Hyperbolic sine

Percentage Accurate: 53.9% → 100.0%

Time: 4.6s

Alternatives: 9

Speedup: 1.9×

• 49.2% 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: 53.9% 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

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.001:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x\_m}^{3} + x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (- (exp x_m) (exp (- x_m)))))
   (*
    x_s
    (if (<= t_0 0.001)
      (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* 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 <= 0.001) {
		tmp = ((0.3333333333333333 * pow(x_m, 3.0)) + (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 <= 0.001d0) then
        tmp = ((0.3333333333333333d0 * (x_m ** 3.0d0)) + (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 <= 0.001) {
		tmp = ((0.3333333333333333 * Math.pow(x_m, 3.0)) + (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 <= 0.001:
		tmp = ((0.3333333333333333 * math.pow(x_m, 3.0)) + (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 <= 0.001)
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + 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 <= 0.001)
		tmp = ((0.3333333333333333 * (x_m ^ 3.0)) + (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, 0.001], N[(N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $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))) < 1e-3

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0 90.8%

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

   if 1e-3 < (-.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 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.001:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1500000:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x\_m}^{3} + x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x\_m}^{6} \cdot 0.027777777777777776}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1500000.0)
    (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* x_m 2.0)) 2.0)
    (sqrt (* (pow x_m 6.0) 0.027777777777777776)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1500000.0) {
		tmp = ((0.3333333333333333 * pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0;
	} else {
		tmp = sqrt((pow(x_m, 6.0) * 0.027777777777777776));
	}
	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 <= 1500000.0d0) then
        tmp = ((0.3333333333333333d0 * (x_m ** 3.0d0)) + (x_m * 2.0d0)) / 2.0d0
    else
        tmp = sqrt(((x_m ** 6.0d0) * 0.027777777777777776d0))
    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 <= 1500000.0) {
		tmp = ((0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0;
	} else {
		tmp = Math.sqrt((Math.pow(x_m, 6.0) * 0.027777777777777776));
	}
	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 <= 1500000.0:
		tmp = ((0.3333333333333333 * math.pow(x_m, 3.0)) + (x_m * 2.0)) / 2.0
	else:
		tmp = math.sqrt((math.pow(x_m, 6.0) * 0.027777777777777776))
	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 <= 1500000.0)
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * 2.0)) / 2.0);
	else
		tmp = sqrt(Float64((x_m ^ 6.0) * 0.027777777777777776));
	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 <= 1500000.0)
		tmp = ((0.3333333333333333 * (x_m ^ 3.0)) + (x_m * 2.0)) / 2.0;
	else
		tmp = sqrt(((x_m ^ 6.0) * 0.027777777777777776));
	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, 1500000.0], N[(N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[Sqrt[N[(N[Power[x$95$m, 6.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e6

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0 90.5%

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

   if 1.5e6 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.4%

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

   4. Taylor expanded in x around inf 66.4%

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

      1. pow1 66.4%

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

      2. sqr-pow 66.4%

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

      3. pow-prod-down 82.9%

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

   6. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{{\left(\left({x}^{6} \cdot 0.1111111111111111\right) \cdot 0.25\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 82.9%

         \[\leadsto \color{blue}{\sqrt{\left({x}^{6} \cdot 0.1111111111111111\right) \cdot 0.25}} \]

      2. associate-*l* 82.9%

         \[\leadsto \sqrt{\color{blue}{{x}^{6} \cdot \left(0.1111111111111111 \cdot 0.25\right)}} \]

      3. metadata-eval 82.9%

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

   8. Simplified 82.9%

      \[\leadsto \color{blue}{\sqrt{{x}^{6} \cdot 0.027777777777777776}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1500000:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{6} \cdot 0.027777777777777776}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000000000:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x\_m}^{3} + x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x\_m}^{9}} \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5000000000.0)
    (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* x_m 2.0)) 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 <= 5000000000.0) {
		tmp = ((0.3333333333333333 * pow(x_m, 3.0)) + (x_m * 2.0)) / 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 <= 5000000000.0) {
		tmp = ((0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * 2.0)) / 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 <= 5000000000.0)
		tmp = Float64(Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * 2.0)) / 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, 5000000000.0], N[(N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $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 < 5e9

   1. Initial program 37.7%

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

   3. Taylor expanded in x around 0 90.0%

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

   if 5e9 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.5%

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

   4. Taylor expanded in x around inf 67.5%

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

      1. div-inv 67.5%

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

      2. metadata-eval 67.5%

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

      3. pow-plus 67.5%

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

      4. *-commutative 67.5%

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

      5. associate-*l* 67.5%

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

      6. pow-plus 67.5%

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

      7. metadata-eval 67.5%

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

      8. metadata-eval 67.5%

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

      9. metadata-eval 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. add-cbrt-cube 90.5%

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

      2. pow3 90.5%

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

      3. pow-pow 90.5%

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

      4. metadata-eval 90.5%

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

   8. Applied egg-rr 90.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000000:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3} + x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{9}} \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.5:\\ \;\;\;\;\frac{x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6 \cdot {x\_m}^{-3}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 2.5) (/ (* x_m 2.0) 2.0) (/ 1.0 (* 6.0 (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 <= 2.5) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = 1.0 / (6.0 * 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 <= 2.5d0) then
        tmp = (x_m * 2.0d0) / 2.0d0
    else
        tmp = 1.0d0 / (6.0d0 * (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 <= 2.5) {
		tmp = (x_m * 2.0) / 2.0;
	} else {
		tmp = 1.0 / (6.0 * 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 <= 2.5:
		tmp = (x_m * 2.0) / 2.0
	else:
		tmp = 1.0 / (6.0 * 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 <= 2.5)
		tmp = Float64(Float64(x_m * 2.0) / 2.0);
	else
		tmp = Float64(1.0 / Float64(6.0 * (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 <= 2.5)
		tmp = (x_m * 2.0) / 2.0;
	else
		tmp = 1.0 / (6.0 * (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, 2.5], N[(N[(x$95$m * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(6.0 * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 37.0%

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

   3. Taylor expanded in x around 0 69.5%

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

   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 65.6%

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

   4. Taylor expanded in x around inf 65.6%

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

      1. pow1 65.6%

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

      2. sqr-pow 65.6%

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

      3. pow-prod-down 81.9%

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

   6. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{{\left(\left({x}^{6} \cdot 0.1111111111111111\right) \cdot 0.25\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 81.9%

         \[\leadsto \color{blue}{\sqrt{\left({x}^{6} \cdot 0.1111111111111111\right) \cdot 0.25}} \]

      2. associate-*l* 81.9%

         \[\leadsto \sqrt{\color{blue}{{x}^{6} \cdot \left(0.1111111111111111 \cdot 0.25\right)}} \]

      3. metadata-eval 81.9%

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

   8. Simplified 81.9%

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

      1. sqrt-prod 81.9%

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

      2. sqrt-pow1 65.6%

         \[\leadsto \color{blue}{{x}^{\left(\frac{6}{2}\right)}} \cdot \sqrt{0.027777777777777776} \]

      3. metadata-eval 65.6%

         \[\leadsto {x}^{\color{blue}{3}} \cdot \sqrt{0.027777777777777776} \]

      4. add-sqr-sqrt 65.6%

         \[\leadsto \color{blue}{\left(\sqrt{{x}^{3}} \cdot \sqrt{{x}^{3}}\right)} \cdot \sqrt{0.027777777777777776} \]

      5. metadata-eval 65.6%

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

      6. add-sqr-sqrt 65.6%

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

      7. metadata-eval 65.6%

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

      8. div-inv 65.6%

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

      9. clear-num 65.6%

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

      10. pow1 65.6%

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

      11. pow1 65.6%

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

      12. div-inv 65.6%

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

      13. pow-flip 65.6%

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

      14. metadata-eval 65.6%

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

   10. Applied egg-rr 65.6%

       \[\leadsto \color{blue}{\frac{1}{6 \cdot {x}^{-3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6 \cdot {x}^{-3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (/ (+ (* 0.3333333333333333 (pow x_m 3.0)) (* 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 * (((0.3333333333333333 * pow(x_m, 3.0)) + (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 * (((0.3333333333333333d0 * (x_m ** 3.0d0)) + (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 * (((0.3333333333333333 * Math.pow(x_m, 3.0)) + (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 * (((0.3333333333333333 * math.pow(x_m, 3.0)) + (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(Float64(0.3333333333333333 * (x_m ^ 3.0)) + 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 * (((0.3333333333333333 * (x_m ^ 3.0)) + (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[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

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

3. Taylor expanded in x around 0 84.6%

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

4. Final simplification 84.6%

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

Alternative 6: 84.4% accurate, 1.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 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 37.0%

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

   3. Taylor expanded in x around 0 69.5%

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

   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 65.6%

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

   4. Taylor expanded in x around inf 65.6%

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

      1. div-inv 65.6%

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

      2. metadata-eval 65.6%

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

      3. pow-plus 65.6%

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

      4. *-commutative 65.6%

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

      5. associate-*l* 65.6%

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

      6. pow-plus 65.6%

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

      7. metadata-eval 65.6%

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

      8. metadata-eval 65.6%

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

      9. metadata-eval 65.6%

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

   6. Applied egg-rr 65.6%

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

4. Final simplification 68.5%

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

Alternative 7: 52.7% 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 1 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 52.5%

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

3. Taylor expanded in x around 0 53.7%

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

4. Final simplification 53.7%

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

Alternative 8: 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 1 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 52.5%

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

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 1 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 52.5%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 2024026 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))