Hyperbolic sine

Percentage Accurate: 55.2% → 99.8%

Time: 6.7s

Alternatives: 8

Speedup: 1.9×

• 50.6% 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: 55.2% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\frac{x\_m \cdot 2 + 0.3333333333333333 \cdot {x\_m}^{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 1e-7)
      (/ (+ (* x_m 2.0) (* 0.3333333333333333 (pow x_m 3.0))) 2.0)
      (/ t_0 2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = exp(x_m) - exp(-x_m);
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = ((x_m * 2.0) + (0.3333333333333333 * pow(x_m, 3.0))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return x_s * tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x_m) - exp(-x_m)
    if (t_0 <= 1d-7) then
        tmp = ((x_m * 2.0d0) + (0.3333333333333333d0 * (x_m ** 3.0d0))) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = x_s * tmp
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) {
	double t_0 = Math.exp(x_m) - Math.exp(-x_m);
	double tmp;
	if (t_0 <= 1e-7) {
		tmp = ((x_m * 2.0) + (0.3333333333333333 * Math.pow(x_m, 3.0))) / 2.0;
	} else {
		tmp = t_0 / 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):
	t_0 = math.exp(x_m) - math.exp(-x_m)
	tmp = 0
	if t_0 <= 1e-7:
		tmp = ((x_m * 2.0) + (0.3333333333333333 * math.pow(x_m, 3.0))) / 2.0
	else:
		tmp = t_0 / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(exp(x_m) - exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 1e-7)
		tmp = Float64(Float64(Float64(x_m * 2.0) + Float64(0.3333333333333333 * (x_m ^ 3.0))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = exp(x_m) - exp(-x_m);
	tmp = 0.0;
	if (t_0 <= 1e-7)
		tmp = ((x_m * 2.0) + (0.3333333333333333 * (x_m ^ 3.0))) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = 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_] := Block[{t$95$0 = N[(N[Exp[x$95$m], $MachinePrecision] - N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 1e-7], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 9.9999999999999995e-8

   1. Initial program 32.7%

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

   3. Taylor expanded in x around 0 92.4%

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

      1. distribute-rgt-in 92.4%

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

      2. fma-define 92.4%

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

      3. associate-*l* 92.4%

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

      4. pow-plus 92.4%

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

      5. metadata-eval 92.4%

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

   5. Simplified 92.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right)}}{2} \]
   6. Step-by-step derivation

      1. fma-undefine 92.4%

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

      2. *-commutative 92.4%

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

   7. Applied egg-rr 92.4%

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

   if 9.9999999999999995e-8 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 10^{-7}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+25}:\\ \;\;\;\;\frac{x\_m \cdot 2 + 0.3333333333333333 \cdot {x\_m}^{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x\_m}^{9}} \cdot 0.16666666666666666\\ \end{array} \end{array} \]

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 1e+25)
    (/ (+ (* x_m 2.0) (* 0.3333333333333333 (pow x_m 3.0))) 2.0)
    (* (cbrt (pow x_m 9.0)) 0.16666666666666666))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1e+25) {
		tmp = ((x_m * 2.0) + (0.3333333333333333 * pow(x_m, 3.0))) / 2.0;
	} else {
		tmp = cbrt(pow(x_m, 9.0)) * 0.16666666666666666;
	}
	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 <= 1e+25) {
		tmp = ((x_m * 2.0) + (0.3333333333333333 * Math.pow(x_m, 3.0))) / 2.0;
	} else {
		tmp = Math.cbrt(Math.pow(x_m, 9.0)) * 0.16666666666666666;
	}
	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 <= 1e+25)
		tmp = Float64(Float64(Float64(x_m * 2.0) + Float64(0.3333333333333333 * (x_m ^ 3.0))) / 2.0);
	else
		tmp = Float64(cbrt((x_m ^ 9.0)) * 0.16666666666666666);
	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, 1e+25], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[x$95$m, 9.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000009e25

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0 90.1%

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

      1. distribute-rgt-in 90.1%

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

      2. fma-define 90.1%

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

      3. associate-*l* 90.1%

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

      4. pow-plus 90.1%

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

      5. metadata-eval 90.1%

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

   5. Simplified 90.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right)}}{2} \]
   6. Step-by-step derivation

      1. fma-undefine 90.1%

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

      2. *-commutative 90.1%

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

   7. Applied egg-rr 90.1%

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

   if 1.00000000000000009e25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.7%

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

      1. distribute-rgt-in 77.7%

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

      2. fma-define 77.7%

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

      3. associate-*l* 77.7%

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

      4. pow-plus 77.7%

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

      5. metadata-eval 77.7%

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

   5. Simplified 77.7%

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

   6. Taylor expanded in x around inf 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-/l* 77.7%

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

      3. metadata-eval 77.7%

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

   8. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{{x}^{3} \cdot 0.16666666666666666} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left({x}^{3} \cdot {x}^{3}\right) \cdot {x}^{3}}} \cdot 0.16666666666666666 \]

      2. pow3 100.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left({x}^{3}\right)}^{3}}} \cdot 0.16666666666666666 \]

      3. pow-pow 100.0%

         \[\leadsto \sqrt[3]{\color{blue}{{x}^{\left(3 \cdot 3\right)}}} \cdot 0.16666666666666666 \]

      4. metadata-eval 100.0%

         \[\leadsto \sqrt[3]{{x}^{\color{blue}{9}}} \cdot 0.16666666666666666 \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\sqrt[3]{{x}^{9}}} \cdot 0.16666666666666666 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+25}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{9}} \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot {x\_m}^{2}\right)}{2} \end{array} \]

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 (+ 2.0 (* 0.3333333333333333 (pow x_m 2.0)))) 2.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * (2.0 + (0.3333333333333333 * pow(x_m, 2.0)))) / 2.0);
}

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 * (2.0d0 + (0.3333333333333333d0 * (x_m ** 2.0d0)))) / 2.0d0)
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 * (2.0 + (0.3333333333333333 * Math.pow(x_m, 2.0)))) / 2.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * (2.0 + (0.3333333333333333 * math.pow(x_m, 2.0)))) / 2.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * (x_m ^ 2.0)))) / 2.0))
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 * (2.0 + (0.3333333333333333 * (x_m ^ 2.0)))) / 2.0);
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[(N[(x$95$m * N[(2.0 + N[(0.3333333333333333 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in x around 0 87.0%

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

4. Final simplification 87.0%

   \[\leadsto \frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{2} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot 2 + 0.3333333333333333 \cdot {x\_m}^{3}}{2} \end{array} \]

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 2.0) (* 0.3333333333333333 (pow x_m 3.0))) 2.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((x_m * 2.0) + (0.3333333333333333 * pow(x_m, 3.0))) / 2.0);
}

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 * 2.0d0) + (0.3333333333333333d0 * (x_m ** 3.0d0))) / 2.0d0)
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 * 2.0) + (0.3333333333333333 * Math.pow(x_m, 3.0))) / 2.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((x_m * 2.0) + (0.3333333333333333 * math.pow(x_m, 3.0))) / 2.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64(x_m * 2.0) + Float64(0.3333333333333333 * (x_m ^ 3.0))) / 2.0))
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 * 2.0) + (0.3333333333333333 * (x_m ^ 3.0))) / 2.0);
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[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in x around 0 87.0%

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

   1. distribute-rgt-in 87.1%

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

   2. fma-define 87.1%

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

   3. associate-*l* 87.1%

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

   4. pow-plus 87.1%

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

   5. metadata-eval 87.1%

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

5. Simplified 87.1%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right)}}{2} \]
6. Step-by-step derivation

   1. fma-undefine 87.1%

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

   2. *-commutative 87.1%

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

7. Applied egg-rr 87.1%

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

8. Final simplification 87.1%

   \[\leadsto \frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2} \]
9. Add Preprocessing

Alternative 5: 83.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot 0.16666666666666666\\ \end{array} \end{array} \]

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 2.4) (/ (* x_m 2.0) 2.0) (* (pow x_m 3.0) 0.16666666666666666))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.4) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = pow(x_m, 3.0) * 0.16666666666666666;
	}
	return x_s * tmp;
}

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
    real(8) :: tmp
    if (x_m <= 2.4d0) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = (x_m ** 3.0d0) * 0.16666666666666666d0
    end if
    code = x_s * tmp
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) {
	double tmp;
	if (x_m <= 2.4) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = Math.pow(x_m, 3.0) * 0.16666666666666666;
	}
	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 <= 2.4:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = math.pow(x_m, 3.0) * 0.16666666666666666
	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 <= 2.4)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64((x_m ^ 3.0) * 0.16666666666666666);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2.4)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = (x_m ^ 3.0) * 0.16666666666666666;
	end
	tmp_2 = 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, 2.4], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 32.7%

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

   3. Taylor expanded in x around 0 74.3%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 72.2%

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

      1. distribute-rgt-in 72.2%

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

      2. fma-define 72.2%

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

      3. associate-*l* 72.2%

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

      4. pow-plus 72.2%

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

      5. metadata-eval 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in x around inf 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-/l* 72.2%

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

      3. metadata-eval 72.2%

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

   8. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{{x}^{3} \cdot 0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 41.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m \cdot 2}{2} \end{array} \]

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 2.0) 2.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((x_m * 2.0) / 2.0);
}

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 * 2.0d0) / 2.0d0)
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 * 2.0) / 2.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((x_m * 2.0) / 2.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(x_m * 2.0) / 2.0))
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 * 2.0) / 2.0);
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[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in x around 0 56.1%

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

4. Final simplification 56.1%

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

Alternative 7: 3.4% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 0.0))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 0.0;
}

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 * 0.0d0
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 * 0.0;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 0.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 0.0)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

4. Applied egg-rr 3.7%

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

5. Final simplification 3.7%

   \[\leadsto 0 \]
6. Add Preprocessing

Alternative 8: 4.3% accurate, 206.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 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 * 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 * 0.25), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

4. Applied egg-rr 2.9%

   \[\leadsto \frac{\color{blue}{0.5}}{2} \]

5. Final simplification 2.9%

   \[\leadsto 0.25 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))