Hyperbolic sine

Percentage Accurate: 53.5% → 100.0%

Time: 7.0s

Alternatives: 7

Speedup: 29.4×

• 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: 53.5% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.0001:\\ \;\;\;\;x\_m + 0.16666666666666666 \cdot {x\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.0001)
      (+ x_m (* 0.16666666666666666 (pow x_m 3.0)))
      (/ 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 <= 0.0001) {
		tmp = x_m + (0.16666666666666666 * pow(x_m, 3.0));
	} 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 <= 0.0001d0) then
        tmp = x_m + (0.16666666666666666d0 * (x_m ** 3.0d0))
    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 <= 0.0001) {
		tmp = x_m + (0.16666666666666666 * Math.pow(x_m, 3.0));
	} 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 <= 0.0001:
		tmp = x_m + (0.16666666666666666 * math.pow(x_m, 3.0))
	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 <= 0.0001)
		tmp = Float64(x_m + Float64(0.16666666666666666 * (x_m ^ 3.0)));
	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 <= 0.0001)
		tmp = x_m + (0.16666666666666666 * (x_m ^ 3.0));
	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, 0.0001], N[(x$95$m + N[(0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $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))) < 1.00000000000000005e-4

   1. Initial program 38.2%

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

   3. Taylor expanded in x around 0 90.4%

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

      1. +-commutative 90.4%

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

      2. distribute-rgt-in 90.4%

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

      3. associate-*l* 90.4%

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

      4. fma-define 90.4%

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

      5. pow-plus 90.4%

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

      6. metadata-eval 90.4%

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

   5. Simplified 90.4%

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

   6. Taylor expanded in x around 0 90.4%

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

      1. distribute-rgt-in 90.4%

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

      2. *-lft-identity 90.4%

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

      3. associate-*r* 90.4%

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

      4. unpow2 90.4%

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

      5. unpow3 90.4%

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

   8. Simplified 90.4%

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

   if 1.00000000000000005e-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. Add Preprocessing

Alternative 2: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;x\_m + 0.16666666666666666 \cdot {x\_m}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x\_m} + -1}{2}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.1)
    (+ x_m (* 0.16666666666666666 (pow x_m 3.0)))
    (/ (+ (exp x_m) -1.0) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.1) {
		tmp = x_m + (0.16666666666666666 * pow(x_m, 3.0));
	} else {
		tmp = (exp(x_m) + -1.0) / 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) :: tmp
    if (x_m <= 2.1d0) then
        tmp = x_m + (0.16666666666666666d0 * (x_m ** 3.0d0))
    else
        tmp = (exp(x_m) + (-1.0d0)) / 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 tmp;
	if (x_m <= 2.1) {
		tmp = x_m + (0.16666666666666666 * Math.pow(x_m, 3.0));
	} else {
		tmp = (Math.exp(x_m) + -1.0) / 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):
	tmp = 0
	if x_m <= 2.1:
		tmp = x_m + (0.16666666666666666 * math.pow(x_m, 3.0))
	else:
		tmp = (math.exp(x_m) + -1.0) / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.1)
		tmp = Float64(x_m + Float64(0.16666666666666666 * (x_m ^ 3.0)));
	else
		tmp = Float64(Float64(exp(x_m) + -1.0) / 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)
	tmp = 0.0;
	if (x_m <= 2.1)
		tmp = x_m + (0.16666666666666666 * (x_m ^ 3.0));
	else
		tmp = (exp(x_m) + -1.0) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 2.1], N[(x$95$m + N[(0.16666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 38.5%

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

   3. Taylor expanded in x around 0 90.4%

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

      1. +-commutative 90.4%

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

      2. distribute-rgt-in 90.4%

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

      3. associate-*l* 90.4%

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

      4. fma-define 90.4%

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

      5. pow-plus 90.4%

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

      6. metadata-eval 90.4%

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

   5. Simplified 90.4%

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

   6. Taylor expanded in x around 0 90.4%

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

      1. distribute-rgt-in 90.4%

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

      2. *-lft-identity 90.4%

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

      3. associate-*r* 90.4%

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

      4. unpow2 90.4%

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

      5. unpow3 90.4%

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

   8. Simplified 90.4%

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

   if 2.10000000000000009 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;x + 0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.25)
    (/ (* x_m (+ x_m 2.0)) (- (+ x_m 1.0) -1.0))
    (/ (+ (exp x_m) -1.0) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * (x_m + 2.0)) / ((x_m + 1.0) - -1.0);
	} else {
		tmp = (exp(x_m) + -1.0) / 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) :: tmp
    if (x_m <= 1.25d0) then
        tmp = (x_m * (x_m + 2.0d0)) / ((x_m + 1.0d0) - (-1.0d0))
    else
        tmp = (exp(x_m) + (-1.0d0)) / 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 tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * (x_m + 2.0)) / ((x_m + 1.0) - -1.0);
	} else {
		tmp = (Math.exp(x_m) + -1.0) / 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):
	tmp = 0
	if x_m <= 1.25:
		tmp = (x_m * (x_m + 2.0)) / ((x_m + 1.0) - -1.0)
	else:
		tmp = (math.exp(x_m) + -1.0) / 2.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(Float64(x_m * Float64(x_m + 2.0)) / Float64(Float64(x_m + 1.0) - -1.0));
	else
		tmp = Float64(Float64(exp(x_m) + -1.0) / 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)
	tmp = 0.0;
	if (x_m <= 1.25)
		tmp = (x_m * (x_m + 2.0)) / ((x_m + 1.0) - -1.0);
	else
		tmp = (exp(x_m) + -1.0) / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 1.25], N[(N[(x$95$m * N[(x$95$m + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 38.5%

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

   3. Taylor expanded in x around 0 68.7%

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

      1. clear-num 68.5%

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

      2. inv-pow 68.5%

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

      3. associate-/r* 68.5%

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

      4. metadata-eval 68.5%

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

   5. Applied egg-rr 68.5%

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

      1. unpow-1 68.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{1}{x}}} \]

      2. remove-double-div 68.7%

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

      3. expm1-log1p-u 67.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

      4. expm1-undefine 5.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1} \]

      5. log1p-undefine 5.8%

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

      6. add-exp-log 7.5%

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

      7. +-commutative 7.5%

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

   7. Applied egg-rr 7.5%

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

      1. sub-neg 7.5%

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

      2. metadata-eval 7.5%

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

      3. flip-+ 21.9%

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

      4. metadata-eval 21.9%

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

      5. difference-of-sqr-1 21.9%

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

      6. associate-+l+ 21.9%

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

      7. metadata-eval 21.9%

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

      8. associate--l+ 83.1%

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

      9. metadata-eval 83.1%

         \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) - -1} \]

      10. +-rgt-identity 83.1%

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

   9. Applied egg-rr 83.1%

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

   if 1.25 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\frac{x \cdot \left(x + 2\right)}{\left(x + 1\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around 0 32.1%

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

   1. mul-1-neg 32.1%

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

   2. unsub-neg 32.1%

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

5. Simplified 32.1%

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

6. Taylor expanded in x around 0 66.2%

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

   1. *-commutative 66.2%

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

8. Simplified 66.2%

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

Alternative 5: 83.3% accurate, 18.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around 0 32.1%

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

   1. mul-1-neg 32.1%

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

   2. unsub-neg 32.1%

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

5. Simplified 32.1%

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

6. Taylor expanded in x around 0 80.2%

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

   1. *-commutative 80.2%

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

8. Simplified 80.2%

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

Alternative 6: 75.6% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around 0 32.1%

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

   1. mul-1-neg 32.1%

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

   2. unsub-neg 32.1%

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

5. Simplified 32.1%

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

6. Taylor expanded in x around 0 59.5%

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

   1. *-commutative 59.5%

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

8. Simplified 59.5%

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

Alternative 7: 52.7% accurate, 206.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in x around 0 81.1%

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

   1. +-commutative 81.1%

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

   2. distribute-rgt-in 81.1%

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

   3. associate-*l* 81.1%

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

   4. fma-define 81.1%

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

   5. pow-plus 81.1%

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

   6. metadata-eval 81.1%

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

5. Simplified 81.1%

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

6. Taylor expanded in x around 0 51.6%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))