Hyperbolic sine

Percentage Accurate: 54.5% → 99.3%

Time: 14.5s

Alternatives: 3

Speedup: 18.7×

• 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 0.0002:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x\_m} + \left(x\_m + -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (- (exp x_m) (exp (- x_m))) 0.0002)
    (/ (* x_m (+ 2.0 (* 0.3333333333333333 (* x_m x_m)))) 2.0)
    (/ (+ (exp x_m) (+ x_m -1.0)) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((exp(x_m) - exp(-x_m)) <= 0.0002) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0;
	} else {
		tmp = (exp(x_m) + (x_m + -1.0)) / 2.0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((exp(x_m) - exp(-x_m)) <= 0.0002d0) then
        tmp = (x_m * (2.0d0 + (0.3333333333333333d0 * (x_m * x_m)))) / 2.0d0
    else
        tmp = (exp(x_m) + (x_m + (-1.0d0))) / 2.0d0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if ((Math.exp(x_m) - Math.exp(-x_m)) <= 0.0002) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0;
	} else {
		tmp = (Math.exp(x_m) + (x_m + -1.0)) / 2.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if (math.exp(x_m) - math.exp(-x_m)) <= 0.0002:
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0
	else:
		tmp = (math.exp(x_m) + (x_m + -1.0)) / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 0.0002)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * Float64(x_m * x_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(x_m) + Float64(x_m + -1.0)) / 2.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if ((exp(x_m) - exp(-x_m)) <= 0.0002)
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0;
	else
		tmp = (exp(x_m) + (x_m + -1.0)) / 2.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(x$95$m * N[(2.0 + N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x$95$m], $MachinePrecision] + N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.0000000000000001e-4

   1. Initial program 36.3%

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

   3. Taylor expanded in x around 0 92.3%

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

      1. unpow2 92.3%

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

   5. Applied egg-rr 92.3%

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

   if 2.0000000000000001e-4 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 + -1 \cdot x\right)}}{2} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \frac{e^{x} - \left(1 + \color{blue}{\left(-x\right)}\right)}{2} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

   5. Simplified 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0002:\\ \;\;\;\;\frac{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + \left(x + -1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 18.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot \left(x\_m \cdot x\_m\right)\right)}{2} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (* x_m (+ 2.0 (* 0.3333333333333333 (* x_m x_m)))) 2.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((x_m * (2.0d0 + (0.3333333333333333d0 * (x_m * x_m)))) / 2.0d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * Float64(x_m * x_m)))) / 2.0))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((x_m * (2.0 + (0.3333333333333333 * (x_m * x_m)))) / 2.0);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(x$95$m * N[(2.0 + N[(0.3333333333333333 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in x around 0 85.6%

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

   1. unpow2 85.6%

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

5. Applied egg-rr 85.6%

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

Alternative 3: 52.0% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in x around 0 85.6%

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

4. Taylor expanded in x around 0 52.2%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))