Hyperbolic sine

Percentage Accurate: 53.7% → 98.9%

Time: 5.2s

Alternatives: 6

Speedup: 1.9×

• 48.8% 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: 53.7% 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 51.6%

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

3. Taylor expanded in x around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. associate-/l* 55.4%

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

   3. metadata-eval 55.4%

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

   4. *-commutative 55.4%

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

   5. log1p-expm1-u 99.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(1 \cdot x\right)\right)} \]

   6. *-un-lft-identity 99.5%

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

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: 84.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around 0 86.1%

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

4. Final simplification 86.1%

   \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{2} \]
5. Add Preprocessing

Alternative 3: 84.6% 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 51.6%

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

3. Taylor expanded in x around 0 86.1%

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

   1. distribute-rgt-in 86.1%

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

   2. fma-define 86.1%

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

   3. associate-*l* 86.1%

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

   4. pow-plus 86.1%

      \[\leadsto \frac{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right)}{2} \]

   5. metadata-eval 86.1%

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

5. Simplified 86.1%

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

   1. fma-undefine 86.1%

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

   2. *-commutative 86.1%

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

7. Applied egg-rr 86.1%

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

8. Final simplification 86.1%

   \[\leadsto \frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2} \]
9. Add Preprocessing

Alternative 4: 68.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5) x (* (pow x 3.0) 0.16666666666666666)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = x
    else
        tmp = (x ** 3.0d0) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = x
	else:
		tmp = math.pow(x, 3.0) * 0.16666666666666666
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.5], x, N[(N[Power[x, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 36.4%

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

   3. Taylor expanded in x around 0 71.3%

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

   4. Taylor expanded in x around 0 70.9%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.1%

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

      1. distribute-rgt-in 66.1%

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

      2. fma-define 66.1%

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

      3. associate-*l* 66.1%

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

      4. pow-plus 66.1%

         \[\leadsto \frac{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right)}{2} \]

      5. metadata-eval 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in x around inf 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-/l* 66.1%

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

      3. metadata-eval 66.1%

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

   8. Applied egg-rr 66.1%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% 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 51.6%

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

3. Taylor expanded in x around 0 55.7%

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

4. Final simplification 55.7%

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

Alternative 6: 52.8% 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 51.6%

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

3. Taylor expanded in x around 0 55.7%

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

4. Taylor expanded in x around 0 55.4%

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

5. Final simplification 55.4%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))