Hyperbolic sine

Percentage Accurate: 53.8% → 99.0%

Time: 6.7s

Alternatives: 5

Speedup: 1.9×

• 49.0% 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: 53.8% 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: 99.0% 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 56.2%

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

3. Taylor expanded in x around 0 50.6%

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

   1. *-commutative 50.6%

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

   2. associate-/l* 50.3%

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

   3. metadata-eval 50.3%

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

   4. *-commutative 50.3%

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

   5. log1p-expm1-u 98.9%

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

   6. *-un-lft-identity 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right) \]
7. Add Preprocessing

Alternative 2: 93.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(2 + 0.0003968253968253968 \cdot {x}^{6}\right)}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x (+ 2.0 (* 0.0003968253968253968 (pow x 6.0)))) 2.0))

C

double code(double x) {
	return (x * (2.0 + (0.0003968253968253968 * pow(x, 6.0)))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (2.0d0 + (0.0003968253968253968d0 * (x ** 6.0d0)))) / 2.0d0
end function

Java

public static double code(double x) {
	return (x * (2.0 + (0.0003968253968253968 * Math.pow(x, 6.0)))) / 2.0;
}

Python

def code(x):
	return (x * (2.0 + (0.0003968253968253968 * math.pow(x, 6.0)))) / 2.0

Julia

function code(x)
	return Float64(Float64(x * Float64(2.0 + Float64(0.0003968253968253968 * (x ^ 6.0)))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x * (2.0 + (0.0003968253968253968 * (x ^ 6.0)))) / 2.0;
end

Wolfram

code[x_] := N[(N[(x * N[(2.0 + N[(0.0003968253968253968 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in x around 0 93.6%

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {x}^{2}\right)\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative 93.6%

      \[\leadsto \frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{x}^{2} \cdot 0.0003968253968253968}\right)\right)\right)}{2} \]

5. Simplified 93.6%

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.016666666666666666 + {x}^{2} \cdot 0.0003968253968253968\right)\right)\right)}}{2} \]

6. Taylor expanded in x around inf 93.0%

   \[\leadsto \frac{x \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {x}^{6}}\right)}{2} \]

7. Final simplification 93.0%

   \[\leadsto \frac{x \cdot \left(2 + 0.0003968253968253968 \cdot {x}^{6}\right)}{2} \]
8. Add Preprocessing

Alternative 3: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot 0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.2) x (* (pow x 7.0) 0.0001984126984126984)))

C

double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = x;
	} else {
		tmp = pow(x, 7.0) * 0.0001984126984126984;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.2d0) then
        tmp = x
    else
        tmp = (x ** 7.0d0) * 0.0001984126984126984d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.2) {
		tmp = x;
	} else {
		tmp = Math.pow(x, 7.0) * 0.0001984126984126984;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.2:
		tmp = x
	else:
		tmp = math.pow(x, 7.0) * 0.0001984126984126984
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.2)
		tmp = x;
	else
		tmp = Float64((x ^ 7.0) * 0.0001984126984126984);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.2)
		tmp = x;
	else
		tmp = (x ^ 7.0) * 0.0001984126984126984;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.2], x, N[(N[Power[x, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.20000000000000018

   1. Initial program 40.7%

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

   3. Taylor expanded in x around 0 66.1%

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

   4. Taylor expanded in x around 0 66.1%

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

   if 4.20000000000000018 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 88.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {x}^{2}\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. *-commutative 88.9%

         \[\leadsto \frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{x}^{2} \cdot 0.0003968253968253968}\right)\right)\right)}{2} \]

   5. Simplified 88.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot \left(0.016666666666666666 + {x}^{2} \cdot 0.0003968253968253968\right)\right)\right)}}{2} \]

   6. Taylor expanded in x around inf 88.9%

      \[\leadsto \frac{\color{blue}{0.0003968253968253968 \cdot {x}^{7}}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 88.9%

         \[\leadsto \frac{\color{blue}{{x}^{7} \cdot 0.0003968253968253968}}{2} \]

      2. associate-/l* 88.9%

         \[\leadsto \color{blue}{{x}^{7} \cdot \frac{0.0003968253968253968}{2}} \]

      3. metadata-eval 88.9%

         \[\leadsto {x}^{7} \cdot \color{blue}{0.0001984126984126984} \]

   8. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{{x}^{7} \cdot 0.0001984126984126984} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot 0.0001984126984126984\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 56.2%

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

3. Taylor expanded in x around 0 50.6%

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

4. Final simplification 50.6%

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

Alternative 5: 52.7% 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 56.2%

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

3. Taylor expanded in x around 0 50.6%

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

4. Taylor expanded in x around 0 50.3%

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

5. Final simplification 50.3%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))