Hyperbolic sine

Percentage Accurate: 54.6% → 99.7%

Time: 3.8s

Alternatives: 8

Speedup: 18.7×

• 50.0% 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: 54.6% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.02\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.02)))
     (/ t_0 2.0)
     (/
      (+
       (* x 2.0)
       (+
        (* 0.3333333333333333 (pow x 3.0))
        (* 0.016666666666666666 (pow x 5.0))))
      2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.02)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + ((0.3333333333333333 * pow(x, 3.0)) + (0.016666666666666666 * pow(x, 5.0)))) / 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 <= 0.02)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + ((0.3333333333333333 * Math.pow(x, 3.0)) + (0.016666666666666666 * Math.pow(x, 5.0)))) / 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 <= 0.02):
		tmp = t_0 / 2.0
	else:
		tmp = ((x * 2.0) + ((0.3333333333333333 * math.pow(x, 3.0)) + (0.016666666666666666 * math.pow(x, 5.0)))) / 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 <= 0.02))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(0.016666666666666666 * (x ^ 5.0)))) / 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 <= 0.02)))
		tmp = t_0 / 2.0;
	else
		tmp = ((x * 2.0) + ((0.3333333333333333 * (x ^ 3.0)) + (0.016666666666666666 * (x ^ 5.0)))) / 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, 0.02]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $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 0.0200000000000000004 < (-.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))) < 0.0200000000000000004

   1. Initial program 8.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -\infty \lor \neg \left(e^{x} - e^{-x} \leq 0.02\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e-8)))
     (/ t_0 2.0)
     (/ (* x (+ 2.0 (* 0.3333333333333333 (* x x)))) 2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (x * (2.0 + (0.3333333333333333 * (x * x)))) / 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 <= 5e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (x * (2.0 + (0.3333333333333333 * (x * x)))) / 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 <= 5e-8):
		tmp = t_0 / 2.0
	else:
		tmp = (x * (2.0 + (0.3333333333333333 * (x * x)))) / 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 <= 5e-8))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(x * Float64(2.0 + Float64(0.3333333333333333 * Float64(x * x)))) / 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 <= 5e-8)))
		tmp = t_0 / 2.0;
	else
		tmp = (x * (2.0 + (0.3333333333333333 * (x * x)))) / 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, 5e-8]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(x * N[(2.0 + N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 4.9999999999999998e-8 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

   if -inf.0 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 4.9999999999999998e-8

   1. Initial program 7.3%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow3 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. fma-def 100.0%

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

   4. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. associate-*r* 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -\infty \lor \neg \left(e^{x} - e^{-x} \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}{2}\\ \end{array} \]

Alternative 3: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2 + 0.016666666666666666 \cdot {x}^{5}}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* x 2.0) (* 0.016666666666666666 (pow x 5.0))) 2.0))

C

double code(double x) {
	return ((x * 2.0) + (0.016666666666666666 * pow(x, 5.0))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * 2.0d0) + (0.016666666666666666d0 * (x ** 5.0d0))) / 2.0d0
end function

Java

public static double code(double x) {
	return ((x * 2.0) + (0.016666666666666666 * Math.pow(x, 5.0))) / 2.0;
}

Python

def code(x):
	return ((x * 2.0) + (0.016666666666666666 * math.pow(x, 5.0))) / 2.0

Julia

function code(x)
	return Float64(Float64(Float64(x * 2.0) + Float64(0.016666666666666666 * (x ^ 5.0))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * 2.0) + (0.016666666666666666 * (x ^ 5.0))) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in x around 0 90.1%

   \[\leadsto \frac{\color{blue}{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}}{2} \]

3. Taylor expanded in x around inf 89.7%

   \[\leadsto \frac{2 \cdot x + \color{blue}{0.016666666666666666 \cdot {x}^{5}}}{2} \]

4. Final simplification 89.7%

   \[\leadsto \frac{x \cdot 2 + 0.016666666666666666 \cdot {x}^{5}}{2} \]

Alternative 4: 83.2% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.5) (not (<= x 2.5)))
   (* x (* x (* x 0.16666666666666666)))
   (/ (* x 2.0) 2.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.5) || !(x <= 2.5)) {
		tmp = x * (x * (x * 0.16666666666666666));
	} 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 * 0.16666666666666666d0))
    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 * 0.16666666666666666));
	} 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 * 0.16666666666666666))
	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(x * Float64(x * Float64(x * 0.16666666666666666)));
	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 * 0.16666666666666666));
	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[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $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 70.8%

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

      1. unpow3 70.8%

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

      2. associate-*r* 70.8%

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

      3. distribute-rgt-out 70.8%

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

      4. *-commutative 70.8%

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

      5. +-commutative 70.8%

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

      6. associate-*l* 70.8%

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

      7. fma-def 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in x around inf 70.8%

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

      1. unpow2 70.8%

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

   7. Simplified 70.8%

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

      1. associate-/l* 70.8%

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

      2. div-inv 70.8%

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

      3. associate-/r* 70.8%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\frac{2}{0.3333333333333333}}{x \cdot x}}} \]

      4. metadata-eval 70.8%

         \[\leadsto x \cdot \frac{1}{\frac{\color{blue}{6}}{x \cdot x}} \]

   9. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{6}{x \cdot x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 70.8%

          \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\frac{6}{x}}{x}}} \]

       2. associate-/r/ 70.8%

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\frac{6}{x}} \cdot x\right)} \]

       3. clear-num 70.8%

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

       4. div-inv 70.8%

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot \frac{1}{6}\right)} \cdot x\right) \]

       5. metadata-eval 70.8%

          \[\leadsto x \cdot \left(\left(x \cdot \color{blue}{0.16666666666666666}\right) \cdot x\right) \]

   11. Applied egg-rr 70.8%

       \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot 0.16666666666666666\right) \cdot x\right)} \]

   if -2.5 < x < 2.5

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0 98.6%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \]

Alternative 5: 83.5% 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 57.3%

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

2. Taylor expanded in x around 0 84.1%

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

   1. unpow3 84.1%

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

   2. associate-*r* 84.1%

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

   3. distribute-rgt-out 84.1%

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

   4. *-commutative 84.1%

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

   5. +-commutative 84.1%

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

   6. associate-*l* 84.1%

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

   7. fma-def 84.1%

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

4. Simplified 84.1%

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

   1. fma-udef 84.1%

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

   2. associate-*r* 84.1%

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

6. Applied egg-rr 84.1%

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

7. Final simplification 84.1%

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

Alternative 6: 51.9% 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 57.3%

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

2. Taylor expanded in x around 0 49.1%

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

3. Final simplification 49.1%

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

Alternative 7: 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 57.3%

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

3. Applied egg-rr 2.7%

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

4. Final simplification 2.7%

   \[\leadsto -1 \]

Alternative 8: 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 57.3%

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

3. Applied egg-rr 3.2%

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

4. Final simplification 3.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023194 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))