Hyperbolic sine

Percentage Accurate: 53.7% → 99.0%

Time: 8.3s

Alternatives: 6

Speedup: 29.4×

• 48.9% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;e^{x\_m} - e^{-x\_m} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x\_m + {x\_m}^{3} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(x\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= (- (exp x_m) (exp (- x_m))) 2e-7)
    (+ x_m (* (pow x_m 3.0) 0.16666666666666666))
    (/ (expm1 x_m) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if ((exp(x_m) - exp(-x_m)) <= 2e-7) {
		tmp = x_m + (pow(x_m, 3.0) * 0.16666666666666666);
	} else {
		tmp = expm1(x_m) / 2.0;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if ((Math.exp(x_m) - Math.exp(-x_m)) <= 2e-7) {
		tmp = x_m + (Math.pow(x_m, 3.0) * 0.16666666666666666);
	} else {
		tmp = Math.expm1(x_m) / 2.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if (math.exp(x_m) - math.exp(-x_m)) <= 2e-7:
		tmp = x_m + (math.pow(x_m, 3.0) * 0.16666666666666666)
	else:
		tmp = math.expm1(x_m) / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (Float64(exp(x_m) - exp(Float64(-x_m))) <= 2e-7)
		tmp = Float64(x_m + Float64((x_m ^ 3.0) * 0.16666666666666666));
	else
		tmp = Float64(expm1(x_m) / 2.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision], 2e-7], N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[x$95$m] - 1), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 1.9999999999999999e-7

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

      2. distribute-rgt-in 87.1%

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

      3. associate-*l* 87.1%

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

      4. fma-define 87.1%

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

      5. pow-plus 87.1%

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

      6. metadata-eval 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in x around 0 87.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. distribute-rgt-in 87.1%

         \[\leadsto \color{blue}{1 \cdot x + \left(0.16666666666666666 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 87.1%

         \[\leadsto \color{blue}{x} + \left(0.16666666666666666 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*r* 87.1%

         \[\leadsto x + \color{blue}{0.16666666666666666 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 87.1%

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

      5. unpow3 87.1%

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

      6. *-commutative 87.1%

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

   8. Simplified 87.1%

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

   if 1.9999999999999999e-7 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

   5. Simplified 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{1}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \frac{e^{x} + \color{blue}{-1}}{2} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{e^{x} + -1}}{2} \]
   9. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{e^{x} - 1}}{2} \]

      3. expm1-define 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{2} \]

   10. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(x\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.25) x_m (/ (expm1 x_m) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = x_m;
	} else {
		tmp = expm1(x_m) / 2.0;
	}
	return x_s * tmp;
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = x_m;
	} else {
		tmp = Math.expm1(x_m) / 2.0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.25:
		tmp = x_m
	else:
		tmp = math.expm1(x_m) / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = x_m;
	else
		tmp = Float64(expm1(x_m) / 2.0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.25], x$95$m, N[(N[(Exp[x$95$m] - 1), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

      2. distribute-rgt-in 87.1%

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

      3. associate-*l* 87.1%

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

      4. fma-define 87.1%

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

      5. pow-plus 87.1%

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

      6. metadata-eval 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in x around 0 69.5%

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

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

         \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

   5. Simplified 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{e^{x} - \color{blue}{1}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \frac{e^{x} + \color{blue}{-1}}{2} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{e^{x} + -1}}{2} \]
   9. Step-by-step derivation

      1. metadata-eval 100.0%

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

      2. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{e^{x} - 1}}{2} \]

      3. expm1-define 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{2} \]

   10. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.9% accurate, 13.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \left(1 + x\_m \cdot \left(0.25 + x\_m \cdot \left(0.08333333333333333 + x\_m \cdot 0.020833333333333332\right)\right)\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (*
   x_m
   (+
    1.0
    (*
     x_m
     (+ 0.25 (* x_m (+ 0.08333333333333333 (* x_m 0.020833333333333332)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * (1.0d0 + (x_m * (0.25d0 + (x_m * (0.08333333333333333d0 + (x_m * 0.020833333333333332d0)))))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * Float64(1.0 + Float64(x_m * Float64(0.25 + Float64(x_m * Float64(0.08333333333333333 + Float64(x_m * 0.020833333333333332))))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * (0.08333333333333333 + (x_m * 0.020833333333333332)))))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * N[(1.0 + N[(x$95$m * N[(0.25 + N[(x$95$m * N[(0.08333333333333333 + N[(x$95$m * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in x around 0 27.8%

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

   1. mul-1-neg 27.8%

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

   2. unsub-neg 27.8%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

5. Simplified 27.8%

   \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

6. Taylor expanded in x around 0 72.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + 0.020833333333333332 \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 72.0%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + \color{blue}{x \cdot 0.020833333333333332}\right)\right)\right) \]

8. Simplified 72.0%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + x \cdot 0.020833333333333332\right)\right)\right)} \]
9. Add Preprocessing

Alternative 4: 82.8% accurate, 18.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \left(1 + x\_m \cdot \left(0.25 + x\_m \cdot 0.08333333333333333\right)\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* x_m (+ 1.0 (* x_m (+ 0.25 (* x_m 0.08333333333333333)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * 0.08333333333333333)))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * (1.0d0 + (x_m * (0.25d0 + (x_m * 0.08333333333333333d0)))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * 0.08333333333333333)))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * 0.08333333333333333)))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * Float64(1.0 + Float64(x_m * Float64(0.25 + Float64(x_m * 0.08333333333333333))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * (1.0 + (x_m * (0.25 + (x_m * 0.08333333333333333)))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * N[(1.0 + N[(x$95$m * N[(0.25 + N[(x$95$m * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in x around 0 27.8%

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

   1. mul-1-neg 27.8%

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

   2. unsub-neg 27.8%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

5. Simplified 27.8%

   \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

6. Taylor expanded in x around 0 85.1%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + 0.08333333333333333 \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 85.1%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.25 + \color{blue}{x \cdot 0.08333333333333333}\right)\right) \]

8. Simplified 85.1%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + x \cdot 0.08333333333333333\right)\right)} \]
9. Add Preprocessing

Alternative 5: 74.8% accurate, 29.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \left(1 + x\_m \cdot 0.25\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m (+ 1.0 (* x_m 0.25)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * 0.25)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * (1.0d0 + (x_m * 0.25d0)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * (1.0 + (x_m * 0.25)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * (1.0 + (x_m * 0.25)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * Float64(1.0 + Float64(x_m * 0.25))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * (1.0 + (x_m * 0.25)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * N[(1.0 + N[(x$95$m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in x around 0 27.8%

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

   1. mul-1-neg 27.8%

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

   2. unsub-neg 27.8%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

5. Simplified 27.8%

   \[\leadsto \frac{e^{x} - \color{blue}{\left(1 - x\right)}}{2} \]

6. Taylor expanded in x around 0 66.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.25 \cdot x\right)} \]
7. Step-by-step derivation

   1. *-commutative 66.8%

      \[\leadsto x \cdot \left(1 + \color{blue}{x \cdot 0.25}\right) \]

8. Simplified 66.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot 0.25\right)} \]
9. Add Preprocessing

Alternative 6: 52.8% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in x around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. distribute-rgt-in 85.4%

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

   3. associate-*l* 85.4%

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

   4. fma-define 85.4%

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

   5. pow-plus 85.4%

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

   6. metadata-eval 85.4%

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

5. Simplified 85.4%

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

6. Taylor expanded in x around 0 54.5%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))