Hyperbolic sine

Percentage Accurate: 54.1% → 99.2%

Time: 5.8s

Alternatives: 5

Speedup: 18.7×

• 49.5% 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: 54.1% 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: 99.2% 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 55.1%

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

2. Taylor expanded in x around 0 52.3%

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

   1. associate-/l* 51.5%

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

   2. associate-/r/ 51.7%

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

   3. metadata-eval 51.7%

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

   4. *-un-lft-identity 51.7%

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

   5. log1p-expm1-u 98.7%

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

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

Alternative 2: 84.3% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

2. Taylor expanded in x around 0 83.4%

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

   1. unpow3 83.4%

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

   2. associate-*r* 83.4%

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

   3. distribute-rgt-out 83.4%

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

   4. *-commutative 83.4%

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

   5. associate-*l* 83.4%

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

   6. fma-def 83.4%

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

4. Simplified 83.4%

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

   1. fma-udef 83.4%

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

   2. distribute-rgt-in 83.4%

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

6. Applied egg-rr 83.4%

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

7. Final simplification 83.4%

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

Alternative 3: 84.3% 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 55.1%

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

2. Taylor expanded in x around 0 83.4%

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

   1. unpow3 83.4%

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

   2. associate-*r* 83.4%

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

   3. distribute-rgt-out 83.4%

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

   4. *-commutative 83.4%

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

   5. associate-*l* 83.4%

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

   6. fma-def 83.4%

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

4. Simplified 83.4%

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

5. Taylor expanded in x around 0 83.4%

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

   1. unpow3 83.4%

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

   2. associate-*r* 83.4%

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

   3. *-commutative 83.4%

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

   4. associate-*r* 83.4%

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

   5. distribute-rgt-out 83.4%

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

   6. associate-*r* 83.4%

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

   7. *-commutative 83.4%

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

   8. fma-def 83.4%

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

7. Simplified 83.4%

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

   1. fma-udef 83.4%

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

9. Applied egg-rr 83.4%

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

10. Final simplification 83.4%

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

Alternative 4: 52.5% 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 55.1%

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

2. Taylor expanded in x around 0 52.3%

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

3. Final simplification 52.3%

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

Alternative 5: 52.4% 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 55.1%

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

2. Taylor expanded in x around 0 52.3%

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

3. Taylor expanded in x around 0 51.7%

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

4. Final simplification 51.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))