Hyperbolic sine

Percentage Accurate: 53.7% → 100.0%

Time: 4.7s

Alternatives: 8

Speedup: 1.9×

• 48.8% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.0004:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot {x\_m}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.0004)
      (/ (* x_m (+ 2.0 (* 0.3333333333333333 (pow 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 <= 0.0004) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * pow(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 <= 0.0004d0) then
        tmp = (x_m * (2.0d0 + (0.3333333333333333d0 * (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 <= 0.0004) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * Math.pow(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 <= 0.0004:
		tmp = (x_m * (2.0 + (0.3333333333333333 * math.pow(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 <= 0.0004)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * (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 <= 0.0004)
		tmp = (x_m * (2.0 + (0.3333333333333333 * (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, 0.0004], 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], 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))) < 4.00000000000000019e-4

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0 91.3%

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

   if 4.00000000000000019e-4 < (-.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 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.0004:\\ \;\;\;\;\frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+24}:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot {x\_m}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({x\_m}^{12} \cdot 0.0007716049382716049\right)}^{0.25}\\ \end{array} \end{array} \]

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+24)
    (/ (* x_m (+ 2.0 (* 0.3333333333333333 (pow x_m 2.0)))) 2.0)
    (pow (* (pow x_m 12.0) 0.0007716049382716049) 0.25))))

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+24) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * pow(x_m, 2.0)))) / 2.0;
	} else {
		tmp = pow((pow(x_m, 12.0) * 0.0007716049382716049), 0.25);
	}
	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+24) then
        tmp = (x_m * (2.0d0 + (0.3333333333333333d0 * (x_m ** 2.0d0)))) / 2.0d0
    else
        tmp = ((x_m ** 12.0d0) * 0.0007716049382716049d0) ** 0.25d0
    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+24) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * Math.pow(x_m, 2.0)))) / 2.0;
	} else {
		tmp = Math.pow((Math.pow(x_m, 12.0) * 0.0007716049382716049), 0.25);
	}
	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+24:
		tmp = (x_m * (2.0 + (0.3333333333333333 * math.pow(x_m, 2.0)))) / 2.0
	else:
		tmp = math.pow((math.pow(x_m, 12.0) * 0.0007716049382716049), 0.25)
	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+24)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * (x_m ^ 2.0)))) / 2.0);
	else
		tmp = Float64((x_m ^ 12.0) * 0.0007716049382716049) ^ 0.25;
	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+24)
		tmp = (x_m * (2.0 + (0.3333333333333333 * (x_m ^ 2.0)))) / 2.0;
	else
		tmp = ((x_m ^ 12.0) * 0.0007716049382716049) ^ 0.25;
	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+24], 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], N[Power[N[(N[Power[x$95$m, 12.0], $MachinePrecision] * 0.0007716049382716049), $MachinePrecision], 0.25], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999998e23

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0 90.0%

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

   if 9.9999999999999998e23 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 79.4%

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

   4. Taylor expanded in x around inf 79.4%

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

      1. add-sqr-sqrt 79.4%

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

      2. sqrt-unprod 87.2%

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

      3. pow1/2 87.2%

         \[\leadsto \color{blue}{{\left(\frac{0.3333333333333333 \cdot {x}^{3}}{2} \cdot \frac{0.3333333333333333 \cdot {x}^{3}}{2}\right)}^{0.5}} \]

      4. pow2 87.2%

         \[\leadsto {\color{blue}{\left({\left(\frac{0.3333333333333333 \cdot {x}^{3}}{2}\right)}^{2}\right)}}^{0.5} \]

      5. div-inv 87.2%

         \[\leadsto {\left({\color{blue}{\left(\left(0.3333333333333333 \cdot {x}^{3}\right) \cdot \frac{1}{2}\right)}}^{2}\right)}^{0.5} \]

      6. *-commutative 87.2%

         \[\leadsto {\left({\left(\color{blue}{\left({x}^{3} \cdot 0.3333333333333333\right)} \cdot \frac{1}{2}\right)}^{2}\right)}^{0.5} \]

      7. associate-*l* 87.2%

         \[\leadsto {\left({\color{blue}{\left({x}^{3} \cdot \left(0.3333333333333333 \cdot \frac{1}{2}\right)\right)}}^{2}\right)}^{0.5} \]

      8. unpow-prod-down 87.2%

         \[\leadsto {\color{blue}{\left({\left({x}^{3}\right)}^{2} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{2}\right)}}^{0.5} \]

      9. pow-pow 87.2%

         \[\leadsto {\left(\color{blue}{{x}^{\left(3 \cdot 2\right)}} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{2}\right)}^{0.5} \]

      10. metadata-eval 87.2%

          \[\leadsto {\left({x}^{\color{blue}{6}} \cdot {\left(0.3333333333333333 \cdot \frac{1}{2}\right)}^{2}\right)}^{0.5} \]

      11. metadata-eval 87.2%

          \[\leadsto {\left({x}^{6} \cdot {\left(0.3333333333333333 \cdot \color{blue}{0.5}\right)}^{2}\right)}^{0.5} \]

      12. metadata-eval 87.2%

          \[\leadsto {\left({x}^{6} \cdot {\color{blue}{0.16666666666666666}}^{2}\right)}^{0.5} \]

      13. metadata-eval 87.2%

          \[\leadsto {\left({x}^{6} \cdot \color{blue}{0.027777777777777776}\right)}^{0.5} \]

   6. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{{\left({x}^{6} \cdot 0.027777777777777776\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 87.2%

         \[\leadsto \color{blue}{\sqrt{{x}^{6} \cdot 0.027777777777777776}} \]

   8. Simplified 87.2%

      \[\leadsto \color{blue}{\sqrt{{x}^{6} \cdot 0.027777777777777776}} \]
   9. Step-by-step derivation

      1. pow1/2 87.2%

         \[\leadsto \color{blue}{{\left({x}^{6} \cdot 0.027777777777777776\right)}^{0.5}} \]

      2. add-sqr-sqrt 87.2%

         \[\leadsto {\color{blue}{\left(\sqrt{{x}^{6} \cdot 0.027777777777777776} \cdot \sqrt{{x}^{6} \cdot 0.027777777777777776}\right)}}^{0.5} \]

      3. pow1/2 87.2%

         \[\leadsto {\left(\color{blue}{{\left({x}^{6} \cdot 0.027777777777777776\right)}^{0.5}} \cdot \sqrt{{x}^{6} \cdot 0.027777777777777776}\right)}^{0.5} \]

      4. pow1/2 87.2%

         \[\leadsto {\left({\left({x}^{6} \cdot 0.027777777777777776\right)}^{0.5} \cdot \color{blue}{{\left({x}^{6} \cdot 0.027777777777777776\right)}^{0.5}}\right)}^{0.5} \]

      5. pow-prod-down 96.8%

         \[\leadsto {\color{blue}{\left({\left(\left({x}^{6} \cdot 0.027777777777777776\right) \cdot \left({x}^{6} \cdot 0.027777777777777776\right)\right)}^{0.5}\right)}}^{0.5} \]

      6. pow-pow 96.8%

         \[\leadsto \color{blue}{{\left(\left({x}^{6} \cdot 0.027777777777777776\right) \cdot \left({x}^{6} \cdot 0.027777777777777776\right)\right)}^{\left(0.5 \cdot 0.5\right)}} \]

      7. swap-sqr 96.8%

         \[\leadsto {\color{blue}{\left(\left({x}^{6} \cdot {x}^{6}\right) \cdot \left(0.027777777777777776 \cdot 0.027777777777777776\right)\right)}}^{\left(0.5 \cdot 0.5\right)} \]

      8. pow-prod-up 96.8%

         \[\leadsto {\left(\color{blue}{{x}^{\left(6 + 6\right)}} \cdot \left(0.027777777777777776 \cdot 0.027777777777777776\right)\right)}^{\left(0.5 \cdot 0.5\right)} \]

      9. metadata-eval 96.8%

         \[\leadsto {\left({x}^{\color{blue}{12}} \cdot \left(0.027777777777777776 \cdot 0.027777777777777776\right)\right)}^{\left(0.5 \cdot 0.5\right)} \]

      10. metadata-eval 96.8%

          \[\leadsto {\left({x}^{12} \cdot \color{blue}{0.0007716049382716049}\right)}^{\left(0.5 \cdot 0.5\right)} \]

      11. metadata-eval 96.8%

          \[\leadsto {\left({x}^{12} \cdot 0.0007716049382716049\right)}^{\color{blue}{0.25}} \]

   10. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{{\left({x}^{12} \cdot 0.0007716049382716049\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+24}:\\ \;\;\;\;\frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{12} \cdot 0.0007716049382716049\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + 0.3333333333333333 \cdot {x\_m}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x\_m}^{9} \cdot 0.004629629629629629}\\ \end{array} \end{array} \]

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 2e+31)
    (/ (* x_m (+ 2.0 (* 0.3333333333333333 (pow 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 <= 2e+31) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * pow(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 <= 2e+31) {
		tmp = (x_m * (2.0 + (0.3333333333333333 * Math.pow(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 <= 2e+31)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64(0.3333333333333333 * (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, 2e+31], 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], 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 < 1.9999999999999999e31

   1. Initial program 37.8%

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

   3. Taylor expanded in x around 0 87.8%

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

   if 1.9999999999999999e31 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.3%

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

   4. Taylor expanded in x around inf 86.3%

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

      1. add-cbrt-cube 98.3%

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

      2. pow1/3 98.3%

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

      3. pow3 98.3%

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

      4. div-inv 98.3%

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

      5. *-commutative 98.3%

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

      6. associate-*l* 98.3%

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

      7. unpow-prod-down 98.3%

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

      8. pow-pow 98.3%

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

      9. metadata-eval 98.3%

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

      10. metadata-eval 98.3%

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

      11. metadata-eval 98.3%

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

      12. metadata-eval 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. unpow1/3 98.3%

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

   8. Simplified 98.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+31}:\\ \;\;\;\;\frac{x \cdot \left(2 + 0.3333333333333333 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{9} \cdot 0.004629629629629629}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% 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 51.2%

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

3. Taylor expanded in x around 0 87.5%

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

4. Final simplification 87.5%

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

Alternative 5: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.5:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{3} \cdot 0.16666666666666666\\ \end{array} \end{array} \]

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 2.5) (/ (* 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 <= 2.5) {
		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 <= 2.5d0) 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 <= 2.5) {
		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 <= 2.5:
		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 <= 2.5)
		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 <= 2.5)
		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, 2.5], 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 < 2.5

   1. Initial program 35.3%

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

   3. Taylor expanded in x around 0 71.9%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.8%

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

   4. Taylor expanded in x around inf 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-/l* 75.8%

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

      3. metadata-eval 75.8%

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

   6. Applied egg-rr 75.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{3} \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% 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 51.2%

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

3. Taylor expanded in x around 0 55.6%

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

4. Final simplification 55.6%

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

Alternative 7: 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 51.2%

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

4. Applied egg-rr 3.6%

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

5. Final simplification 3.6%

   \[\leadsto 0 \]
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 51.2%

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

4. Applied egg-rr 3.0%

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

5. Final simplification 3.0%

   \[\leadsto 0.25 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))