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < -1 or 0.0100000000000000002 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   if -1 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 0.0100000000000000002

   1. Initial program 8.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + 2 \cdot x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -1 \lor \neg \left(e^{x} - e^{-x} \leq 0.01\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)}{2}\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -1 \lor \neg \left(t_0 \leq 5 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 -1.0) (not (<= t_0 5e-15)))
     (/ t_0 2.0)
     (/ (+ (* 0.3333333333333333 (pow x 3.0)) (* x 2.0)) 2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -1.0) || !(t_0 <= 5e-15)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((0.3333333333333333 * pow(x, 3.0)) + (x * 2.0)) / 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 <= (-1.0d0)) .or. (.not. (t_0 <= 5d-15))) then
        tmp = t_0 / 2.0d0
    else
        tmp = ((0.3333333333333333d0 * (x ** 3.0d0)) + (x * 2.0d0)) / 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 <= -1.0) || !(t_0 <= 5e-15)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((0.3333333333333333 * Math.pow(x, 3.0)) + (x * 2.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - math.exp(-x)
	tmp = 0
	if (t_0 <= -1.0) or not (t_0 <= 5e-15):
		tmp = t_0 / 2.0
	else:
		tmp = ((0.3333333333333333 * math.pow(x, 3.0)) + (x * 2.0)) / 2.0
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - exp(Float64(-x)))
	tmp = 0.0
	if ((t_0 <= -1.0) || !(t_0 <= 5e-15))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(x * 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - exp(-x);
	tmp = 0.0;
	if ((t_0 <= -1.0) || ~((t_0 <= 5e-15)))
		tmp = t_0 / 2.0;
	else
		tmp = ((0.3333333333333333 * (x ^ 3.0)) + (x * 2.0)) / 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, -1.0], N[Not[LessEqual[t$95$0, 5e-15]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * N[Power[x, 3.0], $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))) < -1 or 4.99999999999999999e-15 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

   if -1 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 4.99999999999999999e-15

   1. Initial program 7.3%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -1 \lor \neg \left(e^{x} - e^{-x} \leq 5 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \end{array} \]

Alternative 4: 84.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

2. Taylor expanded in x around 0 88.6%

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

3. Final simplification 88.6%

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

Alternative 5: 52.4% 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 54.3%

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

2. Taylor expanded in x around 0 52.2%

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

3. Final simplification 52.2%

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

Alternative 6: 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 54.3%

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

3. Applied egg-rr 3.1%

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

4. Final simplification 3.1%

   \[\leadsto -1 \]

Alternative 7: 3.5% 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 54.3%

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

3. Applied egg-rr 3.5%

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

4. Final simplification 3.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023307 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))