Hyperbolic sine

Percentage Accurate: 53.4% → 99.6%

Time: 4.9s

Alternatives: 7

Speedup: 18.7×

• 48.7% 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: 53.4% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right) + x \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-8)))
     (/ t_0 2.0)
     (/ (+ (* (* x x) (* x 0.3333333333333333)) (* x 2.0)) 2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (((x * x) * (x * 0.3333333333333333)) + (x * 2.0)) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - Math.exp(-x);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (((x * x) * (x * 0.3333333333333333)) + (x * 2.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - math.exp(-x)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e-8):
		tmp = t_0 / 2.0
	else:
		tmp = (((x * x) * (x * 0.3333333333333333)) + (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 <= Float64(-Inf)) || !(t_0 <= 2e-8))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * 0.3333333333333333)) + 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 <= -Inf) || ~((t_0 <= 2e-8)))
		tmp = t_0 / 2.0;
	else
		tmp = (((x * x) * (x * 0.3333333333333333)) + (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, (-Infinity)], N[Not[LessEqual[t$95$0, 2e-8]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333), $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))) < -inf.0 or 2e-8 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e-8

   1. Initial program 9.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)\right)} + 2 \cdot x}{2} \]

      2. expm1-udef 99.5%

         \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)} - 1\right)} + 2 \cdot x}{2} \]

      3. log1p-udef 99.5%

         \[\leadsto \frac{\left(e^{\color{blue}{\log \left(1 + 0.3333333333333333 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x}{2} \]

      4. add-exp-log 99.5%

         \[\leadsto \frac{\left(\color{blue}{\left(1 + 0.3333333333333333 \cdot {x}^{3}\right)} - 1\right) + 2 \cdot x}{2} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + 0.3333333333333333 \cdot {x}^{3}\right) - 1\right)} + 2 \cdot x}{2} \]
   5. Step-by-step derivation

      1. add-exp-log 99.5%

         \[\leadsto \frac{\left(\color{blue}{e^{\log \left(1 + 0.3333333333333333 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x}{2} \]

      2. log1p-udef 99.5%

         \[\leadsto \frac{\left(e^{\color{blue}{\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x}{2} \]

      3. expm1-udef 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)\right)} + 2 \cdot x}{2} \]

      4. expm1-log1p-u 100.0%

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

      5. *-commutative 100.0%

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

      6. unpow3 100.0%

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

      7. associate-*l* 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -\infty \lor \neg \left(e^{x} - e^{-x} \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right) + x \cdot 2}{2}\\ \end{array} \]

Alternative 2: 83.6% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.5) (not (<= x 2.5)))
   (/ (* x (* (* x x) 0.3333333333333333)) 2.0)
   (/ (* x 2.0) 2.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.5) || !(x <= 2.5)) {
		tmp = (x * ((x * x) * 0.3333333333333333)) / 2.0;
	} else {
		tmp = (x * 2.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.5d0)) .or. (.not. (x <= 2.5d0))) then
        tmp = (x * ((x * x) * 0.3333333333333333d0)) / 2.0d0
    else
        tmp = (x * 2.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.5) || !(x <= 2.5)) {
		tmp = (x * ((x * x) * 0.3333333333333333)) / 2.0;
	} else {
		tmp = (x * 2.0) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.5) or not (x <= 2.5):
		tmp = (x * ((x * x) * 0.3333333333333333)) / 2.0
	else:
		tmp = (x * 2.0) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.5) || !(x <= 2.5))
		tmp = Float64(Float64(x * Float64(Float64(x * x) * 0.3333333333333333)) / 2.0);
	else
		tmp = Float64(Float64(x * 2.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.5) || ~((x <= 2.5)))
		tmp = (x * ((x * x) * 0.3333333333333333)) / 2.0;
	else
		tmp = (x * 2.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.5], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 2.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 72.6%

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

      1. unpow3 72.6%

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

      2. associate-*r* 72.6%

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

      3. distribute-rgt-out 72.6%

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

      4. *-commutative 72.6%

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

      5. associate-*l* 72.6%

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

      6. fma-def 72.6%

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

   4. Simplified 72.6%

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

   5. Taylor expanded in x around inf 72.6%

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

      1. unpow2 72.6%

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

   7. Simplified 72.6%

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

   if -2.5 < x < 2.5

   1. Initial program 9.8%

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

   2. Taylor expanded in x around 0 98.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \]

Alternative 3: 83.9% accurate, 15.8× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in x around 0 85.0%

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

   1. expm1-log1p-u 65.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)\right)} + 2 \cdot x}{2} \]

   2. expm1-udef 65.1%

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)} - 1\right)} + 2 \cdot x}{2} \]

   3. log1p-udef 65.1%

      \[\leadsto \frac{\left(e^{\color{blue}{\log \left(1 + 0.3333333333333333 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x}{2} \]

   4. add-exp-log 84.8%

      \[\leadsto \frac{\left(\color{blue}{\left(1 + 0.3333333333333333 \cdot {x}^{3}\right)} - 1\right) + 2 \cdot x}{2} \]

4. Applied egg-rr 84.8%

   \[\leadsto \frac{\color{blue}{\left(\left(1 + 0.3333333333333333 \cdot {x}^{3}\right) - 1\right)} + 2 \cdot x}{2} \]
5. Step-by-step derivation

   1. add-exp-log 65.1%

      \[\leadsto \frac{\left(\color{blue}{e^{\log \left(1 + 0.3333333333333333 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x}{2} \]

   2. log1p-udef 65.1%

      \[\leadsto \frac{\left(e^{\color{blue}{\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)}} - 1\right) + 2 \cdot x}{2} \]

   3. expm1-udef 65.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot {x}^{3}\right)\right)} + 2 \cdot x}{2} \]

   4. expm1-log1p-u 85.0%

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

   5. *-commutative 85.0%

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

   6. unpow3 85.0%

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

   7. associate-*l* 85.0%

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

6. Applied egg-rr 85.0%

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

7. Final simplification 85.0%

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

Alternative 4: 83.9% accurate, 18.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in x around 0 85.0%

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

   1. unpow3 85.0%

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

   2. associate-*r* 85.0%

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

   3. distribute-rgt-out 85.0%

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

   4. *-commutative 85.0%

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

   5. associate-*l* 85.0%

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

   6. fma-def 85.0%

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

4. Simplified 85.0%

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

   1. fma-udef 85.0%

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

   2. associate-*r* 85.0%

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

6. Applied egg-rr 85.0%

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

7. Final simplification 85.0%

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

Alternative 5: 53.3% 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 58.4%

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

2. Taylor expanded in x around 0 48.6%

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

3. Final simplification 48.6%

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

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 58.4%

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

3. Applied egg-rr 2.8%

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

4. Final simplification 2.8%

   \[\leadsto -1 \]

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 58.4%

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

3. Applied egg-rr 3.1%

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

4. Final simplification 3.1%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023283 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))