Hyperbolic sine

Percentage Accurate: 54.2% → 99.1%

Time: 7.6s

Alternatives: 10

Speedup: 17.1×

• 49.5% 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: 54.2% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.0) (/ (* x_m 2.0) 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 <= 0.0) {
		tmp = (x_m * 2.0) / 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 <= 0.0d0) then
        tmp = (x_m * 2.0d0) / 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 <= 0.0) {
		tmp = (x_m * 2.0) / 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 <= 0.0:
		tmp = (x_m * 2.0) / 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 <= 0.0)
		tmp = Float64(Float64(x_m * 2.0) / 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 <= 0.0)
		tmp = (x_m * 2.0) / 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, 0.0], N[(N[(x$95$m * 2.0), $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))) < 0.0

   1. Initial program 36.3%

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

   3. Taylor expanded in x around 0 69.1%

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

   if 0.0 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 99.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-22}:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({x\_m}^{18} \cdot 2.143347050754458 \cdot 10^{-5}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

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-22)
    (/ (* x_m 2.0) 2.0)
    (pow (* (pow x_m 18.0) 2.143347050754458e-5) 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-22) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = pow((pow(x_m, 18.0) * 2.143347050754458e-5), 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 <= 1d-22) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = ((x_m ** 18.0d0) * 2.143347050754458d-5) ** 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 <= 1e-22) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = Math.pow((Math.pow(x_m, 18.0) * 2.143347050754458e-5), 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 <= 1e-22:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = math.pow((math.pow(x_m, 18.0) * 2.143347050754458e-5), 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-22)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64((x_m ^ 18.0) * 2.143347050754458e-5) ^ 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 <= 1e-22)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = ((x_m ^ 18.0) * 2.143347050754458e-5) ^ 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, 1e-22], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Power[x$95$m, 18.0], $MachinePrecision] * 2.143347050754458e-5), $MachinePrecision], 0.16666666666666666], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e-22

   1. Initial program 36.8%

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

   3. Taylor expanded in x around 0 68.6%

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

   if 1e-22 < x

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 67.4%

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

   4. Taylor expanded in x around inf 62.4%

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

      1. add-cbrt-cube 82.5%

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

      2. pow1/3 82.5%

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

      3. pow3 82.5%

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

      4. div-inv 82.5%

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

      5. *-commutative 82.5%

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

      6. associate-*l* 82.5%

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

      7. unpow-prod-down 82.5%

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

      8. pow-pow 82.5%

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

      9. metadata-eval 82.5%

         \[\leadsto {\left({x}^{\color{blue}{9}} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      10. metadata-eval 82.5%

          \[\leadsto {\left({x}^{9} \cdot {\left(0.3333333333333333 \cdot \color{blue}{0.5}\right)}^{3}\right)}^{0.3333333333333333} \]

      11. metadata-eval 82.5%

          \[\leadsto {\left({x}^{9} \cdot {\color{blue}{0.16666666666666666}}^{3}\right)}^{0.3333333333333333} \]

      12. metadata-eval 82.5%

          \[\leadsto {\left({x}^{9} \cdot \color{blue}{0.004629629629629629}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{{\left({x}^{9} \cdot 0.004629629629629629\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 82.5%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}} \]

   8. Simplified 82.5%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}} \]
   9. Step-by-step derivation

      1. pow1/3 82.5%

         \[\leadsto \color{blue}{{\left({x}^{9} \cdot 0.004629629629629629\right)}^{0.3333333333333333}} \]

      2. metadata-eval 82.5%

         \[\leadsto {\left({x}^{9} \cdot 0.004629629629629629\right)}^{\color{blue}{\left(0.16666666666666666 + 0.16666666666666666\right)}} \]

      3. pow-prod-up 82.5%

         \[\leadsto \color{blue}{{\left({x}^{9} \cdot 0.004629629629629629\right)}^{0.16666666666666666} \cdot {\left({x}^{9} \cdot 0.004629629629629629\right)}^{0.16666666666666666}} \]

      4. pow-prod-down 88.7%

         \[\leadsto \color{blue}{{\left(\left({x}^{9} \cdot 0.004629629629629629\right) \cdot \left({x}^{9} \cdot 0.004629629629629629\right)\right)}^{0.16666666666666666}} \]

      5. swap-sqr 88.7%

         \[\leadsto {\color{blue}{\left(\left({x}^{9} \cdot {x}^{9}\right) \cdot \left(0.004629629629629629 \cdot 0.004629629629629629\right)\right)}}^{0.16666666666666666} \]

      6. pow-prod-up 88.7%

         \[\leadsto {\left(\color{blue}{{x}^{\left(9 + 9\right)}} \cdot \left(0.004629629629629629 \cdot 0.004629629629629629\right)\right)}^{0.16666666666666666} \]

      7. metadata-eval 88.7%

         \[\leadsto {\left({x}^{\color{blue}{18}} \cdot \left(0.004629629629629629 \cdot 0.004629629629629629\right)\right)}^{0.16666666666666666} \]

      8. metadata-eval 88.7%

         \[\leadsto {\left({x}^{18} \cdot \color{blue}{2.143347050754458 \cdot 10^{-5}}\right)}^{0.16666666666666666} \]

   10. Applied egg-rr 88.7%

       \[\leadsto \color{blue}{{\left({x}^{18} \cdot 2.143347050754458 \cdot 10^{-5}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-22}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{18} \cdot 2.143347050754458 \cdot 10^{-5}\right)}^{0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.05:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x\_m}^{9} \cdot 0.004629629629629629}\\ \end{array} \end{array} \]

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 2.05)
    (/ (* x_m 2.0) 2.0)
    (cbrt (* (pow x_m 9.0) 0.004629629629629629)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.05) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = cbrt((pow(x_m, 9.0) * 0.004629629629629629));
	}
	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 <= 2.05) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = Math.cbrt((Math.pow(x_m, 9.0) * 0.004629629629629629));
	}
	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 <= 2.05)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = cbrt(Float64((x_m ^ 9.0) * 0.004629629629629629));
	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, 2.05], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Power[x$95$m, 9.0], $MachinePrecision] * 0.004629629629629629), $MachinePrecision], 1/3], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999998

   1. Initial program 36.3%

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

   3. Taylor expanded in x around 0 69.3%

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

   if 2.0499999999999998 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.6%

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

   4. Taylor expanded in x around inf 65.5%

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

      1. add-cbrt-cube 86.7%

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

      2. pow1/3 86.7%

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

      3. pow3 86.7%

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

      4. div-inv 86.7%

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

      5. *-commutative 86.7%

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

      6. associate-*l* 86.7%

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

      7. unpow-prod-down 86.7%

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

      8. pow-pow 86.7%

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

      9. metadata-eval 86.7%

         \[\leadsto {\left({x}^{\color{blue}{9}} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{3}\right)}^{0.3333333333333333} \]

      10. metadata-eval 86.7%

          \[\leadsto {\left({x}^{9} \cdot {\left(0.3333333333333333 \cdot \color{blue}{0.5}\right)}^{3}\right)}^{0.3333333333333333} \]

      11. metadata-eval 86.7%

          \[\leadsto {\left({x}^{9} \cdot {\color{blue}{0.16666666666666666}}^{3}\right)}^{0.3333333333333333} \]

      12. metadata-eval 86.7%

          \[\leadsto {\left({x}^{9} \cdot \color{blue}{0.004629629629629629}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{{\left({x}^{9} \cdot 0.004629629629629629\right)}^{0.3333333333333333}} \]
   7. Step-by-step derivation

      1. unpow1/3 86.7%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}} \]

   8. Simplified 86.7%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.9× speedup

Math

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

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 54.0%

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

3. Taylor expanded in x around 0 82.3%

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

Alternative 5: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-21}:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{{x\_m}^{-3}}\\ \end{array} \end{array} \]

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-21)
    (/ (* x_m 2.0) 2.0)
    (/ 0.16666666666666666 (pow x_m -3.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1e-21) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = 0.16666666666666666 / pow(x_m, -3.0);
	}
	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) :: tmp
    if (x_m <= 1d-21) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = 0.16666666666666666d0 / (x_m ** (-3.0d0))
    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 tmp;
	if (x_m <= 1e-21) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = 0.16666666666666666 / Math.pow(x_m, -3.0);
	}
	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 <= 1e-21:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = 0.16666666666666666 / math.pow(x_m, -3.0)
	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 <= 1e-21)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64(0.16666666666666666 / (x_m ^ -3.0));
	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)
	tmp = 0.0;
	if (x_m <= 1e-21)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = 0.16666666666666666 / (x_m ^ -3.0);
	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_] := N[(x$95$s * If[LessEqual[x$95$m, 1e-21], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.16666666666666666 / N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999908e-22

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 68.8%

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

   if 9.99999999999999908e-22 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0 67.0%

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

   4. Taylor expanded in x around inf 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-/l* 63.2%

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

      3. metadata-eval 63.2%

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

   6. Applied egg-rr 63.2%

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

      1. *-commutative 63.2%

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

      2. metadata-eval 63.2%

         \[\leadsto \color{blue}{\frac{1}{6}} \cdot {x}^{3} \]

      3. associate-/r/ 63.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{6}{{x}^{3}}}} \]

      4. unpow-1 63.2%

         \[\leadsto \color{blue}{{\left(\frac{6}{{x}^{3}}\right)}^{-1}} \]

      5. add-log-exp 93.7%

         \[\leadsto \color{blue}{\log \left(e^{{\left(\frac{6}{{x}^{3}}\right)}^{-1}}\right)} \]

      6. *-un-lft-identity 93.7%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{{\left(\frac{6}{{x}^{3}}\right)}^{-1}}\right)} \]

      7. log-prod 93.7%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{{\left(\frac{6}{{x}^{3}}\right)}^{-1}}\right)} \]

      8. metadata-eval 93.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{{\left(\frac{6}{{x}^{3}}\right)}^{-1}}\right) \]

      9. add-log-exp 63.2%

         \[\leadsto 0 + \color{blue}{{\left(\frac{6}{{x}^{3}}\right)}^{-1}} \]

      10. unpow-1 63.2%

          \[\leadsto 0 + \color{blue}{\frac{1}{\frac{6}{{x}^{3}}}} \]

      11. div-inv 63.2%

          \[\leadsto 0 + \frac{1}{\color{blue}{6 \cdot \frac{1}{{x}^{3}}}} \]

      12. associate-/r* 63.2%

          \[\leadsto 0 + \color{blue}{\frac{\frac{1}{6}}{\frac{1}{{x}^{3}}}} \]

      13. metadata-eval 63.2%

          \[\leadsto 0 + \frac{\color{blue}{0.16666666666666666}}{\frac{1}{{x}^{3}}} \]

      14. pow-flip 63.2%

          \[\leadsto 0 + \frac{0.16666666666666666}{\color{blue}{{x}^{\left(-3\right)}}} \]

      15. metadata-eval 63.2%

          \[\leadsto 0 + \frac{0.16666666666666666}{{x}^{\color{blue}{-3}}} \]

   8. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{0 + \frac{0.16666666666666666}{{x}^{-3}}} \]
   9. Step-by-step derivation

      1. +-lft-identity 63.2%

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

   10. Simplified 63.2%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-21}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.16666666666666666}{{x}^{-3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-21}:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{x\_m}{\frac{6}{x\_m}}\\ \end{array} \end{array} \]

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-21) (/ (* x_m 2.0) 2.0) (* x_m (/ x_m (/ 6.0 x_m))))))

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-21) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = x_m * (x_m / (6.0 / x_m));
	}
	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 <= 1d-21) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = x_m * (x_m / (6.0d0 / x_m))
    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 <= 1e-21) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = x_m * (x_m / (6.0 / x_m));
	}
	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 <= 1e-21:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = x_m * (x_m / (6.0 / x_m))
	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-21)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64(x_m * Float64(x_m / Float64(6.0 / x_m)));
	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 <= 1e-21)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = x_m * (x_m / (6.0 / x_m));
	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, 1e-21], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(x$95$m * N[(x$95$m / N[(6.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999908e-22

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 68.8%

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

   if 9.99999999999999908e-22 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0 67.0%

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

   4. Taylor expanded in x around inf 63.2%

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

      1. clear-num 63.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{0.3333333333333333 \cdot {x}^{3}}}} \]

      2. inv-pow 63.2%

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

      3. associate-/r* 63.2%

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

      4. metadata-eval 63.2%

         \[\leadsto {\left(\frac{\color{blue}{6}}{{x}^{3}}\right)}^{-1} \]

   6. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{{\left(\frac{6}{{x}^{3}}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 63.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{6}{{x}^{3}}}} \]

      2. clear-num 63.2%

         \[\leadsto \color{blue}{\frac{{x}^{3}}{6}} \]

      3. cube-mult 63.2%

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

      4. associate-/l* 63.2%

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

      5. pow2 63.2%

         \[\leadsto x \cdot \frac{\color{blue}{{x}^{2}}}{6} \]

   8. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{x \cdot \frac{{x}^{2}}{6}} \]
   9. Step-by-step derivation

      1. associate-*r/ 63.2%

         \[\leadsto \color{blue}{\frac{x \cdot {x}^{2}}{6}} \]

      2. unpow2 63.2%

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

      3. cube-mult 63.2%

         \[\leadsto \frac{\color{blue}{{x}^{3}}}{6} \]

      4. div-inv 63.2%

         \[\leadsto \color{blue}{{x}^{3} \cdot \frac{1}{6}} \]

      5. metadata-eval 63.2%

         \[\leadsto {x}^{\color{blue}{\left(--3\right)}} \cdot \frac{1}{6} \]

      6. pow-flip 63.2%

         \[\leadsto \color{blue}{\frac{1}{{x}^{-3}}} \cdot \frac{1}{6} \]

      7. metadata-eval 63.2%

         \[\leadsto \frac{1}{{x}^{-3}} \cdot \color{blue}{0.16666666666666666} \]

      8. associate-/r/ 63.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{-3}}{0.16666666666666666}}} \]

      9. div-inv 63.2%

         \[\leadsto \frac{1}{\color{blue}{{x}^{-3} \cdot \frac{1}{0.16666666666666666}}} \]

      10. metadata-eval 63.2%

          \[\leadsto \frac{1}{{x}^{-3} \cdot \color{blue}{6}} \]

   10. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{\frac{1}{{x}^{-3} \cdot 6}} \]
   11. Step-by-step derivation

       1. associate-/r* 63.2%

          \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{-3}}}{6}} \]

       2. pow-flip 63.2%

          \[\leadsto \frac{\color{blue}{{x}^{\left(--3\right)}}}{6} \]

       3. metadata-eval 63.2%

          \[\leadsto \frac{{x}^{\color{blue}{3}}}{6} \]

       4. pow3 63.2%

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

       5. unpow2 63.2%

          \[\leadsto \frac{\color{blue}{{x}^{2}} \cdot x}{6} \]

       6. associate-*l/ 63.2%

          \[\leadsto \color{blue}{\frac{{x}^{2}}{6} \cdot x} \]

       7. associate-/r/ 63.2%

          \[\leadsto \color{blue}{\frac{{x}^{2}}{\frac{6}{x}}} \]

       8. unpow2 63.2%

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

       9. associate-/l* 63.2%

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

   12. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{x \cdot \frac{x}{\frac{6}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-21}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{\frac{6}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.3% 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 54.0%

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

3. Taylor expanded in x around 0 51.5%

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

4. Final simplification 51.5%

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

Alternative 8: 4.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 4 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 4.0))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 4.0;
}

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 * 4.0d0
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 * 4.0;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 4.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 4.0)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 4.0;
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 * 4.0), $MachinePrecision]

Derivation

1. Initial program 54.0%

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

4. Applied egg-rr 2.9%

   \[\leadsto \frac{\color{blue}{8}}{2} \]

5. Final simplification 2.9%

   \[\leadsto 4 \]
6. Add Preprocessing

Alternative 9: 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 54.0%

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

4. Applied egg-rr 2.8%

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

5. Final simplification 2.8%

   \[\leadsto 0.25 \]
6. Add Preprocessing

Alternative 10: 3.5% 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 54.0%

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

4. Applied egg-rr 3.5%

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

5. Final simplification 3.5%

   \[\leadsto 0 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))